Bialgebra cohomology, deformations, and quantum groups - NCBI

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)

way. The pair (A, B) is biseparable if pi:A 0 A -* A and A:B B ® B have, respectively, a section and a retraction in (A, B)-birep. Let 21.A -> A be the bar resolution of A, in which 9J A A0(q+2), and B -- CB be the dual cobar resolution of B. Then in the double complex C, (A, B) = Hom(AB)birep (L.A, IC B) the modules are Cghq(A, B) = Homl(AB)-birep (MqA, ICB) = HomJ(A-4, B®P), the columns are the Hochschild cochain complexes C,(A, BeP) with coboundary ah, and the rows are the "co-Hochschild" complexes Cc(AO, B) with coboundary 5B. (We frequently suppress A and B, writing simply 8h and 5c.) The bialgebra cohomology H;(A, B) is the homology of the total complex C;(A, B) in which Cb = Ep)+q=,,+1Cg" (note degrees) and the coboundary ah is given by 5bJ0` = ah + (_1)6c . The lowest dimensional cohomology group, H, 7(A, B), is then always k. For applications to deformation theory we also consider the subcomplex Ceb(A, B) in which the edges are replaced by zero complexes. Its homology is H'(A, B) and we have H -1 = Ht = 0. An n-cochain FE CE is thus an n-tuple (Fl, . ,F.,) with F, E Cii-i+1. The projections '- C)," and Cl ' C1 then l induce maps from H;(A, B) to the usual Hochschild and coalgebra cohomologies which, for clarity, we denote Hh and Hc. Using the normalized bar and cobar resolutions produces (* induces a subcomplex (W6 (A, B) whose inclusion C2hit a cohomology isomorphism. Likewise, H^(A, B) can be

We introduce cohomology and deformation ABSTRACT theories for a bialgebra A (over a commutative unital ring k) such that the second cohomology group is the space of infinitesimal deformations. Our theory gives a natural identification between the underlying k-modules of the original and the deformed bialgebra. Certain explicit deformation formulas are given for the construction of quantum groups-i.e., Hopf algebras that are neither commutative nor cocommutative (whether or not they arise from quantum Yang-Baxter operators). These formulas yield, in particular, all GLq(n) and SLq(n) as deformations of GL(n) and SL(n). Using a Hodge decomposition of the underlying cochain complex, we compute our cohomology for GL(n). With this, we show that every deformation of GL(n) is equivalent to one in which the comultiplication is unchanged, not merely on elements of degree one but on all elements (settling in the strongest way a decade-old conjecture) and in which the quantum determinant, as an element of the underlying k-module, is identical with the usual one.

=

Section 1. Bialgebra Cohomology and the Laplacian Let A be a k-bialgebra with multiplication ,u and comultiplication A. We write A®" for the q-fold tensor product A k . .. Ok A. The category of left A-modules has an internal tensor product: M i N is just M Ok N with the "diagonal" action a(m (0 n) = Ya(1)m 0 a(2)n, where Aa = Ya(1) (0 a(2). Dually, 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 Ok N with AMEN = (p. 0 Id)T(AM 0 AN), where r is given by r(a 0 m 0 a' 0 n) = a 0 a' 0 m 0 n. Similarly, tensor products of bimodules and bicomodules are again such and we extend these notations to those cases. An (ordered) bialgebra pair (A, B) consists of bialgebras A and B; an A-bimodule structure on B such that A:B -* B ® B is a bimodule map while the actions A Ok B -> B and B Ok A -> B are coalgebra maps; and, dually, a B-bicomodule structure on A for which L: A Q3A -- A is a bicomodule map and the actions are algebra maps. Every bialgebra map f:A -> B yields a pair in which, for example, the left actions A -> B Ok A and A Ok B -> B are (f 0 Id)A and /L(f 0 Id). (The case most important for us is A = B and f = Id.) Bialgebra pairs form a category in which a map (A', B') -> (A, B) consists of bialgebra maps A'-> A and B-> B' that are also, respectively, a B'-bicomodule map and an A'-bimodule map. An A-bimodule M is a birepresentation of (A, B) if it is also a B-bicomodule in a compatible way; i.e., A 0 M -* M is a B-bicomodule map, M -> M ® B is an A-bimodule map, and the analogous conditions hold on the other sides. Both A and B are examples, hence so are AGO and B®P for all q and p. Birepresentations form a category (A, B)-birep in an obvious

computed using normalized cochains. The sequence 0-> C s-> Cams C-/Ce-> 0 induces a long fi - Hb exact sequence which, for n . 0, is .... The maps t and irare -4 Hgh (A, k) (D HC'(k, B) induced by (a, 0,) -> (ca, 0., 0, 5h13) and by the projection C0 -+ C"',+ D C'". For each map of pairs (A', B') -* (A, B) one can define, in analogy with ref. 1, the (A', B') relative cohomology of(A, B) as that of a relative cochain complex Cb(A, BIA', B'). One then has C;(A, Blk, k) = C;(A, B). The definition and the theorem below (which has been obtained independently by A. T. Giaquinto, a student of M.G.) do not actually require that A' be a coalgebra or B' an algebra. THEOREM 1. If (A', B') is biseparable then the inclusions

