Journal of Pure and Applied Algebra 77 (1992), 87–102 North-Holland THE MULTIGRADED NIJENHUIS-RICHARDSON ALGEBRA, 2 9 ITS UNIVERSAL PROPERTY AND APPLICATIONS 9 1 n a P.A.B. Lecomte J P.W. Michor 1 H. Schicketanz ] A Institut fu¨r Mathematik der Universit¨at Wien, Austria Q Institut des Math´ematique, Universit´e de Li`ege, Belgium . h t February 22, 1991 a m [ Abstract. We define two (n+1) graded Lie brackets on spaces of 1 multilinear mappings. The first one is able to recognize n-graded as- v sociative algebras and their modules and gives immediately the correct 7 differential for Hochschild cohomology. The second one recognizes n- 5 graded Lie algebra structures and their modules and gives rise to the 2 notion of Chevalley cohomology. 1 0 2 9 / h t a 1. Introduction m In this paper we will generalize the construction of Nijenhuis and : v Richardson which associates to a given vector space V a graded Lie i X algebra Alt(V) of multilinear alternating mappings V ×...×V → V to r a study Lie algebra structures on V and their deformations, see [9]. Their construction suggests a ”principle” which we present here as the starting point for our investigations. The principle is as follows: Suppose that S is a type of structures on V, defining for example associative algebras, Lie algebras, modules (over a given Algebra A) or Lie bialgebras on V. Then there exists a Z - graded Lie algebra (E = Ek,[ , ]) such that P ∈ S if and only if P ∈ E1 such that k∈Z [P,P] = 0. L In the case where S is the set of Lie algebra structures on V the space E can be identified with Alt(V). Moreover if V is equipped with such 1991 Mathematics Subject Classification. 17B70. Key words and phrases. Nijenhuis-Richardsonbracket, multigradedalgebras, de- formation theory, graded cohomology. Typeset by AMS-TEX 1 2 LECOMTE, MICHOR, SCHICKETANZ a P, the Chevalley-Eilenberg coboundary operator ∂ of the adjoint P representation of (V,P) is just the adjoint action of P on Alt(V) up to a sign. Another application may be found in [6]. There one uses the graded cohomology of the subalgebras of Alt(V) to classify and to construct formal deformations of (V,P). The purpose of this paper is to establish the principle in each of the cited cases. We will do this in more generality which makes the construction even more powerful. Namely, we assume that V is itself graded over Zn, (n = 0,1,2,...) and we will define for each S a graded Lie algebra E which is now graded over Zn+1 and satisfies the principle. If we don’t stress the special choice of n we will speak of multigraded algebras. Having defined the multigraded Lie algebra E, deformation theory and cohomology of S may be treated at the same time using only the space E and its properties. Given a multigraded vector space V we will construct first M(V), a multigraded Lie algebra which is adapted to study the associative structuresonV. UsingthenthemultigradedalternatorαwedefineA(V) to be the image of M(V) by α equipped with the unique bracket making αahomomorphismofmultigradedLiealgebras. MoreoverA(V)satisfies a universal property and describes multigraded Lie algebra structures on V. We call A(V) the multigraded Nijenhuis-Richardson algebra of V since it coincides with Alt(V) for n = 0. Once having established