Lecture 7 - Universal Enveloping Algebras and Related Concepts, I

February 4, 2013

1 Tensor Algebras

Unlike most other topics in our class, which generally require algebraic completeness in one way or another, our ﬁeld F is arbitrary unless otherwise stated.

1.1 Universal associative algebras

Given a vector space V over F, a universal associative algebra over V is a pair (T (V ), i) where T (V ) is an associative algebra and i : V → T (V ) is a vector space homomorphism, with the following property:

Given any associative algebra A and a vector space homomorphism i : V → A, there is a unique algebra homomorphism θ : T (V ) → A such that i0 = θ ◦ i.

In other words, given any such A, i0 we can complete the diagram below as follows:

V 0 i i (1) T (V ) θ A

This “univeral associative algebra” is easy to construct explicitly. Simply set

∞ M Oi T (V ) = V i=0 (2) = F ⊕ V ⊕ (V ⊗ V ) ⊕ ...

1 and let the map i be the canonical injection.

Theorem 1.1 Given a vector space V over F, the pair (T (V ), i) is the universal associative algebra for V .

Pf. Given any algebra A over F and an F-linear map i0 : V → A, deﬁne θ : T (V ) → A on monomials v1 ⊗ · · · ⊗ vk to be

0 0 0 θ(v1 ⊗ v2 ⊗ · · · ⊗ vk) = i (v1)i (v2) . . . i (vk) (3) and extend by linearity. One easily checks this is a homomorphism. To see uniqueness, note that if ϕ : T (V ) → A were another map with ϕ(v) = i0(v) for v ∈ V , then on monomials 0 0 0 ϕ(v1 ⊗ v2 ⊗ · · · ⊗ vk) = ϕ(v1)ϕ(v2) . . . ϕ(vk) = i (v1)i (v2) . . . i (vk) = θ(v1 ⊗ v2 ⊗ · · · ⊗ vk).

1.2 Quotient algebras

The ﬁrst two subalgebras of the T (V ) one encounters are S(V ), the symmetric algebra over V and V V , the antisymmetric algebra over V .

To deﬁne the symmetric algebra, one lets

I = x ⊗ y − y ⊗ x x, y ∈ V (4) be the ideal in T (V ) generated by elements of the stated form, and deﬁnes

S(V ) = T (V ) /I (5)

The algebra S(V ) is abelian, and satisﬁes the obvious universal property (it is the universal abelian algebra over the vector ﬁeld V ). If V = spanF(v1, . . . , vn) then S(V ) is the span of monomials of the form

vi1 vi2 . . . vik where i1 ≤ i2 ≤ · · · ≤ ik (6) As an aside, the product in S(V ), here indicated by juxtaposition, is sometimes indicated by the symbol , as in v w (and v w = w v), and the algebra itself is often denoted K V. (7)

To deﬁne the antisymmetric algebra, one lets

I = x ⊗ y + y ⊗ x x, y ∈ V (8) be the ideal in T (V ) generated by elements of the stated form, and deﬁnes ^ V = T (V ) /I (9)

2 The algebra V V is antisymmetric. The product is usually denoted with the symbol “∧”, and V V is the span of monomials of the form

vi1 ∧ vi2 ∧ · · · ∧ vik where i1 ≤ i2 ≤ · · · ≤ ik. (10) Unlike the case of S(V ), if V is ﬁnite dimensional then V V is ﬁnite dimensional.

2 Cliﬀord algebras

Given a real vector space V with a non-degenerate symmetric bilinear form h·, ·i, let

I = v ⊗ w + w ⊗ v + hv, wi v, w ∈ V (11) be the ideal in T (V ) generated by elements of the given form. Deﬁne Cl(V ) = T (V ) /I. (12)

The only invariant of a bilinear form on a ﬁnite dimensional vector space over R is its signature (p, q). Thus we write

Clp,q. (13)

If bilinear form is positive deﬁnite of signature (p, 0) we just write Clp. By ﬁat, we take Cl0 = R.

For vector spaces over C, symmetric nondegenerate bilinear forms have no signature, only rank. We therefore have the complex cliﬀord algebras

Cln. (14)

2.1 Cl1, Cl0,1, and Cl2

2 We have V ≈ R, and set V = spanR{v} where v > 0 and v = 1 (obviously v = 1, but we want to distinguish this from the generator of the scalars) .

