Journal of Pure and Applied North-Holland
Algebra
89 (1993)
231
231-235
The intrinsic bracket on the deformation complex of an associative algebra Jim Stasheff
For Alex Heller
on his 65th birthday
Abstruct Stasheff, J.. The intrinsic bracket on the deformation Journal of Pure and Applied Algebra 89 (1993) 231-235.
complex
of an associative
algebra,
In his pioneering work on deformation theory of associative algebras. Gcrstcnhaber created a bracket on the Hochschild cohomology Hoch(A. A). but this bracket seemed to be rather a tour de force since it was not induced from a differential graded Lie algebra structure on the underlying complex. Schlcssinger and Stasheff constructed a diffcrcntial graded Lie algebra structure on a complex giving the Harrison cohomology Harr(A, A) of a commutative alpcbra A in characteristic 0. Here WC prcscnt a differential graded Lie algebra structure on a complex giving the Hochschild cohomology Hoch(A. A) and inducing the Gcrstcnhahcr bracket for any associative algebra in any characteristic. Although the principal is the same as in the commutative case. the details as well as the essential idea will hopefully be revealed more transparently.
In his pioneering work on deformation theory of associative algebras, Gerstenhaber [6] created a bracket on the Hochschild cohomology Hoch(A, A), but this bracket seemed to be rather a tour de force since it was not induced from a differential graded Lie algebra structure on the underlying complex. In [16], Schlessinger and Stasheff constructed a differential graded Lie algebra structure on a complex giving the Harrison cohomology Harr(A, A) of a commutative algebra A in characteristic 0. Feedback from our audience has convinced me that the commutative case is more subtle than the general associative case, at least in part because of its use of the free graded Lie COalgebra. The present paper presents the corresponding results for associative algebras in the hope that the essential idea will be revealed more transparently. While this exposition was being developed, I received copies of preprints by C’orre.sporztler1c.e to: J. Stasheff. Hill, NC 27SYY-3250.
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Chapel
232
J. Stasheff
Lecomte et al. [12, 131 which emphasize and generalize the analogous results for Lie algebras, subject to conditions of finite dimensionality, thus avoiding the use of coalgebras and coderivations. Recall the original definition of Hoch(A, [2,5]. For any module V (over a commutative let T’V denote the tensor COalgebra
A) via the standard unital
(bar) construction
ring k of arbitrary
characteristic),
where VW” = V 63. . . @ V, n factors, for y1> 0 and V” = k. Let 71” : T’V-+ V@” denote the projections. We regard T’V as a graded coalgebra with the degree being the number of tensor factors and A(u,~~~~~u,,)=l~(u,~~~~~u,,)+(u,~~~~~u,,)~1 PZ_I + c (u,~...~u,,)~(u,,+,~...~u,l). ,I = I The tensor coalgebra is in fact the cofree connected graded coalgebra cogenerated by V [14], i.e. given a connected graded coalgebra C and f E Hom,(C, V), there exists a unique coalgebra map f : C + T’V such that rr, of= f. The coalgebra map 7 is given by v,, T(c) = A”c, where A” = (A @ l@. . . @ l)A”-’ is the n-fold iterated diagonal. For any homogeneous c E C,,, f(c) is well defined, in fact is a finite sum since A”c=c
l@...@c@...@l
+c
cc,,@.+@cc,,,
,
of where 0 < deg cc,) < deg c. The cofreeness can be regarded as a consequence the fact that T’ is right adjoint to the forgetful functor from connected kcoalgebras to k-modules. Now given an algebra A over k, let BA = T% where A = A as a k-module, (elements being denoted a) with a differential d = d, given (using traditional ‘bar’ notation [2]) by ,I-
d]Z,
. .
I a,,]=
c
I
(-l)‘[.
I a,a,+,( . .] .
!=I
The Hochschild given by
complex
for
Hoch(A,
A) is Hom(BA,
A) with
(6h)[G,, 1. . . 1&,,I = hd[i,, I . . . 1ii,,] + ci,,h[G, I . . . 1CT,,] + (-l)“h[a,, Our essential
observation
is the following:
1.
I ci,,_,1a,, .
coboundary
6
233
The deformation complex of an associative algebra
Proposition.
As a differential graded module, (Hom(BA,
(Coder(BA),
0).
