The Erwin Schrodinger International Pasteurgasse 6/7. Institute for Mathematical Physics A-1090 Wien, Austria

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1 ESI The Erwin Schrodinger International Pasteurgasse 6/7 Institute for Mathematical Physics A-1090 Wien, Austria n{ary Lie and Associative Algebras Peter W. Michor Alexandre M. Vinogradov Vienna, Preprint ESI 402 (1996) December 5, 1996 Supported by Federal Ministry of Science and Research, Austria Available via

2 n-ary LIE AND ASSOCIATIVE ALGEBRAS Peter W. Michor Alexandre M. Vinogradov Erwin Schrodinger International Institute of Mathematical Physics, Wien, Austria Universita di Salerno, Italy November 26, 1996 To Wlodek Tulczyjew, on the occasion of his 65th birthday. Abstract. With the help of the multigraded Nijenhuis{ Richardson bracket and the multigraded Gerstenhaber bracket from [7] for every n 2 we dene n-ary associative algebras and their modules and also n-ary Lie algebras and their modules, and we give the relevant formulas for Hochschild and Chevalley cohomogy. Table of contents 1. Introduction Review of binary algebras and bimodules n-ary G-graded associative algebras and n-ary modules Review of G-graded Lie algebras and modules n-ary G-graded Lie algebras and their modules Relations between n-ary algebras and Lie algebras Hochschild operations and non commutative dierential calculus Remarks on Filipov's n-ary Lie algebras Dynamical aspects Introduction In 1985 V. Filipov [3] proposed a generalization of the concept of a Lie algebra by replacing the binary operation by an n-ary one. He dened an n-ary Lie algebra structure on a vector space V as an operation which associates with each n-tuple (u 1 ; : : :; u n ) of elements in V another element [u 1 ; : : :; u n ] which is n-linear, skew symmetric, and satises the n-jacobi identity: (1) [u 1 ; : : :; u n?1 ; [v 1 ; : : :; v n ]] = X [v 1 ; : : :; v i?1 [u 1 ; : : :; u n?1 ; v i ]; : : :; v n ]: 1991 Mathematics Subject Classication. 08A62, 16B99, 17B99. Key words and phrases. n-ary associative algebras, n-ary Lie algebras. Supported by Project P PHY of `Fonds zur Forderung der wissenschaftlichen Forschung' Typeset by AMS-TEX

3 2 Michor, Vinogradov Apparently Filippov was motivated by the fact that with this denition one can delelop a meaningful structure theory, in accordance with the aim of Malcev's school: To look for algebraic structures that manifest good properties. On the other hand, in 1973 Y. Nambu [13] proposed an n-ary generalization of Hamiltonian dynamics by means of the n-ary `Poisson bracket' (2) ff 1 ; : : :; f n g j : Apparently he looked for a simple model which explains the unseparability of quarks. Much later, in the early 90's, it was noticed by M. Flato, C. Fronsdal, and others, that the n-bracket (2) satises (1). On this basis L. Takhtajan [17] developed sytematically the foundations of of the theory of n-poisson or Nambu-Poisson manifolds. It seems that the work of Filippov was unknown then; in particular Takhtajan reproduces some results from [3] without refereing to it. Recently Alekseevsky and Guha [2] and later Marmo, Vilasi, and Vinogradov [9] proved that n-poisson structures of the kind above are extremely rigid: Locally they are given by n commuting vector elds of rank n, if n > 2; in other words, n-poisson structures are locally given by (2). This rigidity suggests that one should look for alternative n-ary analogs of the concept of a Lie algebra. One of them is proposed below in this paper. It is based on the completely skew symmetrized version of Filippov's Jacobi identity (2). It is shown in [20] that this approach leads to richer and more diverse structures which seem to be more useful for purposes of dynamics. In fact, we were lead in to the constructions of this paper by some expectations about n-body mechanics and the naturality of the machinary developed in [7]. So, our motives were quite dierent from that by Filippov, Nambu and Takhtajian. This paper is essentailly based on our unpublished notes from In view of the recent developments we decided to publish them now. In this paper we consider G- graded n-ary generalizations of the concept of associative algebras, of Lie algebras, their modules, and their cohomologies; all this is produced by the algebraic machinery of [7]. Related (but not graded) concepts are discussed in [4] in terms of operads and their Koszul duality. The recent preprints [1] and [5] propose dynamical models which correspond to the not graded case with even n in our construction. 2. Review of binary algebras and bimodules In this section we review the results from the paper [7] in a slightly dierent point of view Conventions and denitions.. By a grading group we mean a commutative group (G; +) together with a Z-bilinear symmetric mapping (bicharacter) h ; i : G G! Z 2 := Z=2Z. Elements of G will be called degrees, or G-degrees if more precision is necessary. A standard example of a grading group is Z m with hx; yi = P m i=1 xi y i ( mod 2). If G is a grading group we will consider the grading group ZG with h(k; x); (l; y)i = kl( mod 2) + hx; yi. A G-graded vector space is just a direct sum V = L x2g V x, where the elements of V x are said to be homogeneous of G-degree x. We assume that vector spaces are