this multigraded version, the result for module structures follows quite easily. In this way we rediscover Hochschild and Chevalley-Eilenberg Coho- mology for n ≤ 1, where the differential is given by the adjoint action of P on E. Their generalizations for n > 1 are now obvious and yield a canonical description for multigraded cohomology in both cases. Moreover one can study now the theory of formal deformations of multigraded algebras L and their modules. Roughly speaking we de- scribe a mapping from the cohomology of the adjoint representation of A(L) into the set of formal deformations of all possible structures on L which may be used to construct and classify these deformations. Such a point of view has also been emphasized by [11], [5], and [4]. 2.Multigraded associative algebra structures 2.1. Conventions and definitions.. By a multidegree we mean an element x = (x1,... ,xn) ∈ Zn for some n. We call it also n-degree if we want to stress the special choice of n. We shall need also the inner product of multidegrees h , i : Zn ×Zn → Z, given by hx,yi = n xiyi. i=1 An n-graded vector space is just a direct sum V = Vx, where P x∈Zn the elements of Vx are said to be homogeneous of multidegree x. To L avoid technical problems we assume that vector spaces are defined over afieldKofcharacteristic0. InthefollowingX, Y, etcwillalwaysdenote MULTIGRADED NIJENHUIS RICHARDSON ALGEBRA 3 homogeneous elements of some multigraded vector space of multidegrees x, y, etc. By an n-graded algebra A = Ax we mean an n-graded vector x∈Zn space which is also a K algebra such that Ax ·Ay ⊆ Ax+y. L (1) The multigraded algebra (A,·) is said to be multigraded commu- tative if for homogeneous elements X, Y ∈ A of multidegree x, y, respectively,we have X ·Y = (−1)hx,yiY ·X. (2) If X · Y = −(−1)hx,yiY · X holds it is called multigraded anti- commutative. (3) An n-graded Lie algebra is a multigraded anticommutative alge- bra (E,[ , ]), such that the multigraded Jacobi identity holds: [X,[Y,Z]]= [[X,Y],Z]+(−1)hx,yi[Y,[X,Z]] Obviously the space End(V) = Endδ(V) of all endomorphisms δ∈Zn of a multigraded vector space V is a multigraded algebra under compo- sition, where Endδ(V) is the spaLce of linear endomorphisms D of V of multidegree δ, i.e. D(Vx) ⊆ Vx+δ. Clearly End(V) is a multigraded Lie algebra under the multigraded commutator (4) [D ,D ] := D ◦D +(−1)hδ1,δ2iD ◦D . 1 2 1 2 2 1 If A is an n-graded algebra, an endomorphism D : A → A of multi- degree δ is called a multigraded derivation, if for X, Y ∈ A we have (5) D(X ·Y) = D(X)·Y +(−1)hδ,xiX ·D(Y). Let us write Derδ(A) for the space of all multigraded derivations of degree δ of the algebra A, and we put (5) Der(A) = Derδ(A). δ∈Zn M The following lemma is standard: Lemma. If A is an n-graded algebra, then the space Der(A) of multi- graded derivations is an n-graded Lie subalgebra under the n-graded com- mutator. It is clear from the definitions that non-graded algebras and Z-graded algebras are multigraded of multidegree 0 and 1, respectively. 2.2 Associative algebra structures. Let us recall first the construc- tion in the case of non-graded vector spaces which was given in [3], [1]. There a 1-graded Lie algebra (M(V),[ , ]∆) is described for each vec- tor space V with the property that (V,µ) is an associative algebra if and only if µ ∈ M1(V) and [µ,µ]∆ = 0. This algebra is as follows. 