C; (A, BIA', B')-> C,(A, B) and C(,(A, BIA', B') C(A, B)U

induce cohomology isomorphisms. When C is a bialgebra, a k-split exact sequence 0 B C 2 A -O 0 is singular if it is such both as an algebra extension and as a coalgebra extension; that is, Tr is an algebra map (making B a C-bimodule) and L is a coalgebra map (making A a C-bicomodule) which, further, are respectively a Cbicomodule map and a C-bimodule map. Equivalence classes of singular extensions form, under Baer sum, an abelian group that is then naturally isomorphic to Hb(A, B). As a k-module, Cgq(A, B) = Homk(A®q, B®P), so the of 'f A and B®P -* B(P-1) of boundary maps A®(q-1) -> A®" h.B induce ac: CPt-> Caps- 1 and dh: CPq v> CP 1v. The total complex e((A, B) with ahlC.' = (-l)"ah + ac then has homology groups Hf'(A, B) and a Laplacian A = abh + 5ba, whose kernel we call the harmonic cochains.

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478

Mathematics: Gerstenhaber and Schack THEOREM 2. C'(A, A) = kerA ED Ima,. E Im8b when A is finite-dimensional over k = R8. A Some topological conditions are likely needed for this to hold more generally. Since H,(A, B) is contravariant in A and covariant in B, any contravariant functor A from a small category to kbialgebras gives rise to a "global" cohomology, H;(A, A), as in the case of algebras (cf. section 6 of ref. 2). An analog of the cohomology comparison theorem-reducing this cohomology to that of a single bialgebra-may then hold. Our cohomology theory has several extensions. We can define H,(N, M) for (A, B)-birepresentations as the homology of HOm(A)-birep(.N, TCM), where 5A.N is the bimodule bar resolution of N (in which MO4 = AS(q+1) X N!X AO((+')) and ?(M is the dual bicomodule cobar resolution. It is then a theorem that this agrees with our earlier definition when N = A and M = B. For a one-sided theory, call (A, B) a weak pair if B is a left A-module and A is a right B-comodule with compatibility conditions as earlier. If N and M are one-sided (A, B)-representations (in the evident sense) then we define ft;(N, M) as the homology of Hom(A.B)-rep(9I.N, % M). When (A, B) is a pair H;(A, B) = HR(k, k). This applies, in particular, when (A, B) is a weak pair in which A is commutative and B is cocommutative, for any such is a pair. If, further, A and B are graded connected and we use only graded maps then H,(k, k) (=H,(A, B)) is Singer's cohomology (3) with dimensions shifted down by 1. We do not know the relationship between our cohomologies and the natural Yoneda cohomologies. Section 2. Hodge Decomposition

Suppose that k D 0, that R is commutative algebra, and that M is a symmetric R-bimodule (rm = mr for all r E R and m E M). For each n - 0, the group algebra CS,, operates on Ch(R, M) by &,-'F(r1, . . . r, ) = F(r.1, . . ., ,r0,,). We (4) have constructed a family of orthogonal idempotents {e,,(i)}1jO E CS,, characterized as follows: ej(0) = 0 for n + 0, while ej(n) = (Xa)/n! and e,,(i) = 0 for i > n; Yjej(i) = 1; for each i the family {e,,(i)},,.o commutes with 8h [that is, 8he,,(i)F = e,,+l(i)8hF for all n and F]; and {ej(i)},,.Oj:.0 is universal for the latter property; i.e., if {t,, E US,,},,.o commutes with Sh then t,, = Xiltilell(i), where 1-1 is the sign representation QS, Q. Then Ch(R, M) = LI C'6-i, where C' "' = e,,(i)Cg and G H;(R, M) = LI Hh '-(R, M). These "Hodge decompositions" are functorial in M and Hgh" = 0 for n 4 0. THEOREM 3. If (A, B) is a pair with A commutative and B symmetric as an A-bimodule then all B®P are symmetric, Cb (A, B) has a decomposition induced by those of its

columns, and this induces Hodge decompositions Hb(A, B)