4 n-ary Lie and associative algebras 3 dened over a eld K of characteristic 0. In the following X, Y, etc will always denote homogeneous elements of some G-graded vector space of G-degrees x, y, etc. By an G-graded algebra A = L x2g Ax we mean an G-graded vector space which is also a K algebra such that A x A y A x+y. (1) The G-graded algebra (A; ) is said to be G-graded commutative if for homogeneous elements X, Y 2 A of G-degree x, y, respectively,we have X Y = (?1) hx;yi Y X. (2) If X Y =?(?1) hx;yi Y X holds it is called G-graded anticommutative. (3) By an G-graded Lie algebra we mean a G-graded anticommutative algebra (E; [ ; ]) for which the G-graded Jacobi identity holds: [X; [Y; Z]] = [[X; Y ]; Z] + (?1) hx;yi [Y; [X; Z]] Obviously the space End(V ) = L 2G End (V ) of all endomorphisms of a G-graded vector space V is a G-graded algebra under composition, where End (V ) is the space of linear endomorphisms D of V of G-degree, i.e. D(V x ) V x+. Clearly End(V ) is a G-graded Lie algebra under the G-graded commutator (4) [D 1 ; D 2 ] := D 1 D 2? (?1) h1;2i D 2 D 1 : If A is a G-graded algebra, an endomorphism D : A! A of G-degree is called a G-graded derivation, if for X, Y 2 A we have (5) D(X Y ) = D(X) Y + (?1) h;xi X D(Y ): Let us write Der (A) for the space of all G-graded derivations of degree of the algebra A, and we put (5) Der(A) = M 2G Der (A): The following lemma is standard: Lemma. If A is an G-graded algebra, then the space Der(A) of G-graded derivations is an G-graded Lie algebra under the G-graded commutator. L 2.2 Graded associative algebras. Let V = x2g V x be an G-graded vector space. We dene M M(V ) := M (k;) (V ); (k;)2zg where M (k;) (V ) is the space of all k + 1-linear mappings K : V : : : V! V such that K(V x0 : : : V xk ) V x0++xk+. We call k the form degree and the weight degree of K. We dene for K i 2 M (ki;i) (V ) and X j 2 V xj = Xk 2 i=0 (j(k 1 )K 2 )(X 0 ; : : : ; X k1+k 2 ) := (?1) k1i+h1;2+x0++xi?1i K 2 (X 0 ; : : :; K 1 (X i ; : : : ; X i+k1 ); : : :; X k1+k 2 ); [K 1 ; K 2 ] = j(k 1 )K 2? (?1) k1k2+h1 ;2i j(k 2 )K 1 :