4 LECOMTE, MICHOR, SCHICKETANZ Denote by Mk(V) the space of all k+1-linear mappings K : V ×...× V → V and set M(V) := Mk(V). k∈Z M For K ∈ Mki(V) and X ∈ V we define j(K )K ∈ Mk1+k2(M) by i j 1 2 (j(K )K )(X ,... ,X ) := 1 2 0 k1+k2 k2 = (−1)k1iK (X ,...,K (X ,... ,X ),...,X ). 2 0 1 i i+k1 k1+k2 i=0 X The graded Lie bracket of M(V) is then given by [K ,K ]∆ = j(K )K −(−1)k1k2j(K )K . 1 2 1 2 2 1 Proposition. ([3], [1]) (1) (M(V),[ , ]∆) is a 1-graded Lie algebra. (2) If µ ∈ M1(V), so µ : V ×V → V is bilinear, then (V,µ) is an associative algebra if and only if [µ,µ]∆ = 0. (cid:3) Note that M0(V) = End(V) is a Lie subalgebra of M(V), and its bracket is the negative of the usual commutator. The explicit formulas above follow directly from investigating the 1- graded Lie algebra of (1-graded) derivations of certain graded algebras, see [11]. We explain that in the simple case of a finite dimensional V. Then M(V) is canonically isomorphic to the 1-graded Lie algebra Der( V∗) of derivations of the tensor algebra of V∗, a derivation D of degree k being completely determined by its restriction V∗ → k+1V∗ N and hence by a unique K ∈ Mk(V). N 2.3 Multigraded associative algebras. We will give now the multi- graded generalization. Of course on can proceed as before by identifying M(V) as the algebra of derivations of some suitable multigraded alge- bra. But we will generalize 2.2 directly. So let V = Vx be an x∈Zn n-graded vector space. We define L M(V) := M(k,κ)(V), (k,κ)∈Z×Zn M where M(k,κ)(V) is the space of all k+1-linear mappings K : V ×...× V → V such that K(Vx0 × ...× Vxk) ⊆ Vx0+···+xk+κ. We call k the form degree and κ the weight degree of K. In 2.2 the mapping K had degree k and X had degree −1 in M(V), hence the sign (−1)ki. We i define for K ∈ M(ki,κi)(V) and X ∈ Vxj i j (j(K )K )(X ,... ,X ) := 1 2 0 k1+k2 k2 = (−1)k1i+hκ1,κ2+x0+···+xi−1i· i=0 X ·K (X ,...,K (X ,... ,X ),...,X ) 2 0 1 i i+k1 k1+k2 [K ,K ]∆ = j(K )K −(−1)k1k2+hκ1,κ2ij(K )K . 1 2 1 2 2 1 MULTIGRADED NIJENHUIS RICHARDSON ALGEBRA 5 Theorem. Let V be an n-graded vector space. Then we have: (1) (M(V),[ , ]∆) is an (n+1)-graded Lie algebra. (2) If µ ∈ M(1,0,...,0)(V), so µ : V × V → V is bilinear of weight 0 ∈ Zn, then µ is an associative n-graded multiplication if and only if [µ,µ]∆ = 0. Proof. The bracket is (n + 1)-graded anticommutative. The (n + 1)- graded Jacobi identity follows from the formula j([K ,K ]∆) = [j(K ),j(K )], 1 2 1 2 the multigraded commutator in End(M(V)). This is a long but ele- mentary calculation. The second assertion follows by writing out the definitions. (cid:3) 3. Multigraded Lie Algebra Structures 3.1. Multigraded signs of permutations. Let x = (x ,... ,x ) ∈ 1 k (Zn)k be a multi index of n-degrees x = (x1,...,xn) ∈ Zn and let i i i σ ∈ S be a permutation of k symbols. Then we define the multigraded k sign sign(σ,x) as follows: For a transposition σ = (i,i + 1) we put sign(σ,x) = −(−1)hxi,xi+1i; it can be checked by combinatorics that this gives a well defined mapping sign( ,x) : S → {−1,+1}. In fact one k may define directly sign(σ,x) = sign(σ)sign(σ )···sign(σ ), |x1|,...,|x1| |xn|,...,|xn| 1 k 1 k j j where σ is