An Introduction to Quantum Symmetries
Lectures by: R´eamonn O´ Buachalla
(Notes by: Fatemeh Khosravi)
Noncommutative Geometry the Next Generation (19th September-14th October 2016 ) 10:00 - 10:45
Institute of Mathematics Polish Academy of Science(IMPAN) Warsaw
1 Contents
1 From Lie Algebras to Hopf Algebras 3 1.1 Lie Algebras ...... 3 1.2 Quick Summary of Lie Groups ...... 4 1.3 Universal Enveloping Algebras ...... 5 1.4 Coalgebras and Bialgebras ...... 7 1.5 Universal Enveloping Algebras as Bialgebras ...... 8 1.6 Hopf Algebras ...... 9 1.7 Sweedler Notation ...... 10 1.8 Properties of the Antipode ...... 11
2 q-Deforming the Hopf Algebra U(sl2) 12
2.1 The Hopf Algebra Uq(sl2)...... 13
2.2 The Classical (q = 1)-Limit of Uq(sl2) ...... 14 2.3 Representation Theory ...... 15 2.4 q-Integers ...... 16
2.5 The Representations Tω,l ...... 16 2.6 The Generic Case ...... 17 2.7 The Root of Unity Case ...... 17
3 Algebraic Groups and Commutative Hopf Algebras 18 3.1 Algebraic Sets and Radical Ideals ...... 18 3.2 Morphisms ...... 19 3.3 Algebraic Groups ...... 20 3.4 Hopf Algebras ...... 21
3.5 The Hopf Algebra Oq(SLn)...... 21
4 Finite Duals and the Peter–Weyl Theorem for Oq(SL2) 22 4.1 The Hopf Dual of a Hopf Algebra ...... 22 4.2 Quantum Coordinate Functions ...... 24 4.3 A q-Deformed Peter–Weyl Theorem ...... 25 4.4 Dual Pairings of Hopf Algebras ...... 25 4.5 Comodules and Dual Pairings ...... 26
5 Cosemisimplicity and Compact Quantum Groups 27 5.1 Cosemisimple Coalgebras ...... 27
2 5.2 Compact Quantum Group Algebras ...... 30 5.3 Compact Quantum Groups ...... 31 5.3.1 Compact Groups ...... 31 5.3.2 C∗-algebras and the Gelfand–Naimark Theorem ...... 31 5.3.3 Compact Quantum Semigroups ...... 32 5.3.4 Compact Quantum Groups ...... 33 5.4 The Koornwinder-Dijkhuizen Correspondence ...... 34 5.4.1 The Haar State ...... 34
6 Principal Comodule Algebras 35 6.1 Hopf–Galois Extensions ...... 35 6.2 Quantum Homogeneous Spaces ...... 36 6.3 The Podle´sSphere ...... 36 6.4 Faithfull Flatness ...... 38 6.4.1 General Definition ...... 38 6.4.2 Quantum Homogeneous Spaces and Takeuchi’s Equivalence . . . . 38 6.5 Strong Connections and Principal Comodule Algebras ...... 39 6.5.1 First-Order Differential Calculi ...... 40 6.6 Universal Quantum Principal Bundle ...... 40 6.6.1 Strong Connections and Quantum Principal Comodule Algebras . 41 6.6.2 Connections for Associated Bundles ...... 41 6.7 The Podel´sSphere ...... 43
7 Differential Calculi, Complex Structures, and K¨ahler–Dirac Operators 43 7.1 Classifying first-order differential calculus on a Quantum Homogeneous Space ...... 43 7.2 The Podle´sSphere ...... 44 7.3 Complexes and Double Complexes ...... 45 7.4 Differential Calculi ...... 45 7.5 Complex Structures ...... 45 7.6 Hermitian Surfaces and K¨ahler–Dirac Operators ...... 46 7.6.1 Hermitian Surfaces ...... 46 7.7 A Spectral Triple for the Podle´sSphere ...... 47
8 Appendix 47 8.1 Two Different Definitions of Compact Quantum Qroup ...... 47
3 1 From Lie Algebras to Hopf Algebras
1.1 Lie Algebras
We begin with some basic conventions: Unless stated otherwise, all algebras discussed here will be assumed to be unital, and all algebras, and anti-algebras, maps will be assumed to be unital. Moreover, all vector spaces V will be assumed to be over C.
Definition 1.1. A Lie algebra is a vector space g together with a bilinear map [·, ·]: g × g → g, called the Lie bracket, satisfying the following properties:
1. [x, y] = −[y, z], for all x, y ∈ g,
2. (Jacobi Identity)[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0, for all x, y, z ∈ g.
A Lie subalgebra h of a Lie algebra g is a subspace of g which is closed with respect to the Lie bracket.
Example 1.2. The zero vector space, together with the zero bilinear map is a Lie algebra. We call it the 0 Lie algebra.
Definition 1.3. A Lie algebra g is said to be abelian or commutative if
[x, y] = 0, for all x, y ∈ g.
Given a vector space V, let gl(V ) denote the Lie algebra enveloped by the associative algebra of all linear endomorphisms of V . A representation of a Lie algebra g on V is a Lie algebra homomorphism π : g → gl(V ).
Example 1.4. Show that the matrices g = Mn(C), together with the usual commuta- tion bracket, form a Lie algebra.
Definition 1.5. A derivation on an algebra A is a linear map D : A → A such that
D(ab) = D(a)b + aD(b) a, b ∈ A
Exercise 1.6. For an algebra A consider the subspace of gl(A) consisting of all deriva- tions on A. Show that this is a Lie subalgebra of gl(A).
Exercise 1.7. More generally, show that any associative algebra A can be given the structure of a Lie algebra by defining a Lie bracket
[a, b] := ab − ba, a, b ∈ A.
We denote this Lie algebra by L(A).
4 It is easy to see that for any associative algebra A, its Lie algebra L(A) is abelian if and only if A is abelian as an associative algebra. Now we introduce a Lie algebra which its bracket does not arise from an algebra product.
3 Example 1.8. Assume R as a vector space over R. Then the vector product (x, y) 7→ 3 x ∧ y defines the structure of a Lie algebra on R . Explicitly, if x = (x1, x2, x3) and (y1, y2, y3), then
x ∧ y = (x2y3 − x3y2, x3y1 − x1y3, x1y2 − x2y1)
Exercise 1.9. The Lie algebra sl2 is the vector space spanned by the three elements E,H, and F together with the Lie bracket defined by
[E,F ] = H, [H,E] = 2E, [H,F ] = −2F.
Show that this is a Lie bracket, and show that an injection of Lie algebras is give by sl2 ,→ M3(C), where 0 1 0 0 1 0 E 7→ ,F 7→ ,H 7→ . 0 0 1 0 0 −1
Definition 1.10. A homomorphism between two Lie algebras is a linear map that is compatible with the respective Lie brackets:
ρ : g → g0, ρ([x, y]) = [ρ(x), ρ(y)], for all elements x, y ∈ g
1.2 Quick Summary of Lie Groups
Definition 1.11. A Lie group is a group that is also a manifold G such that
G × G → GG → G −1 (g1, g2) 7→ g1g2 g 7→ g are smooth maps.
Note. For a Lie group G, there exists an action G × C∞(G) → C∞(G) such that (g ◦ F )(g0) = F (g−1g0) for all g, g0 ∈ G and F ∈ C∞(G).
n ∞ n ∞ n Definition 1.12. A vector field on R is a derivation D : C (R ) → C (R ). In general a vector field on a manifold M is a derivation D : C∞(M) → C∞(M).
Exercise 1.13. Let Der(M) denote the set of all derivations D : C∞(M) → C∞(M), then there exists a Lie bracket on Der(M).
[X,Y ] = X ◦ Y − Y ◦ X
5 Definition 1.14. A left-invariant vector field on a Lie group is a derivation which is a left G-module map.
Fact. For a Lie group G, the Lie algebra (Der(G), [·, ·]) := Lie(G) is finite dimensional. We call it the Lie algebra of G. A simple consequence of Ado’s theorem (that every finite dimensional real Lie algebra can be embedded in gl(V ), for some V ) is the following.
Theorem 1.15 Every finite dimensional real Lie algebra is a Lie algebra of a Lie group.
Example 1.16. sl2 is the Lie algebra of SL(2).
1.3 Universal Enveloping Algebras
In this subsection we introduce tensor products and the universal enveloping algebra of a Lie algebra. These objects are defined using universal properties, which if you are comfortable enough with category theory, you might want to look up the general definition of.
Lemma 1.17 Given two vector spaces U and V there exists a unique vector space U ⊗V and a bilinear map ϕ : U × V → U ⊗ V such that, for any bilinear map f : U × V → W , for some vector space W , there exists a unique linear map fe : U ⊗ V → W such that the following diagram commutes
i U × V / U ⊗ V
fe f * W.
Exercise 1.18. Prove the lemma by concluding uniqueness directly and by concretely constructing U ⊗ V . Hint: Take the vector space with basis given by the elements of U × V and find a suitable quotient.
Definition 1.19. The tensor algebra of a vector space V is the vector space
∞ M T (V ) := V ⊗k, k=0
0 where we use the convention V = C.
If A and B are two algebras a multiplication can be defined on A ⊗ B by
(a ⊗ b).(a0 ⊗ b0) := aa0 ⊗ bb0, a, a0 ∈ A, b, b0 ∈ B.
6 Endowed with this multiplication we call A ⊗ B the algebra tensor product of A and B.
We now come to the universal enveloping algebra of a Lie algebra, also defined using a universal property
Lemma 1.20 For any Lie algebra g there exists a unique pair (U(g), i) (up to isomor- phism) where U(g) is a unital associative algebra and i : g → L(U(g)) is a Lie algebra ho- momorphism such that for any associative algebra A and Lie algebra homomorphism f : g → L(A), there exists a unique homomorphism of associative algebras fe : U(g) → A such that the following diagram of linear maps is commutative
i g / U(g)
fe f ) A.
Proof. Uniqueness follows from taking A = U](g) for any other U](g) satisfying the universal property. We will prove existence by explicit construction. Take T (g), the tensor algebra of g, and quotient by the two sided-ideal generated by the elements of the form
x ⊗ y − y ⊗ x − [x, y], for x, y ∈ g.
It is easily checked that this quotient, together with the obvious embedding i, satisfies the universal property. This easily implies the following corollary.
Corollary 1.21 For any Lie algebra homomorphism f : g → g0, there exists a unique algebra map U(f): U(g) → U(g0) such that the following diagram is commutative
f g / g0
i i0 U(g) / U(g0). U(f)
Exercise 1.22. Verify that the quotient constructed in the above lemma does indeed satisfy the required universal property.
Exercise 1.23. Show that U(g) = C if and only if g is the 0 Lie algebra.
Definition 1.24. For a Lie algebra g,
• The quotient algebra of T (g) by the two-sided ideal generated by the elements x ⊗ y − y ⊗ x, x, y ∈ g, is called the symmetric algebra of g and denoted by S(g). for every x, y ∈ g,
7 • The quotient algebra of T (g) by the two-sided ideal generated by the elements x ⊗ y + y ⊗ x, x, y ∈ g, is called the exterior algebra of g and denoted by Λ(g).
It is easy to observe that that for any commutative Lie algebra g, the enveloping alge- bra U(g) is the same as the symmetric algebra of g.
1.4 Coalgebras and Bialgebras
We shall now see that this structure is not an isolated example:
Definition 1.25. A coalgebra is a triple (C, ∆, ε), where C is a vector space, and
∆ : C → C ⊗ C; ε : C → C, are linear maps (called the coproduct and counit respectively), satisfying the following axioms:
1. (∆ ⊗ id) ◦ ∆ = (id ⊗ ∆) ◦ ∆, (coassociativity axiom);
2. (ε ⊗ id) ◦ ∆ = (id ⊗ ε) ◦ ∆ = id, (counit axiom).
These two axioms are sometimes presented in the form of the commutative diagrams:
∆ ∆ C / C ⊗ C C / C ⊗ C id ∆ id⊗∆ ∆ id⊗ε C ⊗ C / C ⊗ C ⊗ C C ⊗ C */ C ∆⊗id ε⊗id
Note that these are ”dual” to the axioms of a unital algebra:
m m AAo ⊗ A AAo ⊗ A O O O j O m id⊗m m id⊗η id A ⊗ A o A ⊗ A ⊗ A. A ⊗ A o A, m⊗id η⊗id where η : C → A, (λ → λ1A) is a C-linear map. (More formally, one can define a Hopf algebra as an object in the dual category of vector spaces dual to an algebra.) For (C, ∆, ε) and (C0, ∆0, ε0) coalgebras , a coalgebra morphism f : C → C0 is a linear map for which
∆0 ◦ f = (f ⊗ f) ◦ ∆, and ε = ε0 ◦ f.