5 4 Michor, Vinogradov Theorem. Let V be an G-graded vector space. Then we have: (1) (M(V ); [ ; ] ) is a (Z G)-graded Lie algebra. (2) If 2 M (1;0) (V ), so : V V! V is bilinear of weight 0 2 G, then is an associative G-graded multiplication if and only if j() = 0. (3) If 2 M (1;n) (V ), so : V V! V is bilinear of weight n 2 G, then j() = 0 is equivalent to ((X 0 ; X 1 ); X 2 )? (?1) hn;ni (X 0 ; (X 1 ; X 2 )) = 0 which is the natural notion of an associative multiplication of weigth n 2 G. Proof. The rst assertion is from [7]. writing out the denitions. The second and third assertion follows by In [7] the formulation was as follows: 2 M (1;0) (V ) is an associative G-graded algebra structure if and only if [; ] = 2j() = 0. For 2 M (1;n) (V ) we have [; ] = (1 + (?1) hn;ni )j() Multigraded bimodules. Let V and W be G-graded vector spaces and : V V! V a G-graded algebra structure. A G-graded bimodule M = (W; ; ) over A = (V; ) is given by ; : V! End(W ) of weight 0 such that (1) (2) (3) (4) j() = 0 so A is associative ((X 1 ; X 2 )) = (X 1 ) (X 2 ) ((X 1 ; X 2 )) = (?1) hx1;x2i (X 2 ) (X 1 ) (X 1 ) (X 2 ) = (?1) hx1;x2i (X 2 ) (X 1 ) where X i 2 V xi and denotes the composition in End(W ) Theorem. Let E be the (Z G)-graded vector space dened by E (k;) = 8 >< >: V if k = 0 W if k = 1 0 otherwise. Then P 2 M (1;0) (E) denes a bimodule structure on W if and only if j(p )P = 0. Proof. We dene (X 1 ; X 2 ) := P (X 1 ; X 2 ) (X)Y := P (X; Y ) (X)Y := (?1) hx;yi P (Y; X) where we suppose the X i 's 2 V and Y 2 W to be embedded in E. Then if Z i 2 E be arbitrary we get (j(p )P )(Z 0 ; Z 1 ; Z 2 ) = P ((Z 0 ; Z 1 ); Z 2 )? P (Z 0 ; (Z 1 ; Z 2 )): Now specify Z i 2 V resp. W to get eight independent equations. Four of them vanish identically because of their degree of homogeneity, the others recover the dening equations for the G-graded bimodules.

6 n-ary Lie and associative algebras Corollary. In the above situation we have the following decomposition of the (Z 2 G)-graded space M(E) : M (k;q;) (E) = 8 >< >: 0 for q > 1 L (k+1;) (V; W ) for q = 1 M k+1 M (k;) (V ) (L (k;) (V; End(W )) for q = 0 where L (k;) (V; W ) denotes the space of k-linear mappings V : : : V! W. If P is as above, then P = + + corresponds exactly to this decomposition Hochschild cohomology and multiplicative structures. Let V,W and P be as in Theorem 2.4 and let : W W! W be a G-graded algebra structure, so 2 M (1;?1;0) (E). Then for C i 2 L (ki;ci) (V; W ) we dene C 1 C 2 := [C 1 ; [C 2 ; ] ] = (C 1 ; C 2 ): Since [C 1 ; C 2 ] = 0 it follows that (L(V; W ); ) is (Z G)-graded commutative. Theorem. 1. The mapping [P; ] : M(E)! M(E) is a dierential. Its restriction P to L(V; W ) is a generalization of the Hochschild coboundary operator to the G-graded case: If C 2 L (k;c) (V; W ), then we have for X i 2 V xi ( P C)(X 0 ; : : : ; X k ) = (X 0 )C(X 1 ; : : : ; X k ) X k?1? i=0 (?1) i C(X 0 ; : : :; (X i ; X i+1 ); : : :; X k ) + (?1) k+1+hx0++xk?1+c;xki (X k )C(X 0 ; : : : ; X k?1 ) The corresponding (Z G)-graded cohomology will be denoted by H(A; M). 2. If [P; ] = 0, then P is a derivation of L(V; W ) of (Z G)-degree (1; 0). In this case the product carries over to a (Z G)-graded (cup) product on H(A; M). 3. n-ary G-graded associative algebras and n-ary modules 3.1. Denition. Let V be a G-graded vector space. Let 2 M (n?1;0) (V ), so : V n! V is n-linear of weight 0 2 G. We call an n-ary associative G-graded multiplication of weigth 0 2 G if j() = 0 2 M (2n?2;0) (V ). Remark. We are forced to use j() = 0 instead of [; ] = 0 since the latter condition is automatically satised for odd n Example. If V is 0-graded, then a ternary associative multiplication : V V V! V satises (j())(x 0 ; : : : ; X 5 ) = ((X 0 ; X 1 ; X 2 ); X 3 ; X 4 )+ + (X 0 ; (X 1 ; X 2 ; X 3 ); X 4 ) + (X 0 ; X 1 ; (X 2 ; X 3 ; X 4 )) = 0:

7 6 Michor, Vinogradov 3.3. Denition. Let V and W be G-graded vector spaces. We consider the (ZG)- graded vector space E dened by E (k;) = 8 >< >: V if k = 0 W if k = 1 0 otherwise. Then P 2 M (n?1;0;0) (E) is called an n-ary G-graded module structure on W over an n-ary algebra structure on V if j(p )P = 0. Let us denote the resulting n-ary algebra by A, and the n-ary module by W. The mapping P is the sum of partial mappings = P : V : : : V! V P : W V : : : V! W P : V W V : : : V! W : : : P : V : : : V W V! W P : V : : : V W! W the n-ary algebra structure the rightmost n-ary module structure the leftmost n-ary module structure This decomposition of P corresponds exactly to the last line in the decomposition of M (n?1;0;) of 2.5. The above denition is easily generalized by changing the form degree of W or/and by augmenting the number of W 's. For simplicity we don't discuss this possibility here Example. If V and W are 0-graded then a ternary module satises the following conditions besides the one from 3.2 describing the ternary algebra structure on V : P (P (w 0 ; v 1 ; v 2 ); v 3 ; v 4 ) + P (w 0 ; (v 1 ; v 2 ; v 3 ); v 4 ) + P (w 0 ; v 1 ; (v 2 ; v 3 ; v 4 )) = 0 P (P (v 0 ; w 1 ; v 2 ); v 3 ; v 4 ) + P (v 0 ; P (w 1 ; v 2 ; v 3 ); v 4 ) + P (v 0 ; w 1 ; (v 2 ; v 3 ; v 4 )) = 0 P (P (v 0 ; v 1 ; w 2 ); v 3 ; v 4 ) + P (v 0 ; P (v 1 ; w 2 ; v 3 ); v 4 ) + P (v 0 ; v 1 ; P (w 2 ; v 3 ; v 4 )) = 0 P ((v 0 ; v 1 ; v 2 ); w 3 ; v 4 ) + P (v 0 ; P (v 1 ; v 2 ; w 3 ); v 4 ) + P (v 0 ; v 1 ; P (v 2 ; w 3 ; v 4 )) = 0 P ((v 0 ; v 1 ; v 2 ); v 3 ; w 4 ) + P (v 0 ; (v 1 ; v 2 ; v 3 ); w 4 ) + P (v 0 ; v 1 ; P (v 2 ; v 3 ; w 4 )) = Hochschild cohomology for even n. Let V and W be G-graded vector spaces, and let P 2 M (n?1;0;0) (E) be an n-ary module structure on W over an n-ary G-graded algebra structure on V as in denition 3.3. Theorem. Let n = 2k be even. Then we have: The mapping [P; ] : M(E)! M(E) is a dierential. Its restriction P to L(V; W ) is called the Hochschild coboundary operator. For a cochain C 2 M (k;1;c) = L (k+1;c) (V; W ) and with p = n? 1 we have for X i 2 V xi ( P C)(X 0 ; : : : ; X k+p ) =? px j=0 kx i=0 (?1) pi C(X 0 : : :; P (X i ; : : :; X i+p ); : : :; X k+p ) (?1) k(j+p)+hx0++xj?1 ;ci P (X 0 ; : : :; C(X j ; : : :; X j+k ); : : :; X k+p ):

8 n-ary Lie and associative algebras 7 The corresponding (Z G)-graded cohomology will be denoted by H(A; M). Proof. We have by the (Z 2 G)-graded Jacobi identity [P; [P; Q] ] = [[P; P ] ; Q] + (?1) (n?1)2 [P; [P; Q] ] which implies that [P; ] is a dierential since n? 1 is odd and [P; P ] = j(p )P? (?1) (n?1)2 j(p )P = 2j(P )P = 0. The rest follows from a computation Remark. We get an easy extension of the Hochschild coboundary operator for n-ary algebra structures for odd n if we choose the weigth accordingly. Let P 2 M (n?1;0;p) (E) be an n-ary module structure of weight p on W over an n-ary G- graded algebra structure of weight p on V, similarly as in denition 3.3: We require that j(p )P = 0. Let us suppose that k(n? 1; 0; p)k 2 = (n? 1) 2 + hp; pi is odd. Then by 2.2 we have [P; P ] = 1? (?1) (n?1)2 +hp;pi j(p )P = 2j(P )P = 0; [P; [P; Q] ] = [[P; P ] ; Q] + (?1) (n?1)2 +hp;pi [P; [P; Q] ] = 0; so that we get a dierential. A dual version of this can be seen in 7.2.(3) below Ideals. Let (V; ) be an n-ary G-graded associative algebra. An ideal I in (V; ) is a linear subspace I V such that (X 1 ; : : :; X n ) 2 I whenever one of the X i 2 I. Then factors to an n-ary associative multiplication on the quotient space V=I. This quotient space is again G-graded, if I is a G-graded subspace in the sense that I = L x2g (I \ V x ). Of course any ideal I is an n-ary module over (V; ) which is G-graded if and only if I is G-graded. Conversely, any n-ary module W over (V; ) is an ideal in the n-ary algebra V W = E with the multiplication P from 3.3. Here P (X 1 ; : : :; X n ) = 0 if any two elements X i lie in W, so that E may be regarded as an G-graded or as a (Z G)-graded algebra. It could be called also the semidirect product of V and W Homomorphisms. A linear mapping f : V! W of degree 0 between two G-graded algebras (V; ) and (W; ) is called a homomorphism of G-graded algebras if it is compatible with the two n-ary multiplications: f((x 1 ; : : :; X n )) = (f(x 1 ); : : :; f(x n )) Then the kernel of f is an n-ary ideal in (V; ) and the image of f is an n-ary subalgebra of (W; ) which is isomorphic to V= ker(f). Similarly we can dene the notion of an n-ary V -module homomorphism between two V -modules W 0 and W 1. Then the category of all (G-graded) n-ary V -modules and of their homomorphisms is an abelian category. We did not investigate the relation to the embedding theorem of Freyd and Mitchell. 4. Review of G-graded Lie algebras and modules In this section we sketch the theory from [7] for G-graded Lie algebras from a slightly dierent angle. In this section section we need that the ground eld K has characteristic 0.

