Bialgebra Cohomology, Deformations, and Quantum Groups (Hopf Algebra/Quantum Yang-Baxter Operator/Hodge Decomposition/Laplacian) MURRAY Gerstenhabert and SAMUEL D
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Proc. NatI. Acad. Sci. USA Vol. 87, pp. 478-481, January 1990 Mathematics Bialgebra cohomology, deformations, and quantum groups (Hopf algebra/quantum Yang-Baxter operator/Hodge decomposition/Laplacian) MURRAY GERSTENHABERt AND SAMUEL D. SCHACKt tDepartment of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395; and $Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14214-3093 Communicated by Nathan Jacobson, September 8, 1989 (received for review March 21, 1989) ABSTRACT We introduce cohomology and deformation way. The pair (A, B) is biseparable if pi:A 0 A -* A and A:B theories for a bialgebra A (over a commutative unital ring k) B ® B have, respectively, a section and a retraction in (A, such that the second cohomology group is the space of infini- B)-birep. Let 21.A -> A be the bar resolution of A, in which tesimal deformations. Our theory gives a natural identification 9J A = A0(q+2), and B -- CB be the dual cobar resolution of between the underlying k-modules of the original and the B. Then in the double complex C, (A, B) = Hom(AB)birep deformed bialgebra. Certain explicit deformation formulas are (L.A, IC B) the modules are Cghq(A, B) = Homl(AB)-birep given for the construction of quantum groups-i.e., Hopf (MqA, ICB) = HomJ(A-4, B®P), the columns are the Hoch- algebras that are neither commutative nor cocommutative schild cochain complexes C,(A, BeP) with coboundary ah, (whether or not they arise from quantum Yang-Baxter oper- and the rows are the "co-Hochschild" complexes Cc(AO, B) ators). These formulas yield, in particular, all GLq(n) and with coboundary 5B. (We frequently suppress A and B, SLq(n) as deformations of GL(n) and SL(n). Using a Hodge writing simply 8h and 5c.) The bialgebra cohomology H;(A, decomposition of the underlying cochain complex, we compute B) is the homology of the total complex C;(A, B) in which Cb our cohomology for GL(n). With this, we show that every = Ep)+q=,,+1Cg" (note degrees) and the coboundary ah is is to one in which the comul- deformation of GL(n) equivalent given by 5bJ0` = ah + (_1)6c. The lowest dimensional tiplication is unchanged, not merely on elements of degree one cohomology group, H, 7(A, B), is then always k. For appli- but on all elements (settling in the strongest way a decade-old cations to deformation theory we also consider the subcom- conjecture) and in which the quantum determinant, as an plex Ceb(A, B) in which the edges are replaced by zero element of the underlying k-module, is identical with the usual complexes. Its homology is H'(A, B) and we have H -1 = Ht one. = 0. An n-cochain FECE is thus an n-tuple (Fl, . ,F.,) with F, E Cii-i+1. The projections '- C)," and Cl' C1 thenl induce maps from H;(A, B) to the usual Hochschild and Section 1. Bialgebra Cohomology and the Laplacian coalgebra cohomologies which, for clarity, we denote Hh and and cobar resolutions produces A a with ,u and comultipli- Hc. Using the normalized bar Let be k-bialgebra multiplication a whose inclusion (* induces A. We write A®" for the tensor product A k. .. subcomplex (W6(A, B) C2hit cation q-fold a isomorphism. Likewise, H^(A, B) can be Ok A. The category of left A-modules has an internal tensor cohomology product: M i N is just M Ok N with the "diagonal" action computed using normalized cochains. a(m (0 n) = Ya(1)m 0 a(2)n, where Aa = Ya(1) (0 a(2). Dually, The sequence 0-> C s-> Cams C-/Ce-> 0 induces a long exact sequence which, for n . 