Exercise 1.26. Let C be a coalgebra, and c an element of C. Show that there exist Pm P 0 0 presentations of ∆(c) of the form 1 ⊗ c + i=1 ai ⊗ bi, and of the form c ⊗ 1 + ai ⊗ bi, 0 where ai, bi ∈ ker(ε).
8 P Solve. For every c ∈ C, we can write ∆(c) = i ai ⊗ bi, thus c = (ε ⊗ id)∆(c) = P i ε(ai)bi. Indeed we can rewrite ∆(c) as follows: X X X ∆(c) = ai ⊗ bi + 1 ⊗ c − 1 ⊗ c = ai ⊗ bi + 1 ⊗ c − 1 ⊗ ( ε(ai)bi) i i i X X X = 1 ⊗ c + ai ⊗ bi − ε(ai)1 ⊗ bi = 1 ⊗ c + (ai − ε(ai)1) ⊗ bi i i i
It is easy to observe that ai − ε(ai)1 ∈ ker(ε). For the other equality we use the fact that c = (id ⊗ ε)∆(c).
Exercise 1.27. Show that the counit of a coalgebra is unique.
Solve. Assume we have ε and ε0 as counits of a coalgebra C which satisfy the counit condition. Then for every c ∈ C,
ε0(c) = ε0((ε ⊗ id)δ(c)) = (ε ⊗ ε0)δ(c) = ε((id ⊗ ε0)∆(c)) = ε(c)
Which shows that counit must be unique. Many natural examples of coalgebras also have an algebra structure which is computable with the coalgebra structure in the sense captured by the following definition.
Definition 1.28. A coalgebra (A, ∆, ε) is called a bialgebra if A is a unital algebra, and ∆ and ε are algebra maps with respect to the algebra tensor product A ⊗ A.
A morphism between two bialgebras is a coalgebra morphism and an algebra morphism simultaneously .
1.5 Universal Enveloping Algebras as Bialgebras
Exercise 1.29. Formulate the obvious notion of a direct sum of two Lie algebras and check that it is a Lie algebra.
We omit the proof of the following lemma which amounts to a somewhat technical explicit construction of the inverse of the isomorphism. See [1, Corollary V.2.3] for details.
Lemma 1.30 For two Lie algebras g and g0, an isomorphism is given by
U(ϕ): U(g ⊕ g0) → U(g) ⊗ U(g0), where ϕ is the Lie algebra homorphism
ϕ : g ⊕ g0 → U(g) ⊗ U(g0), (x, y) 7→ i(x) ⊗ 1 + 1 ⊗ i0(y).
Exercise 1.31. Verify that the map ϕ is a Lie algebra homomorphism.
9 We can now define a coalgebra structure on U(g) for any Lie algebra g. A comultiplcia- tion is defined by ∆ := U(ϕ)◦U(δ), where δ : g → g⊕g is the diagonal map δ(x) := (x, x) and U(ϕ) is the isomorphism constructed in the above lemma. A counit is defined by ε := U(0) where 0 is the zero morphism to the zero Lie algebra. Clearly, both ∆ and ε are (unital) algebra maps which act on elements of g according to ∆(x) = 1 ⊗ x + x ⊗ 1, ε(x) = 0. It is now an easy exercise to verify that they satisfy the axioms of a coalgebra on the g and hence on all of U(g). The opposite Lie algebra gop of a Lie algebra g, is isomorphic as a vector space but with Lie bracket [x, y]op := [y, x] = −[x, y]. A Lie algebra homomorphism is given by op : g → gop, x 7→ −x. As a little thought will confirm, we have that U(gop) is isomorphic as an algebra to U(g)op, and hence identifying U(g)op as a vector space with U(g) we have a linear map S : U(g) → U(g), x 7→ U(op)(x).
Exercise 1.32. Show that, for xi ∈ g, i = 1, . . . , k, we have k S(x1 ··· xk) = (−1) xk ··· x1.
Exercise 1.33. Find an explicit expression for ∆(x1 ··· xk).
Exercise 1.34. Show that we have m ◦ (S ⊗ id) ◦ ∆ = ε1, m ◦ (id ⊗ S) ◦ ∆ = ε1.
1.6 Hopf Algebras
Definition 1.35. For a bialgebra (H, ∆, ε, m, 1) an antipode is a linear map S : H → H such that m ◦ (S ⊗ id) ◦ ∆ = m ◦ (id ⊗ S) ◦ ∆ = 1ε. in the other words the antipode axiom tells us that the following diagram must be commute, ∆ ∆ H ⊗ H o H / H
S⊗id 1ε id⊗S H ⊗ H m / HHo m ⊗ H. A Hopf algebra is a bialgebra admitting an antipode.
10 For H,H0 two Hopf algebras with antipodes S and S0 respectively, a Hopf algebra mor- phism between H and H0 is a bialgebra map f : H → H0, such that f ◦ S = S0 ◦ f. An important point is that the antipode need not be a bijective map. This is illustrated by the following lemma, It is stated in terms of the coopposite coalgebra Hcop of a coalgebra H, which is H as a vector space endowed with the same counit, but with comultiplication ∆cop : σ ◦ ∆, where σ is the flip map.
Lemma 1.36 For any Hopf algebra A the following conditions are equivalent:
1. The antipode S of A is invertible as a linear mapping of A.
2. The bialgebra Aop is a Hopf algebra.
3. The bialgebra Acop is a Hopf algebra.
In this case, the inverse S−1 of S is the antipode of Aop and Acop.
Proof. [3, 1.2.4, Proposition 6].
Exercise 1.37. Consider a weakening of the definition of a Hopf algebra to one in which the counit is just assumed to be a linear map. Show that one can derive from this that ε is an algebra map and hence that the two definitions are equivalent.
1.7 Sweedler Notation
We will now introduce a special type of notation for dealing with Hopf algebras that proves very useful in practice. For a coalgebra C, and an element c ∈ C, one very often needs to consider presentations
m X 0 00 ∆(c) = ci ⊗ ci . i=1 Dealing with summations and indices on a regular basis tends to be quite tiresome, so one adopts the shorthand
m X 0 00 ∆(c) = ci ⊗ ci =: c(1) ⊗ c(2). i=1 This is known as Sweedler notation.
Let us now consider the coassociativity axiom in terms of Sweedler notation. For c ∈ C, we have by definition that
(∆ ⊗ id) ◦ ∆(c) = (id ⊗ ∆) ◦ ∆(c).
11 Using Sweedler notation, this is equivalent to
∆(c(1)) ⊗ c(2) = c(1) ⊗ ∆(c(2)), which is in turn equivalent to
(c(1))(1) ⊗ (c(1))(2) ⊗ c(2) = c(1) ⊗ (c(2))(1) ⊗ (c(2))(2).
This allows to extend Sweedler notation by denoting
c(1) ⊗ c(2) ⊗ c(3) := (c(1))(1) ⊗ (c(1))(2) ⊗ c(2) = c(1) ⊗ (c(2))(1) ⊗ (c(2))(2).
In fact, as is easy to see, one can can iterate coassociativity and attach a unique meaning to
c(1) ⊗ c(2) ⊗ · · · ⊗ c(k−1) ⊗ c(k), (for any k ∈ N).
Let us now look at the counit and the antipode axioms in terms of Sweedler notation. By definition we have
(ε ⊗ id) ◦ ∆(c) = (id ⊗ ε) ◦ ∆(c) m ◦ (S ⊗ id) ◦ ∆ = m ◦ (id ⊗ S) ◦ ∆ = 1ε.
In Sweedler notation these are equivalent to
ε(c(1))c(2) = ε(c(2))c(1)
S(c(1))c(2) = c(1)S(c(2)) = ε(c)1.
1.8 Properties of the Antipode
Now we can present a proof of some of basic properties of the antipode by using the Sweedler notation.
Lemma 1.38 For any Hopf algebra H, with antipode S, then it holds that:
1. S is an anti-algebra map;
2. S(1) = 1;
3. ∆ ◦ S = (S ⊗ S) ◦ τ ◦ ∆;
4. ε ◦ S = ε;
5. S is the unique map on H satisfying the antipode axiom.
Proof.
12 1. That S is an anti-algebra map follows from
S(b)S(a) = S(b(1)ε(b(2)))S(a(1)ε(a(2)))
= S(b(1))S(a(1))ε(a(2)b(2))
= S(b(1))S(a(1))(a(2)b(2))(1)S((a(2)b(2))(2))
= S(b(1))S(a(1))a(2)b(2)S(a(3)b(3))
= S(b(1))(ε(a(1))1)b(2)S(a(2)b(3))
= ε(a(1))ε(b(1))S(a(2)b(2)) = S(ab).
2. That S is a unital map follows at once from the relation m◦(S ⊗id)◦∆(1) = ε(1)1 combined with the facts that ∆(1) = 1 ⊗ 1 and ε(1) = 1.
3. The third identity follows from
S(a(2)) ⊗ S(a(1)) = S(a(2)ε(a(3))) ⊗ S(a(1))
= (S(a(2)) ⊗ S(a(1))(ε(a(3))1 ⊗ 1)
= (S(a(2)) ⊗ S(a(1))(∆(a(3)S(a(4))))
= (S(a(2)) ⊗ S(a(1))(a(3) ⊗ a(4))∆(S(a(5)))
= (S(a(2))a(3) ⊗ S(a(1))a(4)(∆(S(a(5))))
= (ε(a(2))1 ⊗ S(a(1))a(3))(∆(S(a(4))))
= (1 ⊗ S(a(1))a(2))(∆(S(a(3))))
= (1 ⊗ ε(a(1))1)(∆(S(a(2)))) = ∆(S(a)).
4. This follows immediately from ε(S(a)) = ε S(a(1)ε(a(2))) = ε(S(a(1))a(2)) = ε(ε(a)1) = ε(a).
5. Let S and S0 be antipodes for H, then for every a ∈ H,
0 S(a) = S((ε ⊗ id)∆(a)) = ε(a(1))S(a(2)) = S (a(1))a(2)S(a(3)) 0 0 0 = S (a(1))ε(a(2)) = S ((id ⊗ ε)∆(a)) = S (a).
2 q-Deforming the Hopf Algebra U(sl2)
In this section we extend our discussion of Hopf algebras to include our first q-deformed example. As it is well known, deformations of a Lie algebra g, in the category of Lie algebras, can exist only if the second cohomology group H2(g, g) is non-zero. As a direct consequence of this fact, all semi-simple Lie algebras must be rigid.
13 This caused some to guess that there could exist no non-trivial deformations of U(sl2) in the category of Hopf algebras. Thus, when Uq(sl2) first appeared in the early 1980’s, it came as quite a surprise. We will adopt the same conventions here as outlined at the beginning of the first lecture, with the additional assumption that q denotes a fixed complex number such that, unless otherwise stated, q 6= −1, 0, 1.
2.1 The Hopf Algebra Uq(sl2)
−1 Definition 2.1. For q ∈ C\{0, ±1}, we define Uq(sl2) to be C E,F,K,K /IUq(sl2), where IUq(sl2) is the two-sided ideal generated by the elements KK−1 − 1,K−1K − 1,KEK−1 − q2E,KFK−1 − q−2F, (1)
K − K−1 [E,F ] − . q − q−1
While it is not immediately clear how, or even if, this is a deformation of U(sl2), we do have the following two familiar looking results. (The proof of each result is a simple exercise in linear algebra and can be found in [3].)
Lemma 2.2 The following two sets are vector space bases for Uq(sl2): l m n l m n {F K E | m ∈ Z; l, n ∈ N0}, {E K F | m ∈ Z; l, n ∈ N0}, which we call the PBW bases.
For a general discussion of PBW bases in the classical setting see [1, Theorem V.2.5].
Lemma 2.3 The quantum Casimir element Kq−1 + K−1q Kq + K−1q−1 C := EF + = FE + q (q − q−1)2 (q − q−1)2 lies in the centre of Uq(sl2). If q is not a root of unity, then the centre of Uq(sl2) is generated by Cq.
Proof. It is enough to show that Cq commute with E,F and K. Here we only show that CqE = ECq, then other result is straightforward. Kq−1 + K−1q Kq + K−1q−1 C := EF + = FE + q (q − q−1)2 (q − q−1)2 KEq−1 + K−1Eq C E = EFE + q (q − q−1)2 EKq + EK−1q−1 EC = EFE + q (q − q−1)2
14 but using the fact that KE = q2EK and EK−1 = q2K−1E since (KEK−1 = q2E), the we get the equality. We should note that the requirement on q not to be a root of unity is a common feature in many results about quantised enveloping algebras. In many ways, the root of unity case and the non-root of unity cases can be quite distinct.
Now in addition to the PBW bases, and the quantum Caisimir, U(sl2) has the following all-important additional structure:
Lemma 2.4 There exists a Hopf algebra structure on Uq(sl2) with comultiplication ∆, counit ε, and antipode S, uniquely determined by
∆(E) = 1 ⊗ E + E ⊗ K, ∆(F ) = K−1 ⊗ F + F ⊗ 1, ∆(K) = K ⊗ K, ∆(K−1) = K−1 ⊗ K−1, ε(E) = ε(F ) = 0, ε(K) = ε(K−1) = 1.