9 8 Michor, Vinogradov 4.1. Multigraded signs of permutations. Let x = (x 1 ; : : : ; x k ) 2 G k be a multi index of G-degrees x i 2 G and let 2 S k be a permutation of k symbols. Then we dene the G-graded sign sign(; x) as follows: For a transposition = (i; i + 1) we put sign(; x) =?(?1) hxi;xi+1i ; it can be checked by combinatorics that this gives a well dened mapping sign( ; x) : S k! f?1; +1g. Let us write x = (x 1 ; : : : ; x k ), then we have the following Lemma. sign( ; x) = sign(; x): sign(; x). 4.2 Multigraded Nijenhuis-Richardson algebra. We dene the G-graded alternator : M(V )! M(V ) by 1 X (1) (K)(X 0 ; : : :; X k ) = sign(; x)k(x 0 ; : : :; X k ) (k + 1)! 2S k+1 for K 2 M (k;) (V ) and X i 2 V xi. By lemma 4.1 we have 2 = so is a projection on M(V ), homogeneous of (Z G)-degree 0, and we set A(V ) = M (k;)2zg A (k;) (V ) = M (k;)2zg (M (k;) (V )): A long but straightforward computation shows that for K i 2 M (ki;i) (V ) (j(k 1 )K 2 ) = (j(k 1 )K 2 ); so the following operator and bracket is well dened: i(k 1 )K 2 : = (k 1 + k 2 + 1)! (k 1 + 1)!(k 2 + 1)! (j(k 1)K 2 ) [K 1 ; K 2 ]^ = (k 1 + k 2 + 1)! (k 1 + 1)!(k 2 + 1)! ([K 1; K 2 ] ) = i(k 1 )K 2? (?1) h(k11);(k2;2)i i(k 2 )K 1 The combinatorial factor is explained in [7], Theorem. 1. If K i are as above, then (i(k 1 )K 2 )(X 0 ; : : : ; X k1+k2 ) = 1 X = sign(; x)(?1) h1;2i (k 1 + 1)!k 2! 2S k 1 +k 2 +1 K 2 ((K 1 (X 0 ; : : :; X k1 ); : : :; X (k1+k 2)): 2. (A(V ); [ ; ]^) is a (Z G)-graded Lie algebra. 3. If 2 A (1;0) (V ), so : V V! V is bilinear G-graded anticommutative mapping of weight 0 2 G, then i() = 0 if and only if (V; ) is a G-graded Lie algebra. Proof. For 1 and 2 see [7]. 3. Let 2 A (1;0) (V ), then from 1 we see that (i())(x 0 ; X 1 ; X 2 ) = 1 sign(; x) ((X 2! 0 ; X 1 ); X 2 )) 2S 3 which is equivalent to the G-graded Jacobi expression of (V; ): X (A(V ); [ ; ]^) is called the (Z G)-graded Nijenhuis-Richardson algebra, since A(V ) coincides for G = 0 with Alt(V ) of [14].

