HOPF ALGEBRAS AND LIE ALGEBRAS UCHICAGO PRO-SEMINAR - JANUARY 9, 2014

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1 HOPF ALGEBRAS AND LIE ALGEBRAS UCHICAGO PRO-SEMINAR - JANUARY 9, 2014 Hopf Algebras Lie Algebras Restricted Lie Algebras Poincaré-Birkhoff-Witt Theorem Milnor-Moore Theorem Cohomology of Lie Algebras Remark 0.1. My sources are More Concise Algebraic Topology by May and Ponto, Homological Algebra by Cartan and Eilenberg, Cohomology operations by Mosher and Tangora. Some definitions are essentially copied from these sources. Remark 0.2. Throughout, I will assume that k is a field, although many of these results hold in greater generality. See More Concise for the general results. 1. Introduction Let A be the Steenrod algebra and A be its dual. We have seen that there is a spectral sequences, called the Adams Spectral Sequence, Ext s,t A pf 2, F 2 q ùñ π t s S^ 2. One would like to compute the E 2 -page. This is the subject of Peter s thesis, which finished in 1964, 50 years ago. In the next three lectures, we will be giving an overview of this topic. First, what does Ext s,t A pf 2, F 2 q even stand for? Let IpAq be the augmentation ideal of A, i.e., IpAq kerpɛq. Let sipaq be a copy of IpAq with all elements given homological degree 1. Define BpA, Aq : A b k T psipaqq b k A. 1

2 2 Write ara 1 a 2... a n sb for a typical element, where ra 1 a 2... a n s is zero if any of the a 1 is are in the kernel of ɛ. Let r s denote the unit in T psipaqq 0. Define ɛ : BpA, Aq Ñ A ar sb ÞÑ ab ara 1 a 2... a n sb ÞÑ 0. Define a differential d : BpA, Aq Ñ BpA, Aq which is an A-A-bimodule morphism, so that dpaxbq p 1q deg a adpxqb for x P T psipaqq and with dra 1... a n s a 1 ra 2... a n s Also, define n 1 i1 λpiq i p 1q λpiq ra 1... a i a i 1... a n s p 1q λpn 1q ra 1... a n 1 sa n, i j1 deg a i degra 1... a i s. BpN, Mq N b A BpA, Aq b A M so that Bpk, kq T psipaqq. Now, H pa, kq HpHom A pbpa, kq, kq Ext A pk, kq HpBpk, kq q. Now, A is an example of a Hopf algebra. For a primitively generated Hopf algebra A, there is some technology to compute Ext A pk, kq. This is what May exploited in his thesis. In fact, there is a filtration on A such that the associated graded E 0 A is a primitively generated Hopf algebra. May develops two things. First, a spectral sequence H pe 0 Aq ùñ E 0 H paq, which allows him to compute E 0 H paq knowing E 0 A. Second, he constructs a reasonably small complex XpLq which allows him to compute H pe 0 Aq. In particular, he applies all this to the Steenrod algebra!

3 HOPF ALGEBRAS AND LIE ALGEBRAS UCHICAGO PRO-SEMINAR - JANUARY 9, Hopf Algebras Definition 2.1. Let k be a commutative ring with unit. A Hopf algebra is a graded associative k-algebra pa, φ, ηq and a coalgebra pa, ψ, ɛq where η : k Ñ A ɛ : A Ñ k φ : A b A Ñ A ψ : A Ñ A b A χ : A Ñ A unit counit or augmentation product coproduct antipode such that φ is a morphism of coalgebras (or equivalently ψ is a morphism of algebras). A Hopf algebra is connected if A n 0 if n 0 and A 0 k. It is projective (resp. finitely generated) if each A i is projective (resp. finitely generated) over k. Definition 2.2. A left A-module is a k-module and an action A b N Ñ N. A left A-comodule is a k-module together with a coaction N Ñ A b N. Example 2.3. (i) Commutative Hopf algebras are cogroup objects in the category of commutative k-algebras. (ii) Let G be a group. Then krgs is a Hopf algebra. The antipode is induced by the inverses and the coproduct by the diagnonal. (iii) If X is a connected homotopy associative H-space, then the homology of X is a Hopf algebra. Let IpAq kerpɛq JpAq cokerpηq. Now note that the natural map IpAq Ñ JpAq is an isomorphism. Let P paq and QpAq be defined by the following exact sequences: and 0 Ñ P paq Ñ JpAq ψ ÝÑ JpAq b JpAq