Hn+l,-l =Rbn-l@ ** *i3 b-° and Hn(A, B) = Ho with HgO(A, B) = H 0 there is a long exact .. k) -- H-'l'(A, sequence . . . - 1iobtained by restricting that in Section 1. A The dual conditions also permit a decomposition, and when both hold H,(A, B) has a double decomposition. We do not know the relation, if any, to harmonic cochains. For a smooth (e.g., polynomial) algebra R the results of ref. 5 imply that the decomposition of Hh(R, M) is Hf"'- = 0 for i < n and H'' = Hn0 = fQ"(R, M) = (A''11(R, R)) OR M, the space of alternating n-cochains that are derivations as functions of each individual variable ("skew multiderivations"-derivations for n = 1). Reasoning as in section 28 of ref. 6, we obtain the following. THEOREM 4. Let A be commutative andfinitely generated and B be symmetric. If A is smooth (as occurs when it is a Hopf algebra) then HW(A, B) is the homology of the complex fl'(A, k) -* fl'(A, B) -Q f7'(A, B02) - . .. in which the coboundary maps are &, and Q'(A, Bon) has degree n - 1. Also Hj-(A, B) is the homology of the subcomplex with fl'(A,

Proc. Natl. Acad. Sci. USA 87 (1990) k) replaced by 0. Both vanish for i >> 0 and HiJ =

j> O.

479

Hf'J for U

Section 3. The Cohomology of Matrix Bialgebras

The bialgebra M = M(n) of n x n matrices is the polynomial ring k[{xj}] in n2 variables with Axij = YrXir 0 Xrj. The Hopf algebras of the general linear group and the special linear group are denoted GL(n) and SL(n) and are respectively obtained by localizing M(n) at det X and then taking the quotient modulo det X - 1. Let Md be the subspace of homogeneous elements of degree d in M(n). Then M = M(n) = k + M1 + M2 + . . . and each Md is a subcoalgebra, dual to the dth symmetric power SdM(n) of the usual algebra M(n) of n x n matrices. This, Theorem 5, and ref. 7 imply that M(n) is coseparable, so that HI'(-, M) = 0 for m + 0. THEOREM 5. Symmetric powers of a separable 0-algebra are separable. a The commutativity of M = M(n) ensures that AiM1, viewed as a k-submodule of MOi, is an M-bicomodule. As WI(M, -) = Homk(A'Ml, -), when A = B = M the complex of Theorem 4 becomes Homk(AiMl, k) > Homf.(AiMl, M) -* Homk(AiMi, M®2) - ... , which is exact because its homology is HC(AiM1, M). Since, as is easily checked, the first map is a monomorphism when i 4 0, we have Hm'(M, M) = 0 for m -1 and Thm = Hfjii. Theorem 3 now yields an isomorphism from Hm'°(M, k) = (Amfll(M, k)) (M k = AmM* to Hg'm°(M, M) = f1i, induced by AmM* = (AmM,)* -* cgb(M, M): y (8S'y, 0 0, ). For use in Section 6 we provide another description of this map: if A -4 B and B 4 A are coalgebra maps with gf = Id then the induced cochain maps f. and g. have g.f. = Id; so for any A-bicomodule Tand y E C;O(T, A) = T* we have g~gy = gl18 foy = gofoy = A Sy. Applying this to the case f = Id 0 q: M -* M 0 M and g = ,u we obtain the following. THEOREM 6. Hm(M, M) = Ofor m + -1 and H"(M, M)= Hbm = AmM* = AmM(n). Every class in Hbm(M, M) is represented by a cocycle (p6m(®My, 0 . 0) for a unique y EE AmMf. Also, Hgm(GL(n), GL(n)) = 0 for m 7 -1 and ftYm(GL(n), GL(n)) = AmM*. U Analogous results may hold for all reductive groups. The dual element to xij E M1 is the matrix unit eu1 E M(n) and extends to the derivation d/axj on M. We have

8cM0'(eij A epq)

=

Xrs(xrierj A xspesq - xjeir A xqseps).