10 n-ary Lie and associative algebras Theorem. Let V and W be G-graded vector spaces. Let E be the (ZG)-graded vector space dened by 8 >< V if k = 0 E (k;) = W if k = 1 >: 0 otherwise. Let P 2 A (1;0;0) (E) then i(p )P = 0 if and only if (a) i() = 0 so (V; ) = g is a G-graded Lie algebra, and (b) ((X 1 ; X 2 ))Y = [(X 1 ); (X 2 )]Y where (X 1 ; X 2 ) = P (X 1 ; X 2 ) 2 V and (X)Y = P (X; Y ) 2 W for X, X i and Y 2 W, and where [ ; ] denotes the G-graded commutator in End(W ). So i(p )P = 0 is by denition equivalent to the fact that M := (W; ) is a G-graded Lie-g module. If P is as above the P := [P; its restriction to M k2z (k;) (g; M) := M k2z A(k;1;) (E) 2 V ]^ : A(E)! A(E) is a dierential and generalizes the Chevalley-Eilenberg coboundary operator to the G-graded case: (@ P C)(X 0 ; : : : ; X k ) = kx i=0 (?1) i(x)+hxi;ci (X i )C(X 0 ; : : :; c Xi ; : : :; X k ) + X i<j(?1) ij(x) C((X i ; X j ); : : :; c Xi ; : : :; c Xj ; : : :) where ( i (x) = hx i ; x x i?1 i + i ij (x) = i (x) + i (x) + hx i ; x j i We denote the corresponding (Z G)-graded cohomology space by H(g; M). If : W W! W is G-graded symmetric (so 2 A (1;?1;) (E)) and [P; ]^ = 0 P acts as derivation of G-degree (1; 0) on the (Z G)-graded commutative algebra ((g; M); ), where C 1 C 2 := [C 1 ; [C 2 ; ]^]^ C i 2 (ki;ci) (g; M): In this situation the product carries over to a (Z G)-graded symmetric (cup) product on H(g; M). Proof. Apply the G-graded alternator to the results of 2.3, 2.4, 2.5, and 2.6.

11 10 Michor, Vinogradov 5. n-ary G-graded Lie algebras and their modules 5.1. Denition. Let V be a G-graded vector space. Let 2 A (n?1;0) (V ), so : V n! V is a G-graded skew symmetric n-linear mapping. We call an n-ary G-graded Lie algebra structure on V if i() = Example. If V is 0-graded, then a ternary Lie algebra structure on V is a skew symmetric trilinear mapping : V V V! V satisfying X 0 = (i())(x 0 ; : : : ; X 4 ) = 1 sign() ((X 0 ; X 1 ; X 2 ); X 3 ; X 4 ) 3! 2! 2S 3 = +((X 0 ; X 1 ; X 2 ); X 3 ; X 4 )? ((X 0 ; X 1 ; X 3 ); X 2 ; X 4 ) + ((X 0 ; X 1 ; X 4 ); X 2 ; X 3 ) + ((X 0 ; X 2 ; X 3 ); X 1 ; X 4 )? ((X 0 ; X 2 ; X 4 ); X 1 ; X 3 ) + ((X 0 ; X 3 ; X 4 ); X 1 ; X 2 )? ((X 1 ; X 2 ; X 3 ); X 0 ; X 4 ) + ((X 1 ; X 2 ; X 4 ); X 0 ; X 3 )? ((X 1 ; X 3 ; X 4 ); X 0 ; X 2 ) + ((X 2 ; X 3 ; X 4 ); X 0 ; X 1 ) 5.3. Denition. Let V and W be G-graded vector spaces. We consider the (ZG)- graded vector space E dened by E (k;) = 8 >< >: V if k = 0 W if k = 1 0 otherwise. Then P 2 A (n?1;0;0) (E) is called an n-ary G-graded Lie module structure on W over an n-ary Lie algebra structure on V if i(p )P = 0. Let us denote the resulting n-ary Lie algebra by g, and the n-ary module by W. Ordering by degree and using the G-graded skew symmetry we see that P is now the sum of only two partial n-linear mappings = P : V : : : V! V the n-ary Lie algebra structure = P : V : : : V W! W the n-ary Lie module structure 5.4. Example. If V and W are 0-graded, then a ternary Lie module satises the following condition besides the one from 5.2 describing the ternary Lie algebra structure on V : 0 = ((v 0 ; v 1 ; v 2 ); v 3 ; w)? ((v 0 ; v 1 ; v 3 ); v 2 ; w) + (v 2 ; v 3 ; (v 0 ; v 1 ; w)) + ((v 0 ; v 2 ; v 3 ); v 1 ; w)? (v 1 ; v 3 ; (v 0 ; v 2 ; w)) + (v 1 ; v 2 ; (v 0 ; v 3 ; w))? ((v 1 ; v 2 ; v 3 ); v 0 ; w) + (v 0 ; v 3 ; (v 1 ; v 2 ; w))? (v 0 ; v 2 ; (v 1 ; v 3 ; w)) + (v 0 ; v 1 ; (v 2 ; v 3 ; w)):