Here graded
Coder(BA) k-linear
BA) consists
C Hom(BA,
of graded
A), 6) is isomorphic
coderivations,
to
i.e. those
maps h such that
Ah[ii, 1 . . 1 ii,,] = c
h[i,
1 . . . 1 ii,,]@[a,+,
+ (-l)“““[&,
1 . . 1 a,,]
I . . . 1i,,]@ h[G,>+, 1 . . . I a,,] ,
where Ihl = deg h is defined by h : ABk+= A@‘“-““. Ignoring the differentials for the moment, our description amounts to the dual and graded versions of two rather familiar algebraic facts: First, if k is a commutative unital ring, V is a k-module and T”V is the tensor algebra on V, then there is a canonical isomorphism Der(T”V) = Hom,(V, T”V), where Der(TV) is the space of k-linear graded derivations T”V-+ T”V. This isomorphism is given by restriction to generators and the inverse by extending a homomorphism V-+ T”V to all of T”V as a derivation. Second, the k-linear endomorphisms End,(V) form an associative algebra (under composition) and, when V itself is a k-algebra, the space of derivations Der(V) C End,(V), while not closed under composition, does form a Lie subalgebra with respect to the commutator of the composition product. (This is close to the point of view of [=I .> It is easy to check that Hom(BA, BA) is a differential graded algebra under composition with Dh = d,h k hd, and that Coder(BA) is a subcomplex. On the other hand, graded coderivations form a graded Lie algebra under graded commutator of compositions, i.e. [e, 41 : = 0 0 c$ - (- ~)““-“‘c$o 0 and D is a derivation with respect to this bracket. Thus (Coder(BA), [ , 1, D) is a differential graded Lie algebra and Hoch(A, A), being isomorphic to H(Coder(BA), D), inherits the structure of a graded Lie algebra. To complete the comparison with Gerstenhaber [6-S], we should remark that our degree is one less than Gerstenhaber’s or Hochschild’s since we measure the change in degree, e.g. IhI = n - 1 for h : A@‘”+ A. Then we need only observe that the composition bracket translates, up to sign, to the Gerstenhaber bracket on Hom(BA, A). Notice, however, that Gerstenhaber’s ‘composition’ product is defined on Hom(BA, A) whereas the composition of elements of Coder(BA) lies, in general, in End,(BA) but not in the subspace of coderivations. It is only the commutator brackets that agree. Exactly the same approach works on other deformation theories which are controlled by cochains in a complex of the form Hom( CA, A) where CA is a suitably cofree coalgebra cogenerated by A. This includes the case in which A = L is a Lie algebra and CA = A’sL is the free graded commutative coalgebra on sL, the Cartan-Chevalley-Eilenberg complex, Here the bracket is known as the Nijenhuis-Richardson bracket [15]. This is the case generalized by [12]. The
234
J. .Stu.shrf/
approach also applies to the case in which A is a commutative algebra and CA is the Harrison complex [l I], which can be regarded as the free Lie coalgebra generated by .sA. In each of these cases, the cochain complex with values in A is isomorphic
to Coder(CA)
of coderivations. The deformation
theory
and the bracket
coincides
of bialgebras
is controlled
with the commutator by Hom(BA,
bracket
L!A), where
RA is the cobar construction [l] on A considered as a coalgebra. At the time (summer 1989) I began to write the current exposition, this was not known to be isomorphic to any Codcr(CA). although the beginnings of a bracket arc evidenced in the work of Gerstenhaber and Schack [‘i. IO]. The paper of Lecomte and Roger [13], howcvcr, does provided a differential graded Lie algebra controlling the deformation theory of Lie bialgebras, explicating Drinfel’d’s complex in his ICM talk [4]. Working under suitable finiteness restrictions. their d.g. Lie algebra is of the form Hom(,l’E. ;1E) - n(E” @ E), the latter inheriting the structure of a graded Poisson algebra from the canonical symplectic form on E” @ E. The obvious extension of this approach to Hom(BA, i2A) should provide the appropriate d.g. Lie algebra controlling bialgebra deformations. Finally, Hom(BA, A), as maps of a coalgebra into an algebra, has a natural cup product:
(f-g)(x) for x E BA and to Coder(BA) Coder(BA), the the cup product
= m(f‘@&Q>
m the multiplication on A. This cup product can be transported where it is related to the commutator bracket as in [6]. In cup product looks as follows: For 8 E Coder”-‘, 4 E Coder’-‘, 0~4 E Coder”+“-’ is given by
(ed)[a,
I
I a,,1
=C*ra, l”‘I% I@%+,l”‘I~,+,Mb,+,>+,l”‘I~,+,,+qlI...I~,,l~ where
Osisn-(p+q).