that permutation of |x |+···+|x | symbols which |xj|,...,|xj| 1 k 1 k moves the i-th block of length |xi| to the position σi, and where sign(σ) j denotes the ordinary sign of a permutation in S . Let us write σx = k (x ,... ,x ), then we have the following σ1 σk Lemma. sign(σ ◦τ,x) = sign(σ,x).sign(τ,σx). (cid:3) 3.2 Multigraded Nijenhuis-Richardson algebra. We define the multigraded alternator α : M(V) → M(V) by 1 (1) (αK)(X ,...,X ) = sign(σ,x)K(X ,...,X ) 0 k σ0 σk (k +1)! σ∈XSk+1 for K ∈ M(k,∗)(V) and X ∈ Vxi. If the ground field is not of charac- i teristic 0 one could omit the combinatorial factor, but one should redo the whole developpment starting from the point of view of derivations again, see the remark at the end of 2.2. However, the combinatorial factors used here are quite essential, judging from our experience in dif- ferential geometry. By lemma 3.1 we have α2 = α so α is a projection 6 LECOMTE, MICHOR, SCHICKETANZ defined on M(V), homogeneous of multidegree 0, and we set A(V) = A(k,κ)(V) (k,κ)∈Z×Zn M : = α(M(k,κ)(V)). (k,κ)∈Z×Zn M AlongbutstraightforwardcomputationshowsthatforK ∈ M(ki,κi)(V) i α(j(αK )αK ) = α(j(K )K ), 1 2 1 2 so the following operator and bracket is well defined: (k +k +1)! 1 2 i(K )K : = α(j(K )K ) 1 2 1 2 (k +1)!(k +1)! 1 2 (k +k +1)! [K ,K ]∧ = 1 2 α([K ,K ]∆) 1 2 1 2 (k +1)!(k +1)! 1 2 = i(K )K −(−1)h(k1κ1),(k2,κ2)ii(K )K 1 2 2 1 The combinatorial factor will become clear in 3.4 . 3.3. Theorem. 1. If K are as above then i (i(K )K )(X ,... ,X ) = 1 2 0 k1+k2 1 = sign(σ,x)(−1)hκ1,κ2i· (k +1)!k ! 1 2 σ∈SXk1+k2+1 ·K ((K (X ,...,X ),...,X ). 2 1 σ0 σk1 σ(k1+k2) 2. (A(V),[ , ]∧) is an (n+1)-graded Lie algebra. 3. If µ ∈ A(1,0,...,0)(V), so µ : V × V → V is bilinear n-graded anticommutative mapping of weight 0 ∈ Zn then [µ,µ]∧ = 0 if and only if (V,µ) is a n-graded Lie algebra. Proof. 1. This follows by a straight forward computation. 2. [ , ]∧ is clearly multigraded anticommutative and the multi- graded Jacobi identity follows directly from the one of [ , ]∆. 3. Let µ ∈ A(1,0,...,0)(V), then 0 = [µ,µ]∧(X ,X ,X ) 0 1 2 3! 1 = sign(σ,x)·[µ,µ]∆(X ,X ,X ) σ0 σ1 σ2 2!3!3! σX∈S3 = sign(σ,x)·µ(µ(X ,µ(X ,X )) σ0 σ1 σ2 σX∈S3 which is equivalent to the multigraded Jacobi identity of (V,µ). (cid:3) We call (A(V),[ , ]∧) the multigraded Nijenhuis-Richardson alge- bra, since A(V) coincides for n = 0 with Alt(V) of [9]. MULTIGRADED NIJENHUIS RICHARDSON ALGEBRA 7 3.4. Universality of the algebra (A(V),[ , ]∧). Let V be a multigraded vector space and denote by E(V) the category of multi- graded Lie algebras (E,[ , ]) such that E(k,∗) = 0 k < −1 E(−1,∗) = V. If E,F ∈ E(V), then a morphism ϕ : E → F is a homomorphism of multigraded Lie algebras satisfying ϕ|E(−1,∗) = id . For example V M(V) and A(V) are elements of E(V). Theorem. A(V) is a final object in E(V), so for each E ∈ E(V) there exists a unique morphism ε : E → A(V). It follows that A(V) is unique up to isomorphism. Proof. Suppose that Z ∈ E(k,z) then we define ε(Z)(X ,... ,X ) = (−1)hz,x0+···+xki[X ,[X ,...,[X ,Z]...], 0 k 0 1 k an element of E(−1,∗) = V for X ∈ Vxi. Because of the multigraded i Jacobi identity ε(Z) is well defined as an element of A(k,z). So we are left to show that (*) ε([Z ,Z ]) = [ε(Z ),ε(Z )]∧ 1 2 1 2 We will do