S(E) = −EK−1,S(F ) = −KF,S(K) = K−1,S(K−1) = K.
Proof. Just as for the classical example of U(sl2), the proof amounts to showing that the maps ∆, ε, and S, vanish on the generators of the ideal, and that they satisfy the axioms of a Hopf algebra on the generators of the algebra. For example, we have
(∆ ⊗ id) ◦ ∆(E) = (∆ ⊗ id)(1 ⊗ E + E ⊗ K) = 1 ⊗ 1 ⊗ E + 1 ⊗ E ⊗ K + E ⊗ K ⊗ K, and
(id ⊗ ∆) ◦ ∆(E) = (id ⊗ ∆)(1 ⊗ E + E ⊗ K) = 1 ⊗ 1 ⊗ E + 1 ⊗ E ⊗ K + E ⊗ K ⊗ K.
Hence, coassociativity holds for the generator E. The other calculations are left as an excercise.
2.2 The Classical (q = 1)-Limit of Uq(sl2)
It is now time to address the question of how Uq(sl2) q-deforms the classical Hopf al- gebra U(sl2). The obvious problem with setting q = 1 is that Uq(sl2) is no longer well-defined. To get around this problem we will need to consider the following reformu- −1 lation of Uq(sl2): Define Ufq(sl2) to be the algebra C E,F,K,K ,G /IeUq(sl2), where
IeUq(sl2) is the ideal generated by the elements (1), and the additional generators
[G, E] − E(qK + q−1K−1), [G, F ] + (qK + q−1K−1)F, [E,F ] − G, (q − q−1)G − K + K−1.
15 Lemma 2.5 We have an algebra isomorphism α : Ufq(sl2) → Uq(sl2), uniquely deter- mined by
(K − K−1) α(E) = E, α(F ) = F, α(K) = K, α(G) = . q − q−1
With respect to the induced Hopf algebra structure on Ufq(sl2), we have
∆(G) = G ⊗ K + K−1 ⊗ G, ε(G) = 0,S(G) = −G.
Proof. The proof is another basic exercise in generators and relations, and as such, we leave it to the reader.
Exercise 2.6. Find the basis of Ufq(sl2) induced by α from the PBW basis.
Now for q = 1, it is clear that Ufq(sl2) is well-defined. Indeed, in Uf1(sl2), we have that K2 = 1, and moreover that K is an element of the centre of the algebra. The other relations reduce to
[E,F ] = G, [G, E] = 2EK, [G, F ] = −2FK.
Hence, we have the following result:
Lemma 2.7 There exists an isomorphism
2 β : Uf1(sl2) → U(sl2) ⊗ (C[K]/ K − 1 ), uniquely defined by
E → E ⊗ K,F → F ⊗ 1,G → H ⊗ K,K → 1 ⊗ K.
As a few basic checks will confirm, the quotient Uf1(sl2)/ hK − 1i is still well-defined as a Hopf algebra, and as such, it is isomorphic to the Hopf algebra U(sl2).
2.3 Representation Theory
In this lecture we discuss the representation theory of Uq(sl2). As we shall see shortly, when q is not a root of unity, the theory closely mirrors the classical case. We also briefly discuss the root of unity situation, which turns out to be more involved. Explicit proofs of the results stated here can be found in [1].
16 2.4 q-Integers
For any fixed value of q, the set of q-integers is composed of elements qa − q−a [a] = = qa−1 + qa−3 + ··· + q−a+1, (a ∈ ). q (q − q−1) Z Some easily verifiable results about the q-integers are: b −a −b a [−a]q = −[a]q, [a + b] = q [a]q + q [b]q = q [a]q + q [b]q, and, for q not a root of unity, we have [a]q 6= 0. Moreover, we can build upon the definition of a q-integer to produce q-analogues of other familiar integer functions: a [a]q! [a]q! := [1]q[2]q ··· [a − 1]q[a]q, := . b q [b]q![a − b]q! While q-integers are ubiquitous in the theory of quantum groups, they are a much older idea, going back to the work of Gauss.
2.5 The Representations Tω,l
Now that we have defined the q-integers, we can introduce a distinguished class of rep- resentations for Uq(sl2):
1 Definition 2.8. For l ∈ 2 N0, and ω ∈ {−1, 1}, let Vl be the (2l + 1)-dimensional vector space with basis {em | m = −l, −l +1, ··· , l −1, l}. We define operators Tω,l(E),Tω,l(F ), and Tω,l(K) acting on Vl by
2m q Tω,l(K)em = ωq em,Tω,l(E)em = [l − m]q[l + m + 1]qem+1,
q Tω,l(F )em = ω [l + m]q[l − m + 1]qem−1.
This gives a representation of Uq(sl2), which we denote by Tω,l.
Exercise 2.9. Check that this does indeed define a representation of Uq(sl2).
We note that the representations of Uq(sl2) have an extra parameter than the repre- sentations of the classical enveloping algebra U(sl2). In can be explained (in the q = 1 case at least) by the fact Uf1(sl2) needs to be quotiented by the ideal hK − 1i in order to arrive at the classical enveloping algebra U(sl2).
For Cq the quantum Casimir defined in Lemma 2.3, a simple calculation will demonstrate that l+1 −l−1 −1 −2 Tω,l(Cq) = ω(q + q )(q − q ) id.
Thus, we see that just as in the classical case, the image of the Casimir under Tω,l acts through scalar multiplication.
17 2.6 The Generic Case
In this section we will assume that q is not a root of unity, which is to say we will work in the generic setting. The following two propositions demonstrate the importance of these representations, and recall the classical case. The proofs are somewhat technical and are omitted, for details see [3, §3.2]
1 Proposition 2.10 For any l ∈ 2 N0, and ω ∈ {−1, 1}, the representation Tω,l is irre- 0 0 ducible. If (ω, l) 6= (ω , l ), then T(ω,l) and T(ω0,l0) are not equivalent since the values of the operators T(ω,l)(Cq) and T(ω0,l0)(Cq) are different.
Proposition 2.11 Any irreducible finite-dimensional representation T of Uq(sl2) is 1 equivalent to one of the representations Tω,l, for l ∈ 2 N0, and ω ∈ {−1, 1}.
Let T be a finite dimensional representation of Uq(sl2). For any complex number λ ∈ C, we set
Vλ := {v ∈ V | T (K)v = λv}.
If Vλ 6= {0}, then we call Vλ a weight space, and λ a weight, of the representation T . The non-zero elements of Vλ are called weight vectors. A weight vector v for which it holds that
T (E)v = 0, and T (K)v = µv, (µ ∈ C), is called a highest weight vector of T , while µ is called a highest weight of T . If V is the linear span of weight spaces of T , then T is called a weight representation.
Proposition 2.12 Every finite dimensional representation of Uq(sl2) is a weight repre- sentation.
We say that a representation of Uq(sl2) is completely reducible if it decomposes as a direct sum of irreducible sub-representations.
Proposition 2.13 Any finite dimensional representation T of Uq(sl2) is completely re- ducible.
2.7 The Root of Unity Case
Throughout this section we assume that the parameter q is a primitive p-th root of unity, where p > 3. The representation theory of Uq(sl2) is then very different from the case 0 0 p when q is not a root of unity. We set p = p if p is odd and p = 2 if p is even.
p0 p0 p0 −p0 Proposition 2.14 1. The elements E ,F ,K ,K belong to Z(Uq(sl2)).
p0 p0 p0 −p0 2. Z(Uq(sl2)) is generated by E ,F ,K ,K and the Casimir element Cq.
18 Corollary 2.15 Every irreducible representation of Uq(sl2) is finite dimensional.
Proof. Since an irreducible representation T maps central elements into scalar operators, it follows from Proposition 2.14(2) that T (Uq(sl2)) coincides with the linear span of operators T (ErKsF t) for r, s, t ≤ p0 − 1. Therefore, if v is a nonzero vector of the representation space V , then T (Uq(sl2))v is a finite-dimensional invariant subspace, so that T (Uq(sl2))v = V .
Proposition 2.16 Every irreducible representation of Uq(sl2) has dimension less than or equal to k0.
3 Algebraic Groups and Commutative Hopf Algebras
In this lecture we will introduce the Hopf algebra Oq(SL2), which is a direct generali- sation of the classical coordinate algebra of the Lie group SL2. We construct Oq(SL2) as a distinguished subalgebra of the vector space dual of Uq(sl2), and in so doing give a natural presentation of the well-known pairing between the two algebras. This important pairing generalises the classical pairing between the universal enveloping algebra of sl2 and the coordinate algebra of SL2.
3.1 Algebraic Sets and Radical Ideals
n Let C = {z = (z1, ..., zn) | zi ∈ C, 1 ≤ i ≤ n} be the n-dimensional affine space over the complex numbers C. Then for every T ⊆ C[z] = C[z1, ··· , zn] we can define
n Z(T ) = {z ∈ C | f(z) = 0 for very f ∈ T }
n n Definition 3.1. An algebraic set in C is a subset Y ⊆ C such that Y = Z(T ) for some set T ⊆ C[z1, ··· , zn]
Lemma 3.2 1. If hT i is the ideal of C[z] generated by T , then Z(T ) = Z(hT i).
2. If T1 ⊆ T2 ⊆ C[z], then Z(T2) ⊆ Z(T1).
By the Hilbert Basis Theorem, there is a finite set {f1, ··· , fr} ⊆ T of polynomials in T such that Z(T ) = Z(hT i) = Z({f1, ··· , fr}).
Definition 3.3. An ideal I of a ring R is radical if whenever an ideal J of R satisfies J 2 ⊆ I, then J ⊆ I.
Remark 3.4 For a commutative ring R, I/R is radical ⇔ ∀f ∈ R, ∀t ≥ 1, f t ∈ I only if f ∈ I.
19 If I is any ideal of a commutative ring R, we define √ I = {g ∈ R | gt ∈ I, for some t ≥ 1}. √ √ Then√ I/R, with√ I ⊆ I. Moreover, if R = C[z1, ··· , zn] for some n ≥ 1,√Z(I) = Z( I). For Z( I) ⊆ Z(I) we used Lemma 3.2(2) and if a ∈ Z(√I) and g ∈ I, then gt(a) = g(a)t = 0, for some t ≥ 1, and so g(a) = 0. Hence a ∈ Z( I). Now define, for n a set Y ⊆ C , the subset
I(Y ) = {f ∈ C[z1, ··· , zn] | f(y) = 0 ∀y ∈ Y }. We have the following obvious facts:
Lemma 3.5 1. I(Y ) is a radical ideal of C[z1, ··· , zn];
2. If Y1 ⊆ Y2, then I(Y2) ⊆ I(Y1);
n 3. If Y ⊆ C , then Y ⊆ Z(I(Y )); √ 4. If I/ C[z1, ··· ,Zn], then I ⊆ I(Z(I))
Theorem 3.6 (Hilbert Nullstellensatz)√ Let I/ C[z1, ··· , zn] for some n ≥ 1. Sup- pose f ∈ I(Z(I)), then f ∈ I.
Corollary 3.7 Let n ≥ 1. The correspondences ( ) ( ) *I Algebraic subset of n Radical ideals of [z , ··· , z ] C ) C 1 n Z are 1-1 and order reversing.
3.2 Morphisms
n Definition 3.8. Let X ⊆ C be an affine algebraic set. The (affine) coordinate ring of X is [z , ··· , z ] O(X) := C 1 n I(X) .
n Remark 3.9 O(X) is the image of C[z1, ··· , zn] under the restriction map from C to X.
Exercise 3.10. Show an equivalent is given by ( ) ( ) Algebraic Commutative finite generated algebras ←→ sets with no nillpotent elements (al = 0) X 7−→ O(X)
20 n m Definition 3.11. Let X ⊆ C and Y ⊆ C be algebraic sets. A morphism ϕ : X → Y is a function of the form
ϕ((x1, ··· , xn)) = (ϕ1(x1, ··· , xn), ··· , ϕm(x1, ··· , xn))
Where ϕi ∈ O(X) for i = 1, ··· , m.
If ϕ : X → Y is a morphism of algebraic sets, we define its comorphism
ϕ∗ : O(Y ) → O(X) f 7→ f ◦ ϕ.
Theorem 3.12 The functor X → O(X) defines a contravariant equivalence of cate- gories. ( ) ( ) Algebraic sets Affine radical commutative algebras ←→ and their morphisms and algebra morphisms
3.3 Algebraic Groups
Definition 3.13. An algebraic group G, is an algebraic set which also has the structure of a group. That is, we have maps
m : G × G → G :(x, y) 7→ xy, τ : G → G : x 7→ x−1.
A morphism of algebraic groups is a morphism of algebraic sets which is also a group homomorphism.