4 4 IpAq b IpAq φ ÝÑ IpAq Ñ QpAq Ñ 0. Definition 2.4. An element x P IpAq is primitive if its image in JpAq lies in P paq. Lemma 2.5. Let A be a Hopf algebra and x P IpAq. Then ψpxq 1 b x x b 1 x1 b x 2. The element x is primitive if and only if ψpxq 1 b x x b 1. Definition 2.6. Let ν be the composite P paq Ñ JpAq IpAq Ñ QpAq. Then, A is primitive or primitively generated, if ν is an epimorphism. It is coprimitive if ν is a monomorphism. Lemma 2.7. Let A be a Hopf algebra, then its k-linear dual A is also a Hopf algebra. Further, P pa q QpAq. If A is projective of finite type, IpAq b IpAq Ñ IpAq Ñ QpAq Ñ 0. is split exact if and only if 0 Ñ P pa q Ñ JpA q Ñ JpA q b JpA q and when this holds, P pa q QpAq. Definition 2.8 (Product Filtration). Let A be a Hopf algebra. The product filtration on A is given by F q A A if q 0, and F q paq IpAq q if q 0. Define Ep,qA 0 pf p A{F p 1 Aq p q. Note that E 0 q, 0 if p 0, E 0 A k and 0, E0 A QA. 1, Lemma 2.9. If k is a field, (or 0 Ñ F q A Ñ A Ñ A{F q A Ñ 0

5 HOPF ALGEBRAS AND LIE ALGEBRAS UCHICAGO PRO-SEMINAR - JANUARY 9, is split for all q), then E 0 A is a primitively generated Hopf algebra. Proof. The elements of E 0 1, A QpAq generate E0 A. We must verify that they are primitive. Let x P QpAq be represented by x P IpAq. Then ψp xq 1 b x x b 1 x1 b x 2 1 b x x b 1 mod IpAq Lie Algebras Definition 3.1. A (graded) Lie algebra over a commutative ring k is a graded k-module L with a morphism of k-module r, s : L b L Ñ L such that there exists an associative k-algebra A and a monomorphism of j : L Ñ A with jprx, ysq jpxqjpyq p 1q deg x deg y jpyqjpxq. Morphisms of Lie algebras are morphism of k-module which commute with the bracket operations. Remark 3.2. Any associative algebra A is a Lie algebra with bracket ra, bs ab p 1q deg a deg b ba. Thus one can say that the morphism j above commutes with the bracket operations on L and A. Remark 3.3. This imposes the following conditions on the Lie bracket: (i) rx, ys p 1q deg x deg y ry, xs (ii) rx, xs 0 if deg x is even or char k 2 (iii) p 1q deg x deg z rx, ry, zss p 1q deg x deg y ry, rz, xss p 1q deg z deg y rz, rx, yss 0 (Jacobi identity) (iv) rx, rx, xss 0 if deg x is odd. One can show that (i)-(iv) are sufficient conditions for a bracket operation on L to give L the structure of a Lie algebra. Example 3.4. (i) Let A be a Hopf algebra. Then P A is a Lie algebra. Definition 3.5. The universal enveloping algebra of a Lie algebra L is an associative algebra U plq, viewed as a Lie algebra, such that there exists a morphism of Lie algebras i : L Ñ UpLq so that, for any morphism of Lie algebras j : L Ñ A into an associative algebra A, the following diagram

6 6 can be filled where f is a morphism of algebras: L j A i UpLq f Lemma 3.6. The algebra UpLq exists and is isomorphic to the free tensor algebra T plq modulo the two sided ideal generated by elements of the form: xy p 1q deg x deg y yx rx, ys x, y P L. The diagonal on L gives rise to a coproduct on UpLq and the map x ÞÑ x on L gives rise to an antipode, making U plq into a primitive Hopf algebra. Example 3.7. Let L be a commutative Lie algebra. That is rx, ys 0 for all x, y in L. The universal enveloping algebra U plq is thus ApLq, the free (graded) commutative algebra generated by L. There is an obvious inclusion L Ñ UpLq which sends L to the projection of T plq 1 in UpLq. Note that the diagononal L Ñ L L induces the map UpLq Ñ UpL Lq UpLq b UpLq which sends x ÞÑ 1 b x x b 1. So L P UpLq. In fact, we will see that if char k 0, this is an equality. That is, no decomposable element is primitive. Indeed, if y is primitive then ψpy n q i jn pi, jqy i b y j, pi jq! (here, pi, jq ) Given a basis, one could use this to check the claim. If char k 0, the i!j! conclusion would fail for p-th powers and we will see that these are the only elements for which it does fail. The key to a good basis for UpLq is the Poincaré-Birkhoff-Witt theorem. To give the statement of this theorem, we need to describe filtrations on Lie algebras. Definition 3.8 (Lie Filtration). Let L be a Lie algebra. Let F p UpLq 0 for p and F p UpLq pr ` Lq p for p 1. This is called the Lie filtration on UpLq. 0, F 0 UpLq R Given any filtered associative algebra A, define E 0 p,qa pf p A{F p 1 Aq p q,