[1]

The operator >2rXrierj carries Xrj to Xri for all r, annihilates all other Xpq, and is adjoint to right multiplication by eij on M(n). Denoting right and left multiplication by R and L, the right side of Eq. 1 is adjoint to R(eij A epq) - L(ej A epq), the terms of which commute. Section 4. Symmetric Powers of Matrix Algebras

The algebra M(n)0d is a USd-bimodule with o--l(ejj, 0 ... 0 0 ei, I'd and (eij, 0 ... 0 eidj,)o = ei,, 0 eidd) = ei,,,j, 0 ... 0 eJI . The minimal central idempotents Ep of CSd correspond to the partitions P of d or, equivalently, to the irreducible representations of Sd We denote the dimension of the irreducible representation corresponding to P by dp and the unit element of M(n) by I,,. THEOREM 7. Ep(I, 0 ... 0 In) is central in SdM(n) for all P and SdM(n) = LI EpSdM(n), the coproduct ranging over the partitions of d into parts c n. Also, EpSdM(n) M(fp(n)) for an integer-valued function fp(n) that, for large n, is a polynomial of degree d with leading coefficient dp/d!. M For d c 3 all fp(n) are polynomials: f(ll)(n) = n(n + 1)/2 and f(2)(n) = n(n - 1)/2; f(,,1l)(n) = n(n + 1)(n + 2)/6, f(2 l)(n) = 2(n - 1)n(n + 1)/6, and f(3)(n) = (n - 2)(n - 1)n/6. The centralizer (M(n)®2)S2M(I1) of 52M(n) inside M(n)02 is just ...

Proc. Natl. Acad. Sci. USA 87 (1990)

Mathematics: Gerstenhaber and Schack

480

the center of 52M(n) and is spanned by E(2) and E(il,) or, equivalently, I,, 0 I,, and Y(eii 0 efi - eqi 0 eji). In contrast (M(n)®3)S>M(11) is not contained in S3M(n). For large n (n 2 d may suffice) we conjecture that EpM(n)®d)Ep EpSdM(n) M(dp) and, so, that (M(n)®d)S M(-1) U M(dp). The functions fdyP(n) for large n resemble Hilbert polynomials in some =

respects.

Section

5.

Deformations and Obstructions

A deformation A, of the bialgebra (A, g, A) is a kltl-bialgebra structure on the kltl-module Altl in which the multiplication and comultiplication are given by the power series g, = , + tu + t22 + ...and A, = A + tAl+t22+ . . , where gi e Cb 2(A, A) and Ai E CH'(A, A) are extended to be k[t]-bilinear. By design, the k-bialgebra A,/t is just A itself. Collecting the coefficients of t" in the defining equations for a bialgebra, one finds, precisely as for the case of algebras (8) that (.Li, A1) E C2(A, A) is a 2-cocycle, called the infinitesimal of the deformation, and that Sb(QL2, A2) is a function of the infinitesimal, denoted Sq(,ul, A1). For any 2-cocycle, Sq(ixl, A1) is a 3-cocycle whose cohomology class in H'(A, A) is the primary obstruction to integrating (kul, A1)-i.e., finding a deformation with (gl, A1) as infinitesimal. If this obstruction vanishes then there are further obstructions, all in ft3(A, A). [We conjecture that 11(A, A) and H;(A, A) have graded Lie brackets and graded commutative cup products related to each other and to the obstruction theory as in ref. 9, where the former concept was introduced.] Two deformations are equivalent if they are isomorphic as bialgebras via a power series whose reduction modulo t is the identity. As usual, equivalent deformations have cohomologous infinitesimals and any cocycle cohomologous to an integrable one is itself integrable. THEOREM 8. Every deformation is equivalent to one in which the unit and counit are unchanged. Deformation preserves the existence ofantipodes (but may change them). A deformation of a Hopf algebra is again a Hopf algebra. THEOREM 9. If every element of 22(A, A) is cohomologous to one of the form (pi, 0) then every deformation of A is equivalent to one in which the comultiplication is unchanged. This is so, in particular, for GL(n) and M(n). When k D 0 every sp E 4j(A, A) is the infinitesimal of a formal automorphism Id + tep +t2'P2 + . . ., namely e'O, but when k is arbitrary the primary obstruction to constructing such an automorphism is the class of the 2-cocycle (tk(Gp 0 'p), ((P

)A)

Section 6. Exponential Formulas and Quantum Matrix Groups As usual, let A be

a

bialgebra. If

D: A 0 A

-*

A

A is

a

coderivation of the coalgebra A A then el"" is a formal coalgebra automorphism. Since Au = (, 0 IL),r(A 0 A), where T(x 0 y 0 z 0 w) = x 0 z 0 y 0 w, we have the following. THEOREM 10. If 4' is a coderivation of A 0 A then lit = pue'0: Alti

O&kitlAlti

Altl satisfies A/it

=

(At

J)T(A 0 A).

0

Hence At = (Altl, put, A) will be a bialgebra if and only if gt is associative, and its comultiplication will then be identical Q with that of A (extended to be kltI-bilinear). Associativity holds, in particular, when = Y