Exercise 3.14. Show that τ is a morphism.
Exercise 3.15. Show that any subgroup of an algebraic group is an algebraic group.
Example 3.16. Let GLn(C) = {A ∈ Mn(C) | det(A) 6= 0} then GLn(C) is not an n2+1 algebraic set but we can be embeded into C such that
n2+1 ι : GLn(C) ,→ C A 7→ (A, det−1(A))
n2+1 take I = {f} where f : C → C such that f(A, λ) = det(A)λ − 1. Then Z(I) = ι(GLn(C)).
21 3.4 Hopf Algebras
Let G be an algebraic group. It is almost obvious that O(G) is a Hopf algebra if we transpose the multiplication and inverse operations from G to O(G). That is,
∆ : O(G) −→ O(G) ⊗ O(G) =∼ O(G × G) by defining ∆(f) to be the function from G × G to C given by
∆(f)((x, y)) = f(xy), for x, y ∈ G; and ε : O(G) → C is given by f 7→ f(1G). Finally, let S : O(G) → O(G) be given by (Sf)(x) = f(x−1). It is a routine exercise to check that all the Hopf algebra axioms are satisfied. Conversely, if O(X) is the coordinate ring of an algebraic set X and O(X) is a Hopf alge- bra, then X is in fact an algebraic group. Namely, if a, b are C-algebra homomorphisms, define ab := (a ⊗ b) ◦ ∆; so ε : O(X) → C is an identity element for this multiplication; and
a−1 := a ◦ S yields an inverse for a.
Theorem 3.17 The functor G → O(G) defines a contravariant equivalence of categories ( ) ( ) affine algebraic groups affine radical commutative Hopf algebras ←→ and their morphisms and Hopf algebra morphisms
3.5 The Hopf Algebra Oq(SLn)
Example 3.18. Considering now SLn as an algebraic group in the obvious way, it is easy to convince oneself that
i O(SLn) = < u | i, j = 1, . . . , n > / < det −1 > . C j n
i Pn i a i The coproduct formula is given ∆(uj) = a=1 ua⊗uj . The counit is given by ε(uj) = δij. Finally, we leave the formula for the antipode as an exercise.
Just like Uq(sl2), the Hopf algebra O(SL2) admits a q-deformation.
Definition 3.19. As an algebra we define
i Oq(SL2) := C < uj | i, j = 1, 2 > /I,
22 where I is the ideal generated by the elements
1 1 1 1 1 2 2 1 2 2 2 2 1 2 2 1 −1 1 2 u1u2 − qu2u1, u1u1 − qu1u1, u1u2 − qu2u1 u1u2 − u2u1 − (q − q )u2u1, 1 2 2 1 1 2 2 1 2 2 2 2 u2u1 − u1u2 u2u2 − qu2u2 u1u2 − qu2u1 and the determinant relation
ad − qbc = 1.
Moreover, a Hopf algebra structure can be defined by this algebra by
2 i X i a i ∆(uj) = ua ⊗ uj , ε(uj) = δij a=1 and
1 2 1 −1 1 2 2 2 1 S(u1) = u2,S(u2) = −q u2,S(u1) = −qu1,S(u2) = u1.
Exercise 3.20. Verify that the maps ∆, ε and S are well-defined and that they satisfy the axioms of a Hopf algebra.
Exercise 3.21. Find a Hopf algebra q-deformation of O(SLn), for n ≥ 2, generating the n = 2 case given above.
4 Finite Duals and the Peter–Weyl Theorem for Oq(SL2)
4.1 The Hopf Dual of a Hopf Algebra
In this section we present the notion of the Hopf dual of a Hopf algebra. We begin with the dual algebra of a coalgebra. Throughout, for any vector space V , we denote ∗ ∗ ∗ ∗ V := HomC(V, C). Moreover, for a linear map L : V → W , we denote by L : W → V the linear map determined by
L∗(f) = f ◦ L, f ∈ W ∗.
For any coalgebra (C, ∆), consider the convolution product
∗ ∗ ∗ ∗ m := ∆ |C∗⊗C∗ : C ⊗ C → C , f ⊗ g 7→ f ∗ g.
Explicitly, we have
(f ∗ g)(c) = (f ⊗ g)∆(c) = f(c(1))g(c(2)).
Lemma 4.1 For (C, ∆) a coalgebra, the convolution product and 1C∗ := ε define the structure of an algebra on C∗.
23 Proof. Elementary exercise.
We would now like to go in the opposite direction and construct a coalgebra from a bialgebra. The naive construction fails, which is to say, the dual of an algebra does not have an automatic coalgebra structure. To see why this is so, first note that if m is the multiplication of A, then the dual of m has domain and codomain as m∗ : A∗ → (A ⊗ A)∗. When A is infinite dimensional, A∗ ⊗ A∗ is a proper subset of (A ⊗ A)∗, and we have no guarantee that the image of m∗ will lie in A∗ ⊗ A∗. We remedy this situation by defining the finite dual of an algebra A to be Ao := {f ∈ A∗ | ∆(f) = m∗(f) ∈ A∗ ⊗ A∗}.
Lemma 4.2 If A is an algebra with multiplication denoted by m and unit 1, then Ao ∗ is a coalgebra with coproduct ∆ = m |Ao , and counit defined by ε(f) = f(1). If A is commutative, then Ao is cocommutative.
Proof. First we need to make sure that ∆(Ao) ⊆ Ao ⊗ Ao. For f ∈ Ao and ∆(f) = P i fi ⊗ gi, we can choose the functionals {fi}i to be linearly independent. Hence there exist elements ai such that fi(aj) = δij. We have X X ∆(gj)(a ⊗ b) = gj(ab) = fi(aj)gi(ab) = f(ajab) i i X X = fi(aja)gi(b) = (fi ◦ Laj )(a)gi(b) i i X = (fi ◦ Laj ) ⊗ gi (a ⊗ b), i where Laj is the linear operator which multiplies by aj on the left. This implies that o o o o gj ∈ A , and so, ∆(A ) ⊆ A ⊗ A . Coassociativity of ∆ follows from (∆ ⊗ id)∆(f)(a ⊗ b ⊗ c) = f((ab)c) = f(a(bc)) = (id ⊗ ∆)∆(f)(a ⊗ b ⊗ c). That the counit axiom is satisfied is obvious. We leave the implication of commutativity from cocommutativity as Corollary 4.3 If H is a Hopf algebra, then Ho is a Hopf algebra with respect to the dual algebra and coalgebra structure and an antipode defined by So(f)(h) = f(S(h)).
Proof. First we need to show that Ho is a bialgebra. For a, b ∈ H and denoting P 0 o ∗ ∆(g) = j gj ⊗ gj, the fact that H is a subalgebra of H If follows from
X 0 0 fg(ab) = figj(a)fjgj(b). i,j
24 That ∆ and ε are algebra maps is easily verified. Coming now to the antipode, we see that o X 0 X o o 0 S (f)(ab) = f(S(b)S(a)) = fi(S(b))fi (S(a)) = S (fi)(b)S (fi )(a). i i o o Hence, S (f) ∈ H . We leave verification of the antipode axiom as an exercise. In general it does not hold that (Ho)o = H. We will see an example below.
4.2 Quantum Coordinate Functions
In this subsection we introduce an equivalent construction of Ho in terms of modules. Note that (3) requires a formulation of the obvious notion of dual module.
Definition 4.4. Let M be a left-module over a Hopf algebra H. For f ∈ M ∗, v ∈ M, M ∗ the functional cf,v ∈ H is defined by M cf,v(x) = f(x.v), x ∈ H. The space of coordinate functions of M is the subspace of H∗ defined by M ∗ C(M) := {cf,v | f ∈ M , v ∈ M}.
n n We see that if M is finite dimensional with a basis {ei}i=1, and dual basis {ebi}i=1, then a basis of C(M) is given by {cM | i, j = 1, ··· , n}, ebi,ej We leave the proof of the following technical lemma as an instructive exercise.
Lemma 4.5 Let M and N be finite dimensional modules over a Hopf algebra H. Let f ∈ M ∗, g ∈ N ∗, and v ∈ M, w ∈ N. Then
M N M⊕N 1. cf,v + cg,w = c(f,g),(v,w) and hence C(M) + C(N) ⊆ C(M ⊕ N)
M N M⊗N 2. cf,vcg,w = cf⊗g,v⊗w, and hence C(M)C(N) ⊆ C(M ⊗ N) M M M ∗ 3. ε(cf,v) = f(v), and S(cf,v) = cv,f . ∗ 4. If {vi}i and {fi}i are dual basis for M and M , then X ∆(cM ) = cM ⊗ cM . f,v f,vi fi,v i This lemma, more specifically the coproduct formula, tells us that for any Hopf alge- bra H, the subset of Ho containing the coordinate functions of the finite dimensional representations of H is a Hopf subalgebra of Ho. In fact, the two algebras are equal.
Lemma 4.6 For any Hopf algebra H, its Hopf dual is equal to its Hopf algebra of coordinate functions.
25 4.3 A q-Deformed Peter–Weyl Theorem
Our motivation for considering coordinate functions is given by the following classical result (which we have expressed in a form suited to our needs):
Theorem 4.7 (Peter–Weyl) It holds that ∼ o O(SL2) = U(sl2) .
In the q-setting this no longer holds. This is a direct consequence of the existence of non-classical representations of Uq(sl2), that is, those for which ω = −1. As we see in the following lemma, not including such representations in the module construction of the Hopf dual solves the problem.
Lemma 4.8 It holds that ∼ M Oq(SL2) = C(V(l,1)). 1 l∈ 2 N0 We call the representations for which ω = 1, Type I representations.
4.4 Dual Pairings of Hopf Algebras
We finish the lecture by introducing a concept that is central to the theory of Hopf algebras . Moreover, we will see that it generalises a very familiar concept from classical Lie theory.
Definition 4.9. A dual pairing of two Hopf algebras H and G is a bilinear map (·, ·): H × G → C such that for all h1, h2 ∈ H, and g1, g2 ∈ G, it holds that 0 0 (h, gg ) = (h(1), g)(h(2), g ), 0 0 (hh , g) = (h, g(1))(h , g(2)),
(h, 1G) = εH (h), (1H , g) = εG(g),
(h, SGg) = (SH h, g)
In fact, it can be shown that the fourth condition follows from the other three
Exercise 4.10. Show that for any Hopf algebra H and its dual Ho, a dual pairing is given by
o < ·, · >: H × H → C, (f, x) 7→ f(x). o Example 4.11. The dual pairing between Uq(sl2) and Uq(sl2) is given explicitly by
1 − 1 2 1 1 2 < u1, K >= q 2 , < u2, K >= q 2 , < u2, F >=< u1, E >= 1, with all other pairings taking value 0.
26 4.5 Comodules and Dual Pairings
We begin with the definition of a comodule, the dual notion to a module.
Definition 4.12. A (right) comodule of a coalgebra C is a pair (V, ∆R), where V is a vector space, and
∆R : V → V ⊗ C, is a linear map for which it holds that
(id ⊗ ∆) ◦ ∆R = (∆R ⊗ id) ◦ ∆R, (id ⊗ ε) ◦ ∆R = id. (A left comodule is defined analogously.) In an extension of Sweedler notation, we will usually denote X ∆R(v) = vi ⊗ ci =: v(0) ⊗ v(1). i
0 0 Definition 4.13. Let (V, ∆R) and (V , ∆R) be comodules of H. A linear mapping f : V → V 0 is called comodule morphism if
0 ∆R ◦ f = (f ⊗ id) ◦ ∆R The set of such maps f will be denoted by Mor(V,V 0). If there exists an invertible map f : V → V 0 , then the comodules V and V 0 are said to be isomorphic and f is called an isomorphism of the corresponding comodules. A linear subspace U of V is called invariant under ∆R if ∆R(U) ⊆ U ⊗ H. The corepresentation ∆R on V is said to be irreducible if V and {0} are the only invariant subspaces.
Exercise 4.14. Present the two conditions of comodule V in the definition as commu- tative diagrams and compare with the commutative diagrams from the definition of a module.
∆ ∆R V / V ⊗ H V / V ⊗ H
∆R id⊗∆ id⊗ε id V ⊗ H / V ⊗ H ⊗ H ' C ∆R⊗id and compare it with the following commutative diagram of modules which comes from the properties of a module M over a unital algebra A, via the map mR : M ⊗ A → M
mR mR MMo ⊗ A MMo ⊗ A O O g O mR id⊗m id⊗η id M ⊗ A o M ⊗ A ⊗ A. M, mR⊗id
27 Example 4.15. It follows from the axioms of a coalgebra that left Oq(SL2) comodules are given by (Vl, ∆), where
i1 il Vl :=C{u1 ··· u1 | ij = 1, 2 for all 1 ≤ j ≤ l}, and ∆ the restriction of the coproduct to Vl.