7 HOPF ALGEBRAS AND LIE ALGEBRAS UCHICAGO PRO-SEMINAR - JANUARY 9, and E` n A p Ep,qA. 0 qn Theorem 3.9 (Poincaré-Birkhoff-Witt). Let L be a Lie algebra over a field k. The map f : ApLq Ñ E`UpLq induced by the natural inclusion of L into ApLq is an isomorphism of Hopf algebras. This implies that (1) for tx i u and ty i u a k-bases for L and L respectively, indexed on totally ordered sets, UpLq is the free k-module on tx i1... x im y r 1 j 1... y rn j n i 1... i m, y 1... y m, r k 1u. (2) for any k-module L satisfying properties of remark (3.3), L Ñ UpLq is an injection and L is a Lie algebra. Theorem 3.10 (Milnor-Moore (Part 1)). Suppose that char k 0. Let L be the categories of Lie algebras and PH be the category of primitive Hopf algebras. Then U : L Ñ PH and P : PH Ñ L are inverse equivalence of categories. That is, (i) P UpLq L as Lie algebras ; (ii) UpP Aq A as primitive Hopf algebras. Corollary If A is a commutative, primitive Hopf algebra, the A is isomorphic as a Hopf algebra to the free commutative algebra generated by P A, i.e., A EpP A q b P pp A q. 4. Restricted Lie Algebras Definition 4.1. Suppose that char R 2. Let L be the k-submodule of L whose elements are concentrated in even degrees and L be the k-submodule of L whose elements are concentrated in odd degrees. If char R 2, let L L and L t0u. Definition 4.2. Suppose that the ring k has prime characteristic p. A restricted Lie algebra is a Lie algebra L with a map ξ : L Ñ L, called the restriction, with ξpl n q L pn, such that there exists an associative algebra A and a monomorphism j : L Ñ A such that ξpxq x p.

8 8 A morphism of restricted Lie algebras is a morphism of Lie algebra which commutes with the restrictions. Remark 4.3. Any associative algebra A is a restricted Lie algebra with restriction ξpaq a p for a P A. Thus one can say that the morphism j above commutes with the restrictions L and A. Definition 4.4. The universal enveloping algebra of a restricted Lie algebra is an algebra V plq together with a morphism i : L Ñ V plq, such that, given any morphism j : L Ñ A of restricted Lie algebras, where A is an associative algebra, the following diagram can be filled with f is a morphism of algebras: L j A i V plq f Lemma 4.5. The algebra V plq exists and is isomorphic to UpLq modulo the two sided ideal generated by elements of the form: Further, V plq is a primitive Hopf algebra. ξpxq x p, x P L. Example 4.6. Let L be a commutative Lie algebra with zero restriction. The universal enveloping algebra V plq is thus BpLq ApLq{J, where J is the ideal generated by tx p x P Lu. Note that, on V plq, the elements x p have filtration degree 0. Hence, the PBW cannot hold as stated in the restricted case. The obvious correction makes it true: Theorem 4.7 (Restricted Poincaré-Birkhoff-Witt). Let L be a Lie algebra over a field k. The map f : BpLq Ñ E`V plq induced by the natural inclusion of L into BpLq is an isomorphism of Hopf algebras. This implies that for tx i u and ty i u a k-bases for L and L is the free k-module on respectively, indexed on totally ordered sets, UpLq tx i1... x im y r 1 j 1... y rn j n i 1... i m, y 1... y m, 1 r k pu. Theorem 4.8 (Milnor-Moore (Part 2)). Suppose that char k 0. Let RL be the categories of restricted Lie algebras and PH be the category of primitive Hopf algebras over k. Then V : RL Ñ PH and P : PH Ñ RL are inverse equivalence of categories. That is, (i) P V plq L as restricted Lie algebras ; (ii) V pp Aq A as primitive Hopf algebras.

9 HOPF ALGEBRAS AND LIE ALGEBRAS UCHICAGO PRO-SEMINAR - JANUARY 9, In this section, suppose that L L. 5. Cohomology of Lie algebras Definition 5.1. A right L module N is a right UpLq-module. Similarly, a left L-module M is a left U plq-module. Let M be a left L-module and N be a right L-module. H pl; Mq Ext U plqpk, Mq H pl; Nq N b U plq k. Let sl denote a copy of L where all of the elements have been given homological degree 1. The standard complex, or equivalently, the Chevalley-Eilenberg or Koszul complex is given by dpu x 1,..., x n q Y plq UpLq b EpsLq, 1 i n 1 i j n p 1q i 1 ux i x 1,..., px i,..., x n p 1q i j u rx i, x j s, x 1,..., px i,..., px j,..., x n. The complex Y plq has an augmentation ɛ : Y plq Ñ k, induce by the augmentation Y 0 plq UpLq Ñ k. In fact, Y plq is a UpLq-free resolution of k as a left UpLq-module. Example 5.2. If r, s is identically zero on L, this is the usual Koszul complex. Further, Hom U plq py plq, kq HompEpLq, kq EpL q and the differentials are all zero Hence, for an abelian Lie algebra, H pl; kq is an exterior algebra.