Hopf Algebras. Zajj Daugherty. March 6, 2012

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1 Hopf Algebras Zajj Daugherty March 6, 2012 Big idea: The Hopf algebra structure is essentially what one needs in order to guarantee that tensor products of modules are also modules Definition A bialgebra B over a field k is a structure which is both a unital (has identity) associative algebra and a coalgebra (has a map B B B) over k, such that these structures are compatible Compatibility just means that the comultiplication and the counit ε are both unital algebra homomorphisms, or equivalently, that the multiplication and the unit of the algebra both be coalgebra morphisms (these are equivalent in that they are epressed by the same diagram) A Hopf algebra is a bialgebra with an anitpode a map which generalizes the inversion map on a group B B B ε B B S id k id S B B η B B mult B mult commutes, where η : k B is the unit map η(c) = c 1 Eample: group algebras A super-simple eample of these are group algebras with normal multiplication, comultiplication given by g g k g, and antipode g S g 1 1

2 Eample: enveloping algebras and quantum groups Let g be a Lie algebra over k The enveloping algebra Ug of g is the algebra generated by g with the relations y y = [, y] for all, y g If V is a g-module (ie a Ug-module), then this action etends to V V by (v 1 v 2 ) = v 1 v 2 + v 1 v 2, for all g, v 1, v 2 V This reflects the Hopf algebra structure on Ug given by counit comultiplication antipode ε 0, S 1 + 1, and, for g Similarly, if U = U h g is the Drinfeld-Jimbo quantum group, then U is also a Hopf algebra A little more about this is below in the section about ribbon Hopf algebras Fun fact, though, is that in either case, ε is the representation corresponding to the trivial module Eample: polynomial functions on a Lie group Let B = O(G) be the algebra of comple-valued polynomial functions on a comple Lie group G, and identify O(G G) with B B Then, B is a Hopf algebra with counit comultiplication ε(f) = f(1), ((f))(g 1, g 2 ) = f(g 1 g 2 ), and antipode (S(f))(g) = f(g 1 ) If g is the Lie algebra of G, then the above two eamples are dual to one another: Define a bilinear form O(G) Ug C by f, = d dt t=0 f(ep(t)) Then, fg, = f g, (), 1, = ε(), (f), y = f, y, ε(f) = f, e, S(f), = f, S(), f, = f, S() Why topologists care Let G be a topological group There are the two familiar maps G G G (multiplication) and G G G (the diagonal in G) Thus a functor F : {top spaces} {vector spaces} which takes cross products into tensor products gives me a vector space F (G) and maps F (G) F (G) F (G) and F (G) F (G) F (G) So if functors like F let us in some sense see the structure of topological spaces reflected in the category of vector spaces, then what they turn groups into Hopf algebras If F linearizes the category of topological spaces somehow, then this says that the linearization of a group is Hopf algebra 2

3 Quasitriangular and ribbon Hopf algebras Let U be a Hopf algebra with coproduct, counit ε, and antipode S For U, write () = (1) (2) and op () = (2) (1) A quasitriangular Hopf algebra (U, R) is a Hopf algebra U with an invertible element R = R R 1 R 2 in U U such that if R 12 = R R 1 R 2 1, R 13 = R R 1 1 R 2, R 23 = R 1 R 1 R 2, then for U, R()R 1 = op (), ( id)(r) = R 13 R 23, and (id )(R) = R 13 R 12 (1) By [Dr, Prop 21], the element u in U defined by u = R S(R 2 )R 1 satisfies uu 1 = S 2 () for all U (2) A ribbon Hopf algebra (U, R, v) is a quasitriangular Hopf algebra (U, R) with an invertible element v satisfying v Z(U), v 2 = us(u), S(v) = v, ε(v) = 1, (v) = (R 21 R) 1 (v v), (3) where R 21 = R R 2 R 1 Eamples Let g be a finite dimensional comple semisimple Lie algebra, and U h g be the Drinfel d-jimbo quantum group corresponding to g Both U = Ug with R = 1 1 and v = 1, and U = U h g with v = e hρ u, (4) are ribbon Hopf algebras (see [LR, Corollary (215)]) If M and N are U-modules then is a U-module with action given by (m n) = (1) m (2) n, where () = (1) (2), (5) for U The trivial U-module is 1 = C1, with 1 = ε()1, for U (6) If V is a U-module then the dual module is V = Hom(V, C) with (ϕ)(m) = ϕ(s()m), for U, (7) and, if e 1,, e n is a basis of V, and e 1,, e n is the dual basis in V then 1 coev V V 1 i e i e i and V V ev 1 ϕ m ϕ(m) (8) 3

4 are U-module homomorphisms For U-modules M and N, the map Ř MN : m n R 2 n R 1 m R is a U-module isomorphism For U modules M, N, P and a U-module isomorphism τ M : M M, P M (N P ) = (N P ) M N P M Ř M,N P = (ŘMN id P )(id N ŘMP ) which, together, imply the braid relation τ M τ M = Ř MN (id N τ M ) = (τ M id N )ŘMN, M N P P = P () P = P () P Ř M N,P = (id M ŘNP )(ŘMP id N ), M N P P (ŘMN id P )(id N ŘMP )(ŘNP id M ) = (id M ŘNP )(ŘMP id N )(id P ŘMN) (9) If M is a U-module and C M : M M m vm then C M N = (ŘMNŘNM) 1 (C M C N ), (10) by the last identity in (3) Remark 1 [Dr, Prop 31] Let (U, R) be a quasitriangular Hopf algebra Then R satisfies the quantum Yang-Bater equation, R 12 R 13 R 23 = R 12 ( id)(r) = ( op id)(r)r 12 = R 23 R 13 R 12 (11) Since R = (ε id id)( id)(r) = (ε id id)r 13 R 23 = (ε id)(r) R, and R = (id id ε)(id )(R) = (id id ε)r 13 R 12 = (id ε)(r) R, it follows that (ε id)(r) = 1 and (id ε)(r) = 1 (12) Since R(S id)(r) = (m id)(id S id)(r 13 R 23 ) = (m id)(id S id)( id)(r) = (ε id)(r) = 1, it follows that (S id)(r) = R 1, and applying this relation to the quasitriangular Hopf algebra (U op, R 21 ) with antipode S 1 gives (id S 1 )(R) = R 1, and thus (S S)(R) = (id S)(S id)(r) = (id S)(id S 1 )(R) = R Summarizing, (S id)(r) = (id S 1 )(R) = R 1 and (S S)(R) = R (13) 4

5 References [CP] V Chari, A Pressley, A guide to quantum groups, Cambridge University Press, Cambridge, 1995 MR [Dr] V G Drinfel d, On almost cocommutative Hopf algebras, Leningrad Math J 1 (1990) MR (91b:16046) [LR] R Leduc and A Ram, A ribbon Hopf algebra approach to the irreducible representations of centralizer algebras: The Brauer, Birman-Wenzl, and Type A Iwahori-Hecke algebras, Advances Math 125 (1997), 1-94 MR (98c:20015) 5