this by induction on k = k +k . For k < −1 this is trivially 1 2 true. Now let k = −1, so we may assume that Z ∈ Vz1. Then 1 ε([Z ,Z ] = [Z ,Z ] = (−1)hz1,z2iε(Z )(Z ) 1 2 1 2 2 1 = i(Z )ε(Z ) = [Z ,ε(Z )]∧ = [ε(Z ),ε(Z )]∧ 1 2 1 2 1 2 by Theorem 3.2 and since ε|V = id . Suppose that (*) is true for V k +k < k. Then for k +k = k we have 1 2 1 2 i(X)ε([Z ,Z ]) = [X,ε([Z ,Z ])]∧ = ε([X,[Z ,Z ]]) 1 2 1 2 1 2 = ε [[X,Z ],Z ]+(−1)h(−1,x),(k1,z1)i[Z ,[X,Z ]] 1 2 1 2 = [ε(cid:0)([X,Z1]),ε(Z2)]∧ +(−1)h(−1,x),(k1,z1)i[ε(Z1),(cid:1)ε([X,Z2])]∧ = [i(X)ε(Z ),ε(Z )]∧ +(−1)h(−1,x),(k1,z1)i[ε(Z ),i(X)ε(Z )]∧ 1 2 1 2 = i(X)[ε(Z ),Z ]∧ 1 2 by induction hypothesis and the fact that i(X) = [X, ]∧ is a derivation of degree (−1,x) of A(V). This proves the induction. Remark that for E = M(V) the morphism ε is given by ε|Mk,∗(V) = (k +1)! α. (cid:3) 8 LECOMTE, MICHOR, SCHICKETANZ 4. Multigraded Modules and Cohomology 4.1. Multigraded bimodules. Let V and W be multigraded vector spaces and µ : V × V → V a multigraded algebra structure. A multi- graded bimodule M = (W,λ,ρ) over A = (V,µ) is given by λ,ρ : V → End(W) of weight 0 such that (1) [µ,µ]∆ = 0 so A is associative (2) λ(µ(X ,X )) = λ(X )◦λ(X ) 1 2 1 2 (3) ρ(µ(X ,X )) = (−1)hx1,x2iρ(X )◦ρ(X ) 1 2 2 1 (4) λ(X )◦ρ(X ) = (−1)hx1,x2iρ(X )◦λ(X ) 1 2 2 1 where X ∈ Vxi and ◦ denotes the composition in End(W). i 4.2. Theorem. Let E be the multigraded vector space defined by V if k = 0 E(k,∗) = W if k = 1  0 otherwise. Then P ∈ M(1,0,...,0)(E) defines a bimodule structure on W if and only  if [P,P]∆ = 0. Proof. We define µ(X ,X ) := P(X ,X ) 1 2 1 2 λ(X)Y := P(X,Y) ρ(X)Y := (−1)hx,yiP(Y,X) where we suppose the X ’s ∈ V and Y ∈ W to be embedded in E. Then i if Z ∈ E be arbitrary we get i [P,P]∆(Z ,Z ,Z ) = 2(j(P)P)(Z ,Z ,Z ) 0 1 2 0 1 2 = 2P((Z ,Z ),Z )−2P(Z ,(Z ,Z )). 0 1 2 0 1 2 Now specify Z ∈ V resp. W to get eight independent equations. Four i of them vanish identically because of their degree of homogeneity, the others recover the defining equations for the multigraded bimodules. (cid:3) 4.3 Corollary. In the above situation we have the following decompo- sition of M(E) : 0 for q > 1 L(k+1,∗)(V,W) for q = 1 M(k,q,∗)(E) =    k+1  M(k,∗)(V)⊕ (L(k,∗)(V,End(W)) for q = 0  where L(k,∗)(V,W)denotes the spaMce of k-linear mappings V ×...×V → W. If P is as above, then P = µ + λ + ρ corresponds exactly to this decomposition. (cid:3) MULTIGRADED NIJENHUIS RICHARDSON ALGEBRA 9 4.4. Hochschild cohomology and multiplicative structures. Let V,W and P be as in Theorem 4.2 and let ν : W × W → W be a multigraded algebra structure, so ν ∈ M(1,−1,0,...,0)(E). Then for C ∈ i L(ki,ci)(V,W) we define C •C := [C ,[C ,ν]∆]∆. 1 2 1 2 Since [C ,D ]∆ = 0 it follows that (L(V,W),•) is multigraded commu- 1 2 tative. It is the usual extension of the product ν from W to the level of cochains, where the necessary combinatorics is hidden in the brackets. Theorem. 