The compatibility condition (Gerstenhaber’s Theorem follows: For 6, E Coder”, 4 E Coder“, $ E Coder’,
5 [6])
translates
as
Again since BA is cofree as a connected graded coalgebra. it suffices to verify this relation on A@‘+“+r+‘, but that is precisely where Gerstenhaber proves his version. Notice,however,thatheshowsseparatelythathiscompositionproductdistributesover the bracket from the right:
The cief~wmrrtior~ comph
but
that,
from the left, it distributes
of‘ m
rrssociutive
235
ulgehrtr
only up to (cochain)
homotopy
(i.e. modulo
6):
where
2%is a sum of terms (h oI f) 0, g. The component
composition-products
not make sense in terms of Hom(C, A) for a general coalgebra C, but reflect the simplicial structure of BA. Indeed, as Gerstenhaber remarks, distributivity
laws are precisely
those of u and WI as developed
by Steenrod
o1do rather these [ 1X].
References [l] J.F. Adams, On the cobar construction, in: Colloque de Topologie Algebrique Louvain (Masson, Paris, 1956) 82-87. 121H. Cartan. %minairc Hcnri CAKTAN (lY54155), ExposC 3. I31 C. Chcwlley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras. Trans. Amer. Math. Sot. 63 (lY38) 8.5~123. IdI V.G. Drinfel’d, Quantum groups, in: Proceedings ICM. Berkeley 1Y86 (1087) 7YX-870. [Sl S. Eilcnbcrg and S. Mac Lane, On the groups H(11, n).l. Ann. of Math. 58 (lY53) 55-106. The cohomology structure of an associative ring, Ann. of Math. 78 (1963) 161M. Gcrstenhaber. X17-288. On the deformation structure of rings and algebras, Ann. of Math. 7Y (1961) [71 M. Gcratcnhaber. 5Y- IO.?. On the deformation structure of rings and algebras. III. Ann. of Math. 88 1x1 M. Gerstenhabcr. (lY68) l-31. and S.D. Schack, Bialgchra cohomology. deformations. and quantum groups, [‘)I M. Gcrstcnhaber Proc. Nat. Acad. Sci. U.S.A. 87 (1YYO) 478-381. and S.D. Schack. Foundations of deformation theory of algebras in gcncral and IlO1 M. Gerstenhaber bialgehras in particular, in: Proceedings of the Conference on Deformation Theory and Ouantization with Application to Physics. Amherst, June IYYO; also: Contemp. Math.. to appear. 1111D.K. Harrison. Commutative algebra and cohomology. Trans. Amer. Math. Sot. 101 (lY62) I Yl-103. [I21 P.A.B. Lecomte. P.W. Michor and H. Schicketanz. The multigraded Nijenhuis-Richardson algebra, its universal property and applications. J. Pure Appl. Algebra 77 (I) (lYY2) 87-102. dcs big&es de Lie, C.R. Acad. Sci. 1131 P.A.B. Lecomtc and C. Roger. Modules et cohomologies Paris Sir. 1 Math. 310 (1990) 305-410. Lit coalgebras. Adv. in Math. 3X (1980) l-.54. [ldl W. Michaelis, (151 A. Nijcnhuis and R.W. Richardson. Deformations of Lie algebra structures. J. Math. Mech. 17 (lY67) XY-10.5. 1161M. Schlcssinger and J.D. Stasheff, The Lit algebra structure of tangent cohomology and deformation theory, J. Pure Appl. Algebra 38 (19%) 313-322. theory and rational homotopy type. Publ. Math. [I71 M. Schlessinger and J.D. Stasheff, Deformation IHES. to appear. Products of cocycles and extcnaions of mappings, Ann. of Math. 48 (1947) [1X1 N.E. Steenrod. 2YO~320.
