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 T!;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 2 g, 2. (Jacobi Identity)[x; [y; z]] + [y; [z; x]] + [z; [x; y]] = 0, for all x; y; z 2 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 2 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 2 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 2 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 2 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 × C1(G) ! C1(G) such that (g ◦ F )(g0) = F (g−1g0) for all g; g0 2 G and F 2 C1(G). n 1 n 1 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 : C1(M) ! C1(M). Exercise 1.13. Let Der(M) denote the set of all derivations D : C1(M) ! C1(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 ' : 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 1 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 2 A; b; b0 2 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.