0, is fi - Hb for left A-comodules M and N with structure maps km:M -> .... A Ok M and AN we define the left comodule M Q) N to be M -4 Hgh (A, k) (D HC'(k,B) The maps t and irare -> and the Ok N with AMEN = (p. 0 Id)T(AM 0 AN), where r is given by induced by (a, 0,) (ca, 0.,0, 5h13) by projec- tion C0 -+ C"',+ D C'". r(a 0 m 0 a' 0 n) = a 0 a' 0 m 0 n. Similarly, tensor For each map of pairs (A', B') -* (A, B) one can define, in products of bimodules and bicomodules are again such and analogy with ref. 1, the (A', B') relative cohomology of(A, B) we extend these notations to those cases. One consists of A as that of a relative cochain complex Cb(A, BIA', B'). An (ordered) bialgebra pair (A, B) bialgebras then has k) = B). The definition and the and B; an A-bimodule structure on B such that A:B -* B ® B C;(A, Blk, C;(A, A -> B B A theorem below (which has been obtained independently by is a bimodule map while the actions Ok B and Ok not -> a B-bicomodule struc- A. T. Giaquinto, a student of M.G.) do actually require B are coalgebra maps; and, dually, a or B' an Q3 -- a and that A' be coalgebra algebra. ture on A for which L: A A A is bicomodule map the inclusions the actions are algebra maps. Every bialgebra map f:A -> B THEOREM 1. If (A', B') is biseparable then C; (A, B')-> C,(A, B) and C(,(A, BIA', B') C(A, B) yields a pair in which, for example, the left actions A -> B Ok BIA', U A and A Ok B -> B are (f 0 Id)A and /L(f 0 Id). (The case induce cohomology isomorphisms. most important for us is A = B and f = Id.) Bialgebra pairs When C is a bialgebra, a k-split exact sequence 0 B C form a category in which a map (A', B') -> (A, B) consists of 2 A -O 0 is singular if it is such both as an algebra extension bialgebra maps A'-> A and B-> B' that are also, respectively, and as a coalgebra extension; that is, Tr is an algebra map a B'-bicomodule map and an A'-bimodule map. (making B a C-bimodule) and L is a coalgebra map (making A An A-bimodule M is a birepresentation of(A, B) if it is also a C-bicomodule) which, further, are respectively a C- a B-bicomodule in a compatible way; i.e., A 0 M -* M is a bicomodule map and a C-bimodule map. Equivalence classes B-bicomodule map, M -> M ® B is an A-bimodule map, and of singular extensions form, under Baer sum, an abelian the analogous conditions hold on the other sides. Both A and group that is then naturally isomorphic to Hb(A, B). B are examples, hence so are AGO and B®P for all q and p. As a k-module, Cgq(A, B) = Homk(A®q, B®P), so the Birepresentations form a category (A, B)-birep in an obvious boundary maps A®(q-1) -> A®" of 'f A and B®P -* B(P-1) of h.B induce ac:CPt-> Caps- 1 and dh:CPq v> CP 1v. The total = + then has The publication costs of this article were defrayed in part by page charge complex e((A, B) with ahlC.' (-l)"ah ac payment. This article must therefore be hereby marked "advertisement" homology groups Hf'(A, B) and a Laplacian A = abh + 5ba, in accordance with 18 U.S.C. §1734 solely to indicate this fact. whose kernel we call the harmonic cochains. 478 Downloaded by guest on September 23, 2021 Mathematics: Gerstenhaber and Schack Proc. Natl. Acad. Sci. USA 87 (1990) 479 THEOREM 2. C'(A, A) = kerA ED Ima,. E Im8b when A is k) replaced by 0. Both vanish for i >> 0 and HiJ = Hf'Jfor finite-dimensional over k = R8. A j>O. U Some topological conditions are likely needed for this to hold more generally. Section 3. The Cohomology of Matrix Bialgebras Since H,(A, B) is contravariant in A and covariant in B, any contravariant functor A from a small category to k- The bialgebra M = M(n) of n x n matrices is the polynomial bialgebras gives rise to a "global" cohomology, H;(A, A), as ring k[{xj}] in n2 variables with Axij = YrXir 0 Xrj. The Hopf in the case of algebras (cf. section 6 of ref. 2). An analog of algebras of the general linear group and the special linear the cohomology comparison theorem-reducing this co- group are denoted GL(n) and SL(n) and are respectively homology to that of a single bialgebra-may then hold. obtained by localizing M(n) at det X and then taking the Our cohomology theory has several extensions. We can quotient modulo det X - 1. Let Md be the subspace of define H,(N, M) for (A, B)-birepresentations as the homol- homogeneous elements of degree d in M(n). Then M = M(n) ogy of HOm(A)-birep(.N, TCM), where 5A.N is the bimodule = k + M1 + M2 + . and each Md is a subcoalgebra, dual bar resolution of N (in which MO4 = AS(q+1) X N!X AO((+')) to the dth symmetric power SdM(n) of the usual algebra M(n) and ?(M is the dual bicomodule cobar resolution. It is then ofn x n matrices. This, Theorem 5, and ref. 7 imply that M(n) a theorem that this agrees with our earlier definition when N is coseparable, so that HI'(-, M) = 0 for m + 0.