∆L : = ∆|Vl : Vl −→ Oq(SL2) ⊗ Vl i1 il i1 il ∆(u1 ··· u1 ) = ∆(u1 ) ··· ∆(u1 ) 2 2 X X = ( ui1 ⊗ ua1 ) ··· ( uil ⊗ ual ) a1 1 al 1 a1=1 al=1 X = ui1 ··· uil ⊗ ua1 ··· ual a1 al 1 1 a1,··· ,al
Moreover, each Vl is irreducible, and all irreducible comodules are isomorphic to Vl, for some l.
A very important, and useful, fact about dually paired Hopf algebras is that they can be used to construct modules from comodules.
Lemma 4.16 Let H and G be a dual pair of Hopf algebras with dual pairing h·, ·i. It holds that every right (left) comodule of H induces a right (left) module of G. Explicitly, if V is a right H-comodule with coaction ∆R, then the corresponding action of G on V is given by
V ⊗ G → V, v ⊗ g 7→ v(0) S(g), v(1) .
Proof. Exercise.
Exercise 4.17. Using the above lemma, and explicit pairing between Oq(SL2) and Uq(sl2) given above, show that < ·, · > turns the comodules Vl into the modules Vl,w.
5 Cosemisimplicity and Compact Quantum Groups
5.1 Cosemisimple Coalgebras
Lemma 5.1 For a (right) C-comodule (V, ∆R), it holds that
1. Every element of V is contained in a finite dimensional subccmodule of V .
2. Any element of C is contained in a finite dimensional subcomodule of C
Proof.
28 1. Exc: Take an element v ∈ V and compare
(∆R ⊗ id) ◦ ∆R(v), and (id ⊗ ∆) ◦ ∆R(v).
2. Applying the first part to V = C and ∆R = ∆, there exists a finite dimensional subspace W ⊆ C such that ∆(W ) ⊆ W ⊗ C. Exc: Show that B, the the linear span of the elements of W and its matrix elements, is a subcoalgebra of C.
Recall that a (right) comodule (V, ∆R) of a coalgebra C is called irreducible if it has no proper invariant subspace. That is no subspace such that ∆R(W ) ⊆ W ⊗ C. We call a coalgebra cosimple if it has no nonzero proper subcoalgebras.
Corollary 5.2 It holds that
1. Every irreducible comodule is finite dimensional.
2. Every cosimple coalgebra is finite dimensional.
Lemma 5.3 (Schur’s Lemma for Comodules)
1. If V and W are non-isomorphic comodules, then Mor(V,W ) = 0.
2. If V and W are isomorphic, then dim(Mor(V,W )) = 1, then any L ∈ Mor(V,W ) is either 0 or invertible. In particular, Mor(V,V ) = Cid.
Proof. Exc
Lemma 5.4 For any irreducible comodule (V, ∆R), we have an injective comodule mor- phism V → C.
∗ Proof. Take a non-zero functional f ∈ V , and verify that T := (f ⊗ id)∆R is a morphism from V to C. Since ε ◦ T = f, we have T 6= 0. Since ker(T ) must be a subcomodule, and V is irreducible, it must be trivial and hence T is injective.
Definition 5.5. A coalgebra is called cosemisimple if it is the sum of its cosimple sub- coalgebras.
Lemma 5.6 Every cosemisimple Hopf algebra is the direct sum of its cosimple subcoal- gebras.
Proof. Since the intersection of two coalgebras is again a coalgebra, any two cosimple coalgebras C and D must satisfy C ∩ D = ∅ or C = D. Hence any sum of cosimple subcoalgebras is a direct sum.
29 Definition 5.7. A linear functional h on a Hopf algebra H is called left invariant (re- spectively right invariant ), if for all a ∈ H, we have
(id ⊗ h) ◦ ∆(a) = h(a)1, ( respectively (h ⊗ id) ◦ ∆(a) = h(a)1).
Definition 5.8. For any left C-comodule V , its space of matrix elements is the coalgebra
∗ C(V ) := spanC{(id ⊗ f)∆L(v) | f ∈ V , v ∈ V } ⊆ C. Lemma 5.9 A comodule is irreducible if and only if its coalgebra of matrix elements is irreducible, and C(V ) = C(W ) if and only if V is equivalent to W . Moreover, C(V ) decomposes as left C-comodule into dimC(V ) copies of V .
Theorem 5.10 For a Hopf algebra H the following are equivalent:
1. H is cosemisimple.
2. Every comodule is completely reducible, which is to say, a direct sum of irreducible subcomodules.
3. There exists a unique left and right invariant linear functional on H satisfying h(1) = 1.
4. We have the Peter–Weyl decomposition M H ' C(V ). V where the summation is taken over equivalence classes of irreducible comodules, Irr(H).
Proof. We will prove just that 4 implies 3, and refer to [2] for the other implications. From the direct sum decomposition, we have a well defined map h : H → C defined L as projection onto the sub-coalgebra C. Since each summand in V ∈Irr(H) C(V ) is by definition a subcoalgebra , this is a left and right invariant functional. To see that it is unique, consider another right invariant functional eh. By assumption we must have
Cα 3 eh(a(1))a(2) = eh(a)1.
However, 1 ∈/ Cα 6= C since otherwise it would not be cosimple, implying that eh(a) = 0, for all a ∈ Cα 6= C1. Hence, eh(1) = 1, we must have eh = h. The left-covariant case is equivalent.
Exercise 5.11. Using the classification of comodules of Oq(SL2) and the fourth char- acterisation of cosemisimplicity given above, show that Oq(SL2) is cosemisimple.
30 Example 5.12. The Hopf algebra Uq(sl2) is not cosemisimple. Its only cosimple sub- coalgebra is C1.
Lemma 5.13 For a cosemisimple Hopf algebra H, it holds that
h(S(a)) = h(a), for all a ∈ H.
Proof. Since the antipode S of H is injective, one easily verifies that h˜(·) := h(S(·)) is also a left-invariant functional on H. Since h(l) = 1, the uniqueness of h yields h˜ = h. That is, we have h(S(a)) = h(a), a ∈ H.
5.2 Compact Quantum Group Algebras
For a ∗-algebra A we define a ∗-algebra structure on A ⊗ A by
A ⊗ A → A ⊗ A, a ⊗ b 7→ a∗ ⊗ b∗.
Definition 5.14. A ∗-bialgebra C is a bialgebra which is also ∗-algebra and for which
∗ ∗ ∗ ∗ ∗ ∆(c ) = ∆(c) , or equivalently (c(1) ⊗ c(2)) = c(1) ⊗ c(2).
A Hopf ∗-algebra is a ∗-bialgebra which is also a Hopf algebra
Exercise 5.15. Show that ε in a ∗-coalgebra must be a conjugate linear map, that is, we must have ε(a∗) = ε(a).
Lemma 5.16 In a Hopf ∗-algebra H, the antipode is always invertible and it holds that
S−1 = ∗ ◦ S ◦ ∗.
Proof. From the antipode axiom we have, for any a ∈ H, that
∗ ∗ ∗ ∗ a(1)S(a(2)) = ε(a)1 = S(a(1))a(2).
Applying the ∗-map to both sides we get
∗ ∗ ∗ ∗ (S(a(2))) a(1) = ε(a)1 = a(2)(S(a(1))) .
Hence, ∗S∗ is an antipode for Hop the Hopf algebra with opposite multiplication, and −1 so, by Lemma 1.36, we have that S = ∗ ◦ S ◦ ∗.
Definition 5.17. A compact quantum group algebra is a cosemisimple Hopf ∗-algebra H such that for h the Haar measure of H, it holds that
h(aa∗) > 0, for all a ∈ H.
31 5.3 Compact Quantum Groups
5.3.1 Compact Groups
Recall that a topological group G is a group endowed with a Hausdorff topology with respect to which the group multiplication (x, y) 7→ xy and the inverse map x 7→ x−1 are continuous. If G is compact as a topological space, then we say that G is a compact group. Topological, and compact semigroups are defined similarly.
5.3.2 C∗-algebras and the Gelfand–Naimark Theorem
Definition 5.18. A Banach ∗-algebra is a ∗-algebra A together with a complete sub- multiplicative norm such that ka∗k = kak(a ∈ A) . If, in addition, A has a unit such that k1k = 1, we call A a unital Banach ∗-algebra.A C∗-algebra is a Banach ∗-algebra such that ka∗ak = kak2 (a ∈ A)
∗ Example 5.19. If X is a locally compact Hausdorff space, then C0(X) is a C -algebra with involution f 7→ f.
Example 5.20. If H is a Hilbert space, then B(H), the space of all bounded operators over the Hilbert space H, is a C∗-algebra. We will see that every C∗-algebra can be thought of as a C∗-subalgebra of some B(H).
∗ Definition 5.21. A representation of a C -algebra A is a pair (Hφ, φ) where Hφ is a Hilbert space and φ : A → B(Hφ) is a ∗-homomorphism. We say (Hφ, φ) is faithful if φ is injective.
Theorem 5.22 (Gelfand-Naimark) If A is a C∗-algebra, then it has a faithful rep- resentation.
Theorem 5.23 Suppose that (H, φ) and (K, ψ) are representations of the C∗-algebras A and B, respectively. Then there exists a unique ∗-homomorphism π : A⊗B → B(H ⊗K) such that π(a ⊗ b) = φ(a) ⊗ ψ(b)(a ∈ A, b ∈ B). Moreover, if φ and ψ are injective, so is π.
If γ is a C∗-norm on A ⊗ B, we denote the C∗-completion of A ⊗ B with respect to γ by A ⊗γ B.
Theorem 5.24 For every C∗-algebras A and B there exist minimal and maximal tensor product norms on A ⊗ B such that for every other C∗-norm γ on A ⊗ B, we have
ka ⊗ bkmin ≤ γ(a ⊗ b) ≤ ka ⊗ bkmax (a ∈ A, b ∈ B)
Definition 5.25. We say a C∗-algebra A is nuclear if, for each C∗-algebra B, there is only one C∗-norm on A ⊗ B.
32 Notation. For every nuclear C∗-algebra A and for every C∗-algebra B we will denote the unique C∗-norm on their tensor product by ⊗ to emphasize that it is not algebraic nuc tensor product.
Theorem 5.26 (Takesaki) Every commutative C∗-algebra is nuclear.
Exercise 5.27. For every locally compact Hausdorff spaces X and Y , ∼ C0(X) ⊗ C0(Y ) = C0(X × Y ) nuc
Hint. Use Stone-Weierstrass theorem.
Exercise 5.28. Let A, B, C, D C∗-algebras, and ϕ : A → C, ψ : B → D are two ∗- algebra homomorphisms, ϕ ⊗ ψ : A ⊗ B → C ⊗ D is continuous with respect to k · kmin, and hence admits a unique extension to a continuous linear mapping from A ⊗ B to min C ⊗ D. We denote this map again by ϕ ⊗ ψ. min
5.3.3 Compact Quantum Semigroups
Theorem 5.22 tells us that we can recover X, as a topological space, from C(X). How- ever, it is clear that the semigroup structure of S cannot be recovered; two compact semigroups can be homeomorphic as topological spaces without being homomorphic as semigroups. With a view to constructing a noncommutative generalisation of compact groups, we shall attempt below to express the group structure of S in terms of C(S). Consider the ∗-algebra homomorphism
∆ : C(S) → C(S × S), f 7→ ∆(f), where ∆(f)(r, s) = f(rs), r, s ∈ S; we call it the composition with the multiplication of S. As a straightforward examination will verify,
[(id ⊗ ∆)∆(f)](r, s, t) = f(r(st)), and [(∆ ⊗ id)∆(f)](r, s, t) = f((rs)t), for all r, s, t ∈ S. Thus, since C(S) separates the points of S, the associativity of its multiplication is equivalent to the equation
(id ⊗ ∆)∆ = (∆ ⊗ id)∆. (2)
Exercise 5.29. A product m : S × S → S is associative if and only if (id ⊗ ∆)∆ = (∆ ⊗ id)∆.
This motivates the following definition:
33 Definition 5.30. A pair (A, ∆) is a compact quantum semigroup if A is a unital C∗- algebra and ∆ is a comultiplication on A, that is, if ∆ is a unital ∗-algebra homomorphism from A to A ⊗ A that satisfies equation (2). min
We say that (C(S), ∆) is the classical compact quantum semigroup associated to S.
Exercise 5.31. We define an abelian compact quantum semigroup to be a compact quantum semigroup whose C∗-algebra is abelian. Use the Gelfand–Naimark Theorem to show that abelian compact quantum semigroups are in bijective correspondence with compact semigroups. Extend this to an equivalence of categories.