1. The mapping [P, ]∆ : M(E) → M(E) is a differ- ential. We denote its restriction to L(V,W) by δ . This generalizes P the Hochschild coboundary operator to the multigraded case: If C ∈ L(k,c)(V,W) then we have for X ∈ Vxi i (δ C)(X ,... ,X ) = λ(X )C(X ,... ,X ) P 0 k 0 1 k k−1 − (−1)iC(X ,...,µ(X ,X ),...,X ) 0 i i+1 k i=0 X +(−1)k+1+hx0+···+xk−1+c,xkiρ(X )C(X ,... ,X ) k 0 k−1 The corresponding cohomology will be denoted by H(A,M), where A is the multigraded associative algebra (V,µ), and where M is the multi- graded A-bimodule (W,λ,ρ) 2. If [P,ν]∆ = 0 then δ is a derivation of L(V,W) of multidegree P (1,0,...0). In this case the product • carries over to a multigraded (cup) product on H(A,M). Proof. The fact that δ is a differential follows directly from the multi- P graded Jacobi identity since the degree ofδ is(1,0,...,0). The formula P is easily checked by writing out the definitions. Applying the multi- graded Jacobi identity once again one gets immediately that δ is a P derivation if and only if [P,ν]∆ = 0. Bywriting out the definitions one shows that [P,ν]∆ = 0 isequivalent to the following equations: λ(X)ν(Y ,Y ) = ν(λ(X)Y ,Y )) 1 2 1 2 ρ(X)ν(Y ,Y ) = (−1)hx,y1iν(Y ,ρ(X)Y ) 1 2 1 2 ν(ρ(X)Y ,Y ) = (−1)hx,y1iν(Y ,λ(X)Y ) 1 2 1 2 in particular we have (λ−ρ) : V → Der(W,ν). (cid:3) 4.5 Multigraded Lie modules and Chevalley cohomology. We obtain a corresponding result for Lie modules by applying the multi- graded alternator α to M(E), just as we did in section 3 to obtain the Nijenhuis-Richardson bracket. 10 LECOMTE, MICHOR, SCHICKETANZ Theorem. Let P ∈ A(1,0,...,0)(E) then [P,P]∧ = 0 if and only if (a) [µ,µ]∧ = 0 so (V,µ) = g is a multigraded Lie algebra, and (b) π(µ(X ,X ))Y = [π(X ),π(X )]Y 1 2 1 2 where µ(X ,X ) = P(X ,X ) ∈ V and π(X)Y = P(X,Y) ∈ W for 1 2 1 2 X, X ∈ V and Y ∈ W, and where [ , ] denotes the multigraded i commutator in End(W). So [P,P]∧ = 0 is by definition equivalent to the fact that M := (W,π) is a multigraded Lie-g module. If P is as above the mapping ∂ := [P, ]∧ : A(E) → A(E) is a P differential and its restriction to Λ(k,∗)(g,M) := A(k,1,∗)(E) k∈Z k∈Z M M generalizes the Chevalley-Eilenberg coboundary operator to the multi- graded case: k (∂ C)(X ,... ,X ) = (−1)αi(x)+hxi,ciπ(X )C(X ,...,X ,...,X ) P 0 k i 0 i k i=0 X + (−1)αij(x)C(µ(X ,X ),...,X ,..c.,X ,...) i j i j i<j X where c c α (x) = hx ,x +···+x i+i i i 1 i−1 α (x) = α (x)+α (x)+hx ,x i ( ij i i i j We denote the corresponding cohomology space by H(g,M). If ν : W × W → W is multigraded symmetric (so ν ∈ A(1,−1,∗)(E)) and [P,ν]∧ = 0 then ∂ acts as derivation of multidegree (1,0,...,0) on P the multigraded commutative algebra (Λ(g,M),•), where C •C := [C ,[C ,ν]∧]∧ C ∈ Λ(ki,ci)(g,M). 1 2 1 2 i In this situation the product • carries over to a multigraded symmetric (cup) product on H(g,M). Proof. Apply the multigraded alternator α to the results of 4.1, 4.2, 4.3, and 4.4. (cid:3) The formulas we obtained here are not that surprising since they are standard in the non-graded case. The new feature of our approach lies in the fact that we can formulate deformation equations and cohomol- ogy at once inside a multigraded Lie algebra (which we denoted M(E), A(V) respectively). Then all the ”different” results we obtained are con- sequences of only ”one” fact, namely the multigraded Jacobi identity. In the line of [11] it seems to us that this procedure should be somehow extended to other structures defined on a (multigraded) vector space, for example coalgebras, comodules and then of course to bialgebras such as Hopf algebras and Lie bialgebras. The latter one was discussed in [7].