By deﬁnition

2 Cl1 = R ⊕ V ⊕ (V ⊗ V ) ⊕ .../ {v + 1 = 1} (15) which is the algebra of complex numbers.

Cl1 ≈ C. (16)

Next, if the inner product is negative deﬁnite so v2 = 1 instead of −1, we have two ± idempotents π ∈ Cl0,1 given by 1 1 π+ = (1 + v) and π− = (1 − v) (17) 2 2

3 one checks that π+π+ = π+ π−π− = π− (18) π+π− = π−π+ = 0 π+ + π− = 1.

These induce a splitting of the algebra

Cl0,1 ≈ R ⊕ R (19) where π+ is the projection onto the ﬁrst factor and π− is the projection onto the second factor.

Now consider V ≈ R2 with a positive deﬁnite inner product, and let {i, j} be an orthonormal basis of R. Then as a vector space

Cl2 ≈ spanR{ 1, i, j, ij } (20) with the relations being

i2 = −1, j2 = −1, ij + ji = 0 (21)

Therefore CL2 is the quaternions,

Cl2 ≈ H. (22)

2.2 Further cliﬀord algebras

The various Clp,q give rise to a veritable forrest of algebras. The ﬁrst nine of the Cln are

n Cln 0 R 1 C 2 H 3 ⊕ H H (23) 4 H(2) 5 C(4) 6 R(8) 7 R(8) ⊕ R(8) 8 R(16)

Beyond this, we have the “Bott periodicity” phenomenon Cln+8 ≈ Cln ⊗ Cl8, which is related to the theorem of the same name giving a period-8 relation in KO-theory.

4 3 Universal enveloping algebras

Any algebra A over a ﬁeld F has a canonical Lie algebra structure, which we’ll denote AL, given by the commutator bracket. Given a Lie algebra g, a universal enveloping algebra for g is a pair (U(g), i) where i : g → U(g)L is a Lie algebra homomorphism, with the following property:

0 If A is any associative algebra and i : g → AL is a lie algebra homomorphism, then there is a unique homomorphism θ of associative algebras so that i0 = θ ◦ i.

In other words, a unique homomorphism θ can be found making the following diagram commute: g i0 i (24) U(g) θ A

First, if a universal enveloping algebra exists, it is unique. To see this, assume (B(g), j) is another algebra with the universal property. Then there are maps θ, ϕ so that

g g j j i and i (25) ϕ U(g) θ B(g) U(g) B(g) whereupon j = θ ◦ i and i = ϕ ◦ j, so i = ϕ ◦ θ ◦ i. However in that case the maps η = Id and η = ϕ ◦ θ both satisfy the diagram

g i i (26) η U(g) U(g) so that, by uniqueness, ϕ◦θ = Id. Likewise θ◦ϕ = Id so that B(g) and U(g) are isomorphic and i and j are conjugate (basically, B(g) is just U(g) with a re-arrangement of bases).

It is not diﬃcult to establish the existence of U(g). Consider T (g), the tensor algebra over the g as a vector space, and set

I = x ⊗ y − y ⊗ x − [x, y] x, y ∈ g , (27) the ideal generated by the stated relation. Then deﬁne U(g) = T (g) /I. (28)

5 Theorem 3.1 The algebra U(g) with the canonical homomorphism i : g → U(g)L is the universal enveloping algebra.

Pf. Given a map i0 : g → A, we can consider it a vector space map, giving an associative algebra homomorphism θ0 : T (V ) → A. But θ0 passes to a map θ : U(g) → A provided I ⊆ ker θ0, which is certainly the case as θ0(x)θ0(y) − θ0(y)θ0(x) − θ0([x, y]) = 0 owing to the 0 0 0 fact that i = θ ◦ i and i is a Lie algebra homomorphism. It is not immediately clear whether U(g) is inﬁnite- or ﬁnite-dimensional, what its vector-space basis is, or even whether the homomorphism i : g → U(g)L is injective.

Next time we discuss the Poincare-Birkhoﬀ-Witt theorem, which is the basic structure theorem on universal enveloping algebras.