5.3.4 Compact Quantum Groups
It is natural to ask whether or not we can identify a natural subfamily of the family of compact quantum semigroups whose abelian members are in one-to-one correspondence with the compact groups. Pleasingly, it turns out that we can. Let A be a unital C∗-algebra and consider the two subsets of A ⊗ A given by min
∆(A)(1 ⊗ A) = {∆(a)(1 ⊗ b): a, b ∈ A}, (3) and ∆(A)(A ⊗ 1) = {∆(a)(b ⊗ 1) : a, b ∈ A}. (4) When A = C(G), for some compact semigroup G, then the linear spans of (3) and (4) are both dense in C(G) ⊗ C(G) if and only if G is a group. With a view to showing this nuc consider T the unique automorphism of C(G) ⊗ C(G) for which T (f)(s, t) = f(st, t), for nuc f ∈ C(G × G), s, t ∈ G. If we assume that G is a group, then this map is invertible; its inverse T −1 is the unique endomorphism for which T −1(f)(s, t) = f(st−1, t). Since T is continuous, T (C(G) ⊗ C(G)) is dense in C(G) ⊗ C(G). If we now note that nuc
T (g ⊗ h)(s, t) = g ⊗ h(st, t) = g(st)h(t) = [∆(g)(1 ⊗ h)](s, t), for all g, h ∈ C(G), then we see that the linear span of ∆(C(G))(1 ⊗ C(G)) is indeed dense in C(G) ⊗ C(G). A similar argument will establish that ∆(C(G))(C(G) ⊗ 1) is nuc dense in C(G) ⊗ C(G). Conversely, suppose that the linear spans of (3) and (4) are each nuc dense in C(G) ⊗ C(G), and let s1, s2, t ∈ G such that s1t = s2t. As a little thought will nuc verify, the points (s1, t) and (s2, t) are not separated by the elements of the linear span of ∆(C(G))(1 ⊗ C(G)). Thus, the density of the latter in C(G) ⊗ C(G) implies that s1 = s2. Similarly, the density of ∆(C(G))(C(G) ⊗ 1) in C(G) ⊗ C(G) implies that if nuc st1 = st2, then t1 = t2. Now, it is well known that a compact semigroup that satisfies the left- and right-cancellation laws is a compact group. Consequently, G is a compact group. This motivated Woronowicz [?] to make the following definition.
34 Definition 5.32. A compact quantum semigroup (A, ∆) is said to be a compact quan- tum group if the linear spans of (1 ⊗ A)∆(A) and (A ⊗ 1)∆(A) are each dense in A ⊗ A. min
Motivated by the classical case, the density conditions on the linear spans of (1⊗A)∆(A) and (A ⊗ 1)∆(A) are sometimes called Woronowicz’s left- and right-cancellation laws respectively.
5.4 The Koornwinder-Dijkhuizen Correspondence
Proposition 5.33 Let H be a compact quantum group algebra. For a ∈ H, let kak∞ ∗ ∗ denote the supremum of p(a), for all C -seminorms on H. Then k · k∞ is a C -norm on H.
∗ We will denote H, for the C -completion of H with respect to k · k∞.
Theorem 5.34 The coproduct of H extends to a universal ∗-homomorphism ∆ : H → H ⊗ H such that (H, ∆) is a compact quantum group. Moreover we have a bijection min ( ) ( ) Compact Quantum Compact Quantum ←→ . Group Algebras Groups
5.4.1 The Haar State
Let G be a locally compact topological group, and let µ be a non-zero regular Borel measure on G. We call µ a left Haar measure if it is invariant under left translation, that is, if µ(gB) = µ(B), for all g ∈ G, and for all Borel subsets B.A right Haar measure is defined similarly. It is well known that every locally compact topological group G admits a left and a right Haar measure that are unique up to positive scalar multiples. If the left and right Haar measures coincide, then G is said to be unimodular. It is a standard result that every compact group is unimodular. Thus, since we shall only consider compact groups here, we shall speak of the Haar measure. Furthermore, the regularity of µ ensures that µ(G) is finite. This allows us to work with the normalised Haar measure, that is, the measure µ for which µ(G) = 1. We define the Haar integral to be the integral over G with respect to µ. It is easily seen that the left- and right- invariance of the measure imply left- and right-invariance of the integral, that is, Z Z Z f(hg)dµ = f(gh)dµ = f(g)dµ, (5) G G G for all h ∈ G, and for all integrable functions f. (Note that since there is no risk of confusion, we have suppressed explicit reference to µ.) Considered as a linear mapping on C(G), R is easily seen to be a positive linear mapping of norm one. In general, if ϕ is a positive linear mapping of norm one on a C∗-algebra,
35 then we call it a state. Now, as direct calculation will verify, Z Z (id ⊗ )∆(f)(h) = f(hg)dµ, G and Z Z ( ⊗ id)∆(f)(h) = f(gh)dµ. G Thus, the left- and right-invariance of the integral is equivalent to the equation Z Z Z (id ⊗ )∆(f) = ( ⊗ id)∆(f) = fdµ. G G G
This motivates the following definition: Let (A, ∆) be a compact quantum group and let h be a state on A. We call h a Haar state on A if
(id ⊗ h)∆(a) = (h ⊗ id)∆(a) = h(a), for all a ∈ A. The following result is of central importance in the theory of compact quantum groups. It was first established by Woronowicz under the assumption that A was separable [?], and it was later proved in the general case by Van Daele [?].
Theorem 5.35 If (A, ∆) is a compact quantum group, then there exists a unique Haar state on A.
A state ϕ on a C∗-algebra A is called faithful if ker(ϕ) = {0}. Obviously, the Haar state of any classical compact quantum group is faithful. However, this does not carry over to the noncommutative setting. It is a highly desirable property that h be faithful. Necessary and sufficient conditions for this to happen are given in [?].
6 Principal Comodule Algebras
6.1 Hopf–Galois Extensions
For a right H-comodule V with coaction ∆R, we say that an element v ∈ V is coinvariant coH if ∆R(v) = v ⊗ 1, we denote the subspace of all coinvariant elements by V , and call it the coinvariant subspace of the coaction. (We define a coinvariant subspace of a left-coaction analogously.) For a right H-comodule algebra P , its coinvariant subspace M := P coH is clearly a subalgebra of P . If the mapping
ver = (m ⊗ id) ◦ (id ⊗ ∆R): P ⊗M P → P ⊗ H, is an isomorphism, then we say that P is a Hopf–Galois extension of H.
36 6.2 Quantum Homogeneous Spaces
For H a Hopf algebra, a homogeneous right H-coaction on G is a coaction of the form (id ⊗ π)∆, where π : G → H is a surjective Hopf algebra map. A quantum homogeneous space B := GcoH is the coinvariant subspace of such a coaction. If G is a principal comodule algebra then we call it a homogenous principal comodule algebra.
Lemma 6.1 For a quantum homogeneous space π : G → H, and ∆ the coproduct of G, it holds that ∆(B) ⊆ G ⊗ B, and so, we can consider B as a left G-comodule.
Proof. Let g ∈ B, then (id ⊗ π)∆(g) = g ⊗ 1. We must show that the second leg of ∆(g) ∈ B. Using Sweedler notation,
ε(g(1))g(2) ⊗ 1H = (ε ⊗ id)∆(g) ⊗ 1H
= g ⊗ 1H = (id ⊗ π)∆(g) = (id ⊗ π)∆((ε ⊗ id)∆(g))
= ε(g(1))(id ⊗ π)∆(g(2)) we conclude that g(2) ⊗ 1 = (id ⊗ π)∆(g(2)), so g(2) ∈ B.
6.3 The Podle´sSphere
−1 −1 −1 Let O(U1) = C < t, t > / < tt = t t = 1 > and define the Hopf algebra structure i i i i i −i on O(U1) such that ∆(t ) = t ⊗ t , ε(t ) = 1, S(t ) = t then there is a surjective Hopf algebra map π : Oq(SU2) → O(U1) is uniquely defined by
1 2 −1 1 2 π(u1) = t, π(u2) = t , π(u2) = π(u1) = 0.
Then we call the associated coinvariant subalgebra the Podle´ssphere and denote it by 2 coO(U ) Oq(S ) = Oq(SU2) 1 .
Definition 6.2. A Z-grading on an algebra A is a direct sum decomposition M A ' Al, l∈Z
0 such that AlAl0 ⊆ Al+l0 , for all l, l ∈ Z. We call an element of A homogeneous if it is contained in Al, for some l ∈ Z.
Lemma 6.3 A O(U1)-coaction on an algebra A is equivalent to a Z-grading on A.
Proof. Let A be an algebra ∆R : A → A ⊗ O(U1) be a coaction then we for every l n ∈ Z, we define Al = {a ∈ A|∆R(a) = a ⊗ t } then follow from the axioms of a coaction, and the Hopf algebra structure of O(U1). Firstly, for a ∈ A, we will always P li P have ∆R(a) = ai ⊗ t . Applying id ⊗ ε to both sides gives a = i ai. We now use the P lj li P li li axiom (∆R ⊗id)◦∆R = (id⊗∆)◦∆R on a then we get i,j aij ⊗t ⊗t = i ai ⊗t ⊗t .
37 Now consider the set of elements which are the sums of the terms with the same tli as P lj li the third tensor factor. This set is linearily independent. Thus, j aij ⊗ t = fi ⊗ t . P lj Since ∆R(fi) = j fij ⊗ t , we have the required result.
Lemma 6.4 The grading M Oq(SU2) ' Oq(SU2)l l∈Z corresponding to the O(U1)-coaction on Oq(SU2) is described by
1 i 1 j 2 k Oq(SU2)l := spanC{(u1) (u2) (u1) | − i + j − k = l}.
Proof. Recall that a basis of Oq(SU2) is given by
1 i 1 j 2 k spanC{(u1) (u2) (u1) |i, j, k ∈ N0} 1 i 2 h 2 l spanC{(u2) (u1) (u2) |i, h, l ∈ N0} look at the following calculation for i = 1, 2.
i i ∆R(u1) = (id ⊗ π) ◦ ∆(u1) i 1 i 2 = (id ⊗ π)(u1 ⊗ u1 + u2 ⊗ u1) i 1 i 2 i −1 = u1 ⊗ π(u1) + u2 ⊗ π(u1) = u1 ⊗ t
i i we can also find ∆R(u2) = u2 ⊗ t and because ∆R is a homomorphism then we can compute ∆R for an element of basis
1 i 1 j 2 k 1 i 1 j 2 k −i+j−k ∆R((u1) (u2) (u1) ) = (u1) (u2) (u1) ⊗ t so according to the definition of Al in Lemma 6.3, we have
1 i 1 j 2 k Oq(SU2)l := spanC{(u1) (u2) (u1) | − i + j − k = l}.
2 Corollary 6.5 A generating set of the Podle´ssphere Oq(S ) is given by
1 1 1 1 2 1 b+ := u1u2, b− := u1S(u2), b0 := u1u2.
2 Proof. Recall that the Podle´ssphere Oq(S ) is the covariant subalgebra of the action O(U1) on Oq(SU2) so
2 Oq(S ) = {x ∈ Oq(SU2)|(id ⊗ π)∆(x) = x ⊗ 1} 1 i 1 j 2 k 1 i 1 j 2 k 1 i 1 j 2 k = spanC{(u1) (u2) (u1) | ∆R((u1) (u2) (u1) ) = (u1) (u2) (u1) ⊗ 1} 1 i 1 j 2 k = spanC{(u1) (u2) (u1) | − i + j − k = 0} = Oq(SU2)0
38 then it is easy to see that all element of Oq(SU2)0 are generated by the folloeing elements:
1 2 2 2 1 1 b0 := u1u2, b+ := u1u2, b− = u1u2
2 Then Oq(S ) is C < b0, b+, b− > module with the relations
±2 ±2 b±b0 = q b0b± + (1 − q )b±, 2 −2 −1 q b−b+ = q b+b− + (q − q )(b0 − 1), 2 b0 = b0 + qb−b+.
2 Hence Oq(S ) can be defined as the free algebra with these generators and the inherited relations as shown.
1 Corollary 6.6 In case q = 1, in terms of x, y, z defined by b± = ±(x±ιy) and b0 = z+ 2 then from the last relation 1 1 (z + )2 = (z + ) − (x − ιy)(x + ιy) 2 2
2 2 2 1 which implies that x + y + z = 4 .
6.4 Faithfull Flatness
6.4.1 General Definition
Recall that P is said to be faithfully flat as a right module over M if the functor G⊗M − : MMod → CMod, from the category of left M-modules to the category of complex vector spaces, maps a sequence to an exact sequence if and only if the original sequence is exact. The definition of faithfully flat as a right M-module is analogous.
6.4.2 Quantum Homogeneous Spaces and Takeuchi’s Equivalence
In the homogeneous case the category of associated vector bundles has a particularly G H G nice form. Consider the abelian categories MMod0 and Mod0. The objects in MMod0 are M-bimodules E (with left and right actions denoted by juxtaposition) endowed with + + a left G-coaction ∆L such that EM ⊆ M E, and
0 0 0 0 ∆L(mem ) = m(1)e(−1)m(1) ⊗ m(2)e(0)m(2), for all m, m ∈ M, e ∈ E. (6)
G The morphisms in MMod0 are those M-bimodule homomorphisms which are also homo- H morphisms of left G-comodules. The objects in Mod0 are left H-comodules V endowed H with the trivial right M-action (v, m) 7→ ε(m)v. The morphisms in Mod0 are the left H H-comodule maps. (Note that Mod0 is equivalent under the obvious forgetful functor to H Mod, the category of left H-comodules.)
39 G + H If E ∈ MMod0, then E/(M E) becomes an object in Mod0 with the trivial right M action and the right H-coaction
∆L[e] = π(e(−1)) ⊗ [e(0)], e ∈ E, (7)
+ G H where [e] denotes the coset of e in E/(M E). We define a functor Φ : MMod0 → Mod0 + G as follows: Φ(E) := E/(M E), and if g : E → F is a morphism in MMod0, then Φ(g) : Φ(E) → Φ(F) is the map to which g descends on Φ(E). H If V ∈ Mod0, then G H V , the cotensor product of G and V , , defined by
G H V := ker(∆R ⊗ id − id ⊗ ∆L : G ⊗ V → G ⊗ H ⊗ V ),
G becomes an object in MMod0 by defining an M-bimodule structure X X X X m gi ⊗ vi = mgi ⊗ vi, gi ⊗ vi m = gim ⊗ vi, (8) i i i i and a left G-coaction
X i i X i i i ∆L g ⊗ v = g(1) ⊗ g(2) ⊗ v . i i
H G We define a functor Ψ : Mod0 → MMod0 as follows: Ψ(V ) := G H V, and if γ is a H morphism in Mod0, then Ψ(γ) := id ⊗ γ. G It was shown in [?, Theorem 1] that a unit-counit equivalence of the categories MMod0 H and Mod0, which we call Takeuchi’s equivalence, is given by the functors Φ and Ψ and the natural transformations h X i X C:Φ ◦ Ψ(V ) → V, gi ⊗ vi 7→ ε(gi)vi, (9) i i
U: E → Ψ ◦ Φ(E), e 7→ e(−1) ⊗ [e(0)]. (10)
G We define the dimension of an object E ∈ Mmod0 to be the vector space dimension of Φ(E).
6.5 Strong Connections and Principal Comodule Algebras
Combining the notion of a Hopf–Galois extension with faithful flatness gives the following definition.
Definition 6.7. A principal right H-comodule algebra is a right H-comodule alge- bra which is Hopf Galois and faithfully flat as a right and left M-module.
As we will now see, principal right H-comodule algebras admit an equivalent formula- tion in terms of a noncommutative geometric generalisation of the classical notion of a principal connection. This will require us to discuss differential calculi.
40 6.5.1 First-Order Differential Calculi
A first-order differential calculus over a unital algebra A is a pair (Ω1, d), where Ω1 is an A-A-bimodule and d : A → Ω1 is a linear map for which the Leibniz rule holds
d(ab) = a(db) + (da)b, a, b, ∈ A,
1 1 and for which Ω = spanC{adb | a, b ∈ A}. We call an element of Ω a one-form. An 1 1 morphism between two first-order differential calculi (Ω (A), dΩ) and (Γ (A), dΓ) is a 1 1 bimodule map ϕ :Ω (A) → Γ (A) such that ϕ ◦ dΩ = dΓ. The direct sum of two first- 1 1 1 1 order differential calculi (Ω (A), dΩ) and (Γ (A), dΓ) is the calculus (Ω (A)⊕Γ (A), dΩ + dΓ). We say that a first-order calculus is connected if it holds that ker(d) = C1. 1 1 The universal first-order differential calculus over A is the pair (Ωu(A), du), where Ωu(A) is the kernel of the product map m : A ⊗ A → A endowed with the obvious bimodule structure, and du is defined by
1 du : A → Ωu(A), a 7→ 1 ⊗ a − a ⊗ 1.
1 By [?, Proposition 1.1], there exists a surjective morphism from Ωu(A) onto any other calculus over A
6.6 Universal Quantum Principal Bundle
We now get our first glimpse of how calculi relate to the algebraic conditions of Hopf– Galois extensions.
To any H-comodule algebra (P, ∆R) we can associate the sequence
1 ι 1 ver + 0 −→ P Ωu(M)P −→Ωu(P )−→P ⊗ H −→ 0, (11)
1 1 where Ωu(M) is the restriction of Ωu(P ) to M, ι is the inclusion map, and
1 + + ver : Ωu(P ) → G ⊗ H , adb 7→ adb(0) ⊗ b(1)
Lemma 6.8 A H-comodule algebra P is a Hopf–Galois of P co(H) if and only if the above sequence is exact.
∼ 1 ∼ + Proof. Note that P ⊗ P = Ωu(P ) ⊗ P , P ⊗ H = P ⊗ H ⊕ P , as left P -modules. This implies that the sequence
1 ι 1 ver + 0 −→ P Ωu(M)P −→ Ωu(P ) −→ P ⊗ H −→ 0 is exact if and only if the sequence
1 can 0 −→ P Ωu(M)P −→ P ⊗ P −→ P ⊗ P ⊗ H −→ 0
41 is exact. Here can is the lift of the canonical map defined by the commutative diagram
π P ⊗ P / P ⊗M P
can can % x P ⊗ H
1 in which π is the defining projection of the tensor product P ⊗M P . Since P Ωu(M)P = kerπ, the second sequence is exact if and only if the canonical map can is bijective.
6.6.1 Strong Connections and Quantum Principal Comodule Algebras
A principal connection for a quantum principal H-bundle P ←-M is a left P -module 1 1 1 projection Π : Ω (P ) → Ω (P ) such that ker(Π) = P Ω (M)P and ∆R◦Π = (Π⊗id)◦∆R. A principal connection Π is called strong if (id − Π)(Ω1(P )) ⊆ P Ω1(M).
Lemma 6.9 When P is a Hopf–Galois extension of B, a principal connection is equiv- alent to a right H covariant splitting of ver, that is a right H-comodule map
+ 1 s : P ⊗ H → Ω (P ), such that ver ◦ s = idP ⊗H+ ,
+ with respect to the H-comodule structure on P ⊗ H given by idP ⊗ ∆H .
Proof. Exercise.
Theorem 6.10 The universal principal quantum bundle of a comodule algebra M := P co(H) admits a connection if and only if it is a principal comodule algebra.
6.6.2 Connections for Associated Bundles
For a principal comodule algebra P , we call a left G-module of the form G H V , for V a left H-comodule, an associated vector bundle.
Exercise 6.11. Show that any associated bundle is a left G-comodule.
Definition 6.12. For an algebra A, together with a first-order differential calculus Ω1 over A, a connection for a left A-module F is a linear map ∇ : F → F ⊗ Ω1 such that
∇(af) = da ⊗ f + a∇f, a ∈ A, f ∈ F.
Proposition 6.13 A module over an algebra A is projective if and only if it admits a connection with respect to the universal calculus.
42 1 Proof. Let (Ωu(A), du) be the universal A-bimodule with du(a) := 1 ⊗ a + a ⊗ 1. Define right A-module homomorphisms
1 j m 0 → E ⊗A Ωu(A) → E ⊗ A → E → 0 by j(s ⊗ du(a)) := s ⊗ a − sa ⊗ 1 and m(s ⊗ a) := sa. This yields a short exact sequence of right A-modules (think of E ⊗ A as a free A-module generated by a vector space 1 basis of E). Any linear map ∇ : E → E ⊗A Ωu(A) gives a linear section of m by f(s) := s ⊗ 1 + j(∇s). Then f(sa) − f(s)a = j(∇(sa) − ∇sa − s ⊗ dua), so f is an A-module homomorphisms precisely when ∇ Leibniz rule. If that happens, f splits the exact sequence and embeds E as a direct summand of the free A-module E ⊗ A, so E is projective.
Lemma 6.14 Show that an injective map is given by
n n 1 1 X i i X i i µ :Ω (B) ⊗B E → Ω (B)G ⊗ V, ω ⊗B ⊗g ⊗ v 7→ ωg ⊗ v . i=1 i=1
1 ∼ 1 1 Proof. For A ⊆ B the isomorphism Ω (B)A = Ω (B) ⊗B A is given by m :Ω (B) ⊗B A → Ω1(B)A ⊆ Ω1(A) because if we consider the map
1 ι : B ⊗ A → Ω (B) ⊗B A, b ⊗ a 7→ −db ⊗ a. for a ∈ A, b ∈ B. It acts on dba ∈ Ω1(B)A ⊆ B ⊗ A, as
ι(cdba) = ι(c ⊗ ba − cb ⊗ a)
= −dc ⊗B ba + dc ⊗B ba + cdb ⊗B a
= cdb ⊗B a. hence m ◦ ι(cdba) = cdba and ι ◦ m(cdb ⊗B a) = cdb ⊗B a, showing that ι is the inverse of m as required.
Lemma 6.15 For an associated bundle E := P H V , the image of the map X X ∇ : E → Ω1(M)P ⊗ V, f i ⊗ vi 7→ (id − Π) ⊗ id df i ⊗ vi , (12) i i
1 1 is contained in the µ(Ω (M) ⊗M E). Moreover, the induced map ∇ : E → Ω (B) ⊗B E is a connection for E.
43 6.7 The Podel´sSphere
Lemma 6.16 For a quantum homogeneous space π : G → H, a linear map i : H → G which is a right and left H-comodule map and satisfies π ◦ i = idH , a strong connection is defined by
1 1 Π:Ω (G) → Ω (G), adb 7→ ab(0)S(i(b(1))(1)) ⊗ i((b(1))(2)) − ab ⊗ 1.
We now construct a bicovariant splitting map for the case of the Pode´ssphere.
Lemma 6.17 A bicovariant splitting is given by
−k 1 k k 2 k i(1) = 1, i(t ) = (u1) , i(t ) = (u2) , k ∈ N.
Proof. According to section 6.3 π : Oq(SU2) → O(U1) is a Hopf algebra map and O (SU ) ∼ L O (SU ) . Let ∆ : V → O(U ) ⊗ V defined by ∆ (v) = tl ⊗ v then q 2 = l∈Z q 2 l L l 1 l L we will show that ∼ Oq(SU2)H Vl = Oq(SU2)l.
7 Differential Calculi, Complex Structures, and K¨ahler– Dirac Operators
We recall here the basics of the theory of differential calculi, the notions in terms of which holomorphic structures is based. We begin with first-order calculi and then recall the maximal prolongation of a first-order differential calculus to a differential calculus. Throughout this section every quantum homogeneous space will be assumed to be faithfully flat.
7.1 Classifying first-order differential calculus on a Quantum Homo- geneous Space
We say that a first-order differential calculus Ω1(M), over a quantum homogeneous 1 space M, is covariant if there exists a (necessarily unique) map ∆L :Ω (M) → G ⊗ Ω1(M), such that
∆L(mdn) = ∆(m)(id ⊗ d)∆(n), m, n ∈ M. 1 G Any covariant calculus Ω (M) is naturally an object in MModM . Moreover, the universal 1 1 calculus over M is covariant, and covariance of any Ω (M) ' Ωu(M)/N is equivalent to 1 G G N being a sub-object of Ωu(M) in MModM . (Note that d is not a morphism in MModM .) Theorem 7.1 For a quantum homogeneous space B, considering B+ as an object in ModH according to its obvious right B-module structure, and the right H-comodule struc- + ture ∆R(m) = m(2) ⊗ S(π(m(1))), for m ∈ B , it holds that:
44 1. Covariant first-order differential calculi over B are in bijective correspondence with sub-objects of B+. 2. The sub-object corresponding to the calculus Ω1 is n o (1) X + X I := ε(mi)mi midni = 0 . i i
3. Denoting V 1 := B+/I(1), (which we call the cotangent space of Ω1) we have an isomorphism σ :Φ Ω1 → V 1, mdn 7→ ε(m)m+.
1 Proof. Applying the functor Φ to the collection of sub-objects of Ωu gives a correspon- 1 dence between left-covariant calculi over B and sub-objects of Φ(Ωu). The theorem then follows from the easily verifiable fact that an isomorphism is given by 1 + + Φ Ωu → B , mdn 7→ ε(m)n . For the special case of a trivial quantum homogeneous space, this result reduces to Woronowicz’s celebrated theorem classifying left-covariant calculi over a Hopf algebra G [?, Theorem 1.5]. For such a calculus Ω1(G), we follow the standard convention of denoting its cotangent space by Λ, and calling it the space of left-invariant 1-forms of the calculus.
7.2 The Podle´sSphere
We call a first-order differential calculus irreducible if it has no non-trivial sub-bimodules.
Theorem 7.2 There exist exactly two finite dimensional irreducible non-isomorphic co- 2 (1,0) variant first order differential calculi over Oq(S ). We denote these two calculi by Ω and Ω(0,1), and their corresponding ideals by I(1,0) and I(0,1) respectively.
Description 1: Regarding E,F ∈ Uq(sl2) as functionals on Oq(SU2) through the dual pairing (1,0) 2 (1,0) 2 I = ker(E : Oq(S ) → C),I = ker(E : Oq(S ) → C).
Description 2: Generating sets for the ideals I(1,0) and I(0,1) are given respectively by (1,0) − + − (0,1) (1,0) + − + I := {b , b0, b b },I := {I := {b , b0, b b }. Hence we see that a basis of V (1,0) and V (0,1)) is given by e+ := [b+] and e− := [b−] respectively.
(1,0) (0,1) Exercise 7.3. Show that, as bimodules, Ω and Ω are isomorphic to E−2 and E2 respectively.
45 7.3 Complexes and Double Complexes
For (S, +) a commutative semigroup, an S-graded algebra is an algebra A equipped L s s with a decomposition A = s∈S A , where each A is a linear subspace of A, and AsAt ⊆ As+t, for all s, t ∈ S. If a ∈ As, then we say that a is a homogeneous element of degree s.A homogeneous mapping of degree t on A is a linear mapping L : A → A such that if a ∈ As, then L(a) ∈ As+t. We say that a subspace B of A is homogeneous if it L s s s admits a decomposition B = s∈S B , with B ⊆ A , for all s ∈ S. A pair (A, d) is called a complex if A is an N0-graded algebra, and d is a homogeneous mapping of degree 1 such that d2 = 0. A triple (A, ∂, ∂) is called a double complex if A 2 is an N0-graded algebra, ∂ is homogeneous mapping of degree (1, 0), ∂ is homogeneous mapping of degree (0, 1), and
2 ∂2 = ∂ = 0, ∂ ◦ ∂ = −∂ ◦ ∂.
Observe that we can associate to any double complex (A, ∂, ∂) the complex (A, ∂ + ∂).
7.4 Differential Calculi
A complex (A, d) is called a differential algebra if d is a graded derivation, which is to say, if it satisfies the graded Leibniz rule
d(αβ) = d(α)β + (−1)kαdβ, (for all α ∈ Ak, β ∈ A).
The operator d is called the differential of the differential algebra.
Definition 7.4. A differential calculus over an algebra A is a differential algebra (Ω•, d) 0 k such that Ω = A, and Ω = spanC{a0da1 ∧ · · · ∧ dak | a0, . . . , ak ∈ A}. We use ∧ to denote the multiplication between elements of a differential calculus when both are of order greater than 0. The notions of differential map, covariance, and a ∗-calculus have naturally/easily guessed differential versions, see [?, §?] for details • 1 We say that a differential calculus (Γ , dΓ) extends a first-order calculus (Ω , dΩ) if there 1 1 exists a bimodule isomorphism ϕ :Ω → Γ such that dΓ = ϕ ◦ dΩ.
Lemma 7.5 Every first-order calculus admits an extension Ω• which is maximal in the sense that there there exists a unique differential map from Ω• onto any other extension of Ω1. We call this extension the maximal prolongation of the first-order calculus.
7.5 Complex Structures
The definition of a complex structure is introduced in two steps. First the definition of an almost complex structure is recalled and then the notion of integrability presented.
46 Definition 7.6. An almost complex structure for a differential ∗-calculus Ω•, over a 2 L (a,b) • 2 ∗-algebra A, is an -algebra grading 2 Ω for Ω such that, for all (a, b) ∈ N0 (a,b)∈N0 N0
k L (a,b) 1. Ω = a+b=k Ω , 2. ∗(Ω(a,b)) = Ω(b,a).
Let ∂ and ∂ be the unique homogeneous operators of order (1, 0), and (0, 1) respectively, defined by
∂|Ω(a,b) := projΩ(a+1,b) ◦ d, ∂|Ω(a,b) := projΩ(a,b+1) ◦ d,
a+b+1 (a+1,b) where projΩ(a+1,b) , and projΩ(a,b+1) , are the projections from Ω onto Ω and Ω(a,b+1) respectively. When d = ∂ + ∂, we say that the almost complex structure is integrable. We usually call an integrable almost complex structure a complex structure
7.6 Hermitian Surfaces and K¨ahler–Dirac Operators
7.6.1 Hermitian Surfaces
Definition 7.7. A surface is called orientable if there exists an isomorphism Ω2 ' B, which is to day, if Ω2 is a free module.
Definition 7.8. A noncommutative complex surface is an algebra A together with an orientable differential calculus Ω• of total dimension 2, endowed with a choice of complex structure Ω(•,•) such that Ω2 = Ω(1,1). If A is a quantum homogeneous space and the calculus and complex structure are covariant, then we say that the complex structure is covariant.
The volume map associated to any such κ is the left B-module map
vol : Ω2 → M, vol(bκ) = b, b ∈ B.
Definition 7.9. An Hermitian surface is a complex structure together with a choice of form κ ∈ Ω(1,1). If the complex surface is covariant and κ is left coinvariant, then we say that the Hermitian structure is covariant.
Definition 7.10. The Hodge map associated to an Hermtian form is the map
± ∗h(1) = κ, ∗h(ω) = ±iω, for ωinΩ . metric associated to
Observe that if g(ω, ν) is contained in the cone of positive elements aa∗, then an inner product is given by
h·, ·i := h ◦ g :Ω• ⊗ Ω• → B.
47 We denote the Hilbert space completion of Ω(0,•) with respect to <, > by L2(Ω(0,•)). ∗ Moreover, we denote D/ := ∂ + ∂ and call it the K¨ahler-Dirac operator
Proposition 7.11 If g is positive, then
1. If G has no zero divisors, then B is faithfully represented on L2(Ω(0,•)) through multiplication.
2. The operators ∂ and ∂ are adjointable
3. The commutator [D/, b] ∈ B(L2(Ω(0,•))), for all b ∈ B.
7.7 A Spectral Triple for the Podle´sSphere
2 (0,2) Proposition 7.12 The Laplacian ∆∂ = D/ acts on Ω as the Casimir.
Corollary 7.13 The Dirac operator has discrete unbounded spectrum.
Proof. Follows from the description of the eigenvalues of the Casimir given earlier.
Corollary 7.14 For any λ ∈res(D/ ), the operator (D/ − λ1)−1 is compact.
2 2 (0,•) Corollary 7.15 The triple (Oq(S ),L (Ω ), D/ ) is a spectral triple.
8 Appendix
8.1 Two Different Definitions of Compact Quantum Qroup
In this section first we introduce the concept of compact quantum matrix group and we refer the readers to [Woro. paper] for more details. Then we give a new definition of compact quantum group based on the definition of compact matrix quantum group with a family of generators anf we will show that this definition is equivalent with the definition of compact quantum group in the sense of Woronowicz.
Definition 8.1. Let A be a unital C∗-algebra and u is an N × N matrix with entries belonging to A. Then G = (A, u) is a compact matrix quantum group (compact matrix pseudogroup) if the following three axioms are satisfied.
1. A is the smallest C∗-algebra containing all matrix elements of u.
2. There exists a ∗-algebra homomorphism ∆ : A → A ⊗ A such that ∆(ukl) = P min r ukr ⊗ url for any k, l = 1, ··· ,N.
48 3. There exists a linear antimultiplicative mapping κ : A → A where A denotes the dense ∗-subalgebra of A generated by matrix elements of u, such that κ(κ(a∗)∗) = a for all a ∈ A and X κ(ukr)url = δrl1 X ukrκ(url) = δrl1. for any k, l = 1, 2, ··· ,N.
We can prove that the last axiom may be replaced by the following: 30. u and uT (the transpose of u) are invertible.
Definition 8.2. The first definition of compact quantum group ∗ Let G = (A, ∆), where A is a unital C -algebra and ∆ : A → A ⊗ A is a unital ∗-algebra min homomorphism. We say that G is a compact quantum group if 1. (id ⊗ ∆)∆ = (∆ ⊗ id)∆, 2. The sets {(b ⊗ 1)∆(a), a, b ∈ A} and {(1 ⊗ b)∆(a), a, b ∈ A} are linearly dense subsets of A ⊗ A. min Definition 8.3. The second definition of compact quantum group A compact quantum group C∗-algebra is a unital C∗-algebra A such that the following two conditions hold:
1. There is a unital ∗-homomorphism ∆ : A → A ⊗ A which satisfies the relation min (∆ ⊗ id)∆ = (id ⊗ ∆)∆.
ν ν 2. There is a family of matrices with entries in A, {u = (uij)|i, j = 1, ··· , dν, ν ∈ I} where dν ∈ N and I is an index set, such that:
ν P ν ν • ∆(ukl) = r ukr ⊗ url for ν ∈ I, i, j = 1, 2, ··· , dν. • uν and its transpose (uν)T , ν ∈ I, are invertible matrices over A. • The ∗-subalgebra A of A generated by the entries uν is dense in A.
Definition 8.4. Let G = (A, ∆) be a compact quantum group and H be a Hilbert space. Then u is a unitary representations of G acting on H if u is a unitary element of M(K(H) ⊗ A) such that (id ⊗ ∆)u = u12u13.
Example 8.5. Let G = (A, ∆) be a compact quantum group and u = (ukl)k,l=1,··· ,N be an N × N matrix with entries belonging to A. u is an N dimensional unitary represen- P tation of G if u is a unitary element of MN (A) = MN (C) ⊗ A and ∆(ukl) = r ukr ⊗ url for all k, l = 1, ··· ,N.
49 It is well known that any finite-dimensional representation of a C∗-algebra decomposes into a direct sum of irreducible representations. We conclude that any finite-dimensional unitary representation of a compact quantum group is a direct sum of irreducible repre- sentations.
Theorem 8.6 Let u be a unitary representation of a compact quantum group G = (A, ∆) acting on a Hilbert space of any dimension. Then u is a direct sum of finite-dimensional irreducible representations.
Definition 8.2 ⇒ Definition 8.3: Let = (A, ∆) be a compact quantum group and (uν) be the complete family of G ν∈Gb irreducible unitary representations of G: The representations (uν) are pairwise inequivalent and any unitary irreducible rep- ν∈Gb ν resentation of G is equivalent to u for one ν ∈ Gb. Let Hν be the carrier Hilbert space ν of u and dν = dimHν. We know that all dν < ∞. Introducing an orthonormal basis in ν ν ν Hν we may identify u with a matrix (ukl)k,l=1,2,··· ,dν , where ukl are elements of A such that:
ν X ν ν ∆(ukl) = ukr ⊗ url r X ν ν∗ ukrulr = δk,l1 r X ν∗ ν urk url = δk,l1 r where the summation indexe r, k runs over 1, 2, ··· , dν and k, l = 1, 2, ··· , dν. Let A be the set of all linear combinations of matrix elements of finite-dimensional representations of G. Then ab ∈ A for any a, b ∈ A (the tensor product of unitary representations is a unitary representation). A is a dense ∗-subalgebra of A and {uν} ν∈Gb is a basis (in the sense of the vector space theory) of A. Let ε : A → C and κ : A → A be linear mappings introduced by the formulae
ν ε(ukl) := δkl ν ν∗ κ(ukl) := ulk for any ν ∈ Gb and k, l = 1, 2, ··· , dν. Using these formulas, we can easily prove that all the conditions of the second definition is satisfied.
Definition 8.3 ⇒ Definition 8.2: If we have the second definition then it is enough to show that {(a ⊗ 1)∆(b), a, b ∈ A} and {(1 ⊗ a)∆(b), a, b ∈ A} are dense subsets of A ⊗ A. Define the following maps from min
50 A ⊗ A to itself.
T : A ⊗ A → A ⊗ A,T (a ⊗ b) = (a ⊗ 1)∆(b) S : A ⊗ A → A ⊗ A,S(a ⊗ b) = (1 ⊗ a)∆(b)
So it is enough to show that these two maps are bijective. we define the maps
T −1 : A ⊗ A → A ⊗ A,T −1(a ⊗ b) = (a ⊗ 1)(κ ⊗ id)∆(b) −1 −1 − S : A ⊗ A → A ⊗ A,S (a ⊗ b) = (b ⊗ 1)σA(id ⊗ κ 1)∆(a) and we will show that these maps are exactly the inverse of T and S respectively. α β It is sufficient to perform computations for a = ukl and b = urs, where α, β ∈ Gb, k, l = 1, ··· , dα and r, s = 1, ··· , dβ. For example
dβ −1 X α β β T (T (a ⊗ b)) = T (uklκ(urm) ⊗ ums) m=1
dβ X α β β β = uklκ(urm)umn ⊗ uns = a ⊗ b. m,n=1
References
[1] C. Kassel, Quantum Groups, Springer–Verlag, New York–Heidelberg-Berlin, (1995).
[2] M. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., (1969).
[3] A. Klimyk and K. Schmdgen, Quantum groups and their representations, Springer (1997).
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