12 n-ary Lie and associative algebras Theorem. If P is as in 5.3 above and if n is even then the P := [P; ]^ : A(E)! A(E) is a dierential. Its restriction to M k2z M (k;) (V; W ) := k2z A(k;1;) (E) generalizes the Chevalley-Eilenberg coboundary operator to the G-graded case: For C 2 A (c;1;) (E) = (c;) (V; W ) we have (@ P C)(X 1 ; : : :; X k+n ) = [P; C]^(X 1 ; : : :; X k+n ) = =?1 (n?1)!(k+1)! + 1 n!k! X X 2S k+n sign(; x)(?1) hx1++x(n?1) ;i (X 1 ; : : :; X (n?1) ):C(X n ; : : :; X (k+n) )+ 2S k+n sign(; x)c((x 1 ; : : :; X (n) ); X (n+1) ; : : :; X (k+n) ) We denote the corresponding cohomology space by H(g; M). If : W W! W is G-graded symmetric (so 2 A (1;?1;) (E)) and [P; ]^ = 0 P acts as derivation of (ZG)-degree (1; 0) on the (ZG)-graded commutative algebra ((g; M); ), where C 1 C 2 := [C 1 ; [C 2 ; ]^]^ C i 2 (ki;ci) (g; M): In this situation the product carries over to a (Z G)-graded symmetric (cup) product on H(g; M). Proof. We have by the (Z 2 G)-graded Jacobi identity [P; [P; Q]^]^ = [[P; P ]^; Q]^ + (?1) (n?1)2 [P; [P; Q]^]^ which implies that [P; ]^ is a dierential since n? 1 is odd and [P; P ]^ = j(p )P? (?1) (n?1)2 j(p )P = 2j(P )P = 0. The rest follows from a computation Ideals. Let (V; ) be an n-ary G-graded Lie algebra. An ideal I in (V; ) is a linear subspace I V such that (X 1 ; : : :; X n ) 2 I whenever one of the X i 2 I. Then factors to an n-ary Lie algebra structure on the quotient space V=I. This quotient space is again G-graded, if I is a G-graded subspace in the sense that I = L x2g (I \ V x ). Of course, any ideal I is an n-ary module over (V; ) which is G-graded if and only if I is G-graded. Conversely, any n-ary module W over (V; ) is an ideal in the n-ary algebra V W = E with the multiplication P from 5.3. Here P (X 1 ; : : :; X n ) = 0 if any two elements X i lie in W, so that E may be regarded as an G-graded or as a (Z G)-graded Lie algebra. It could be called also the semidirect product of V and W.

13 12 Michor, Vinogradov 5.7. Homomorphisms. A linear mapping f : V! W of degree 0 between two G- graded algebras (V; ) and (W; ) is called a homomorphism of G-graded Lie algebras if it is compatible with the two n-ary multiplications: f((x 1 ; : : :; X n )) = (f(x 1 ); : : :; f(x n )) Then the kernel of f is an n-ary ideal in (V; ) and the image of f is an n-ary subalgebra of (W; ) which is isomorphic to V= ker(f). Similarly, we can dene the notion of an n-ary V -module homomorphism between two V -modules W 0 and W Relations between n-ary algebras and Lie algebras 6.1. The n-ary commutator. Let 2 M (n?1;0) (V ), so : V : : : V! V is an n-ary multiplication. The G-graded alternator from 4.2 transforms into an element := n! 2 A (n;0) (V ); which we call the n-ary commutator of. From 4.2 we also have: If is n-ary associative, then is an n-ary Lie algebra structure on V. Denition. An n-ary (ZG)-graded multiplication 2 M (n?1;0) (V ) is called n-ary Lie admissible if is an n-ary (Z G)-graded Lie algebra structure. By 5.1 this is the case if and only if i()() = (2n?1)! (j()) = 0; i. e. the alternation of the (n!) 2 n-ary associator j()() vanishes. For the binary version of this notion see [12] and [11]. An n-ary multiplication is called n-ary commutative if = Induced mapping in cohomology. Let V and W be G-graded vector spaces and let E be the (Z G)-graded vector space E (k;) = 8 >< >: V if k = 0 W if k = 1 0 otherwise as in 3.3. Let P 2 M (n?1;0) (E) be an n-ary G-graded module structure on W over an n-ary algebra structure on V, i. e. j(p )P = 0. Then P = n! P 2 A (n?1;0) (E) is an n-ary G-graded Lie module structure on W over V and some multiple of denes a homomorphism from the Hochschild cohomology of (V; ) with values in W into the Chevalley cohomology of (V; ) with values in the Lie module V. 7. Hochschild operations and non commutative dierential calculus L7.1. Let V be a G-graded vector space. We consider the tensor algebra V = 1 k=0 V k which is now (Z G)-graded such that the degree of X 1 X i is (i; x 1 + L + x i ). Put also Vn 1 = kn V k : Obviously, Vo = V :

14 n-ary Lie and associative algebras 13 The Hochschild operator K associated with K 2 M (k;) (V ) (as in 2.2) is a map K : V k! V 1 given by K = 0 on V k and K (X 0 X l ) := = Xl?k i=0 (?1) ki+h;x0++xi?1i X 0 X i?1 K(X i X i+k ) X l In the natural (Z G)-grading of L(V ; V ) the operator K has degree (?k; ). The mapping is called the Hochschild operation since for an associative multiplication : V V! V the operator is the dierential of the Hochschild homology. For K i 2 M (ki;i) (V ) with k i > 0 the composition K1 K2 is well{dened as a map from V k 1+k 2 to V 1 : 7.2. Proposition. For K i 2 M (ki;i) (V ) we have (1) in general K1 K2 6= j(k1)k 2, (2) [ K1 ; K2 ] = K1 K2? (?1) k1k2+h1;2i K2 K1 = [K1;K 2], (3) [ K ; K ] = 2 K K = 2 j(k)k if and only if k deg( K )k 2 = k 2 + h; i 1 mod 2. Proof. We get K1 X K2 (X 1 X s ) = = j+k 2<i (?1) k1i+h1;x0++xi?1i+k2j+h2;x0++xi?1i X 0 X K 2 (X j X j+k2 ) K 1 (X i X i+k1 ) X s + (?1) k1i+h1;x0++xi?1i+k2j+h2;x0++xi?1i i?k 2ji X 0 K 2 (X j K 1 (X i X i+k1 ) X j+k1+k 2 ) X s + X j>i(?1) k1i+h1;x0++xi?1i+k2j+h2;x0++xi?1i+k1k2+h1;2i X 0 K 1 (X i X i+k1 ) K 2 (X j X j+k2 ) X s : From this all assertions follow Rudiments of a non commutative dierential calculus. An intrinsic characterization of the Hochschild operators can be given as follows. For X 2 V x we consider the left and right multiplication operators X l ; X r 2 L(V m ; V n ) (1;x) which are given by X l (X 1 X k ) := X X 1 X k ; X r (X 1 X k ) := (?1) k+hx;x1++xki X 1 X k X: Then we have [X l ; Y r ] = 0 in L(V m ; V n ) for all X; Y 2 V.

15 14 Michor, Vinogradov Proposition. An operator A 2 L(V k ; V ) 1 is of the form A = K for an uniquely dened K 2 M(V ) (k;) if and only if AjV k = O and [X0; l [X1; r A]] = 0 in L(V k ; V ) 1 for all X i 2 V. Proof. A computation. In view of the theory developed in [19] (see also [6], [18]) the Hochschild operators K can be naturaly interpreted as the rst order dierential operators in the current non{commutative context Example. An element e 2 V is the left (resp., right) unit of a binary multiplication on V if and only if [ ; e l ] = id (on V ) (resp., 1 [ ; e r ] = id): Dierential calculus touched in 7.3 can be put in the following general cadre Denition. Let A be a G-graded associative (binary) algebra. For A; B 2 A let A l ; B r : A! A be the left and (signed) right multiplications, A l (B) = (?1) ha;bi B r (A) = AB. Then we have [A l ; B r ] = A l B r? (?1) ha;bi B r A l = 0: A dierential operator A! A of order (p; q) is an element 2 L(A; A) such that [X l 1; [: : :; [X l p; [Y r 1 ; [: : :; [Y r q ; ] : : :] = 0 for all X i ; Y j 2 A; which we also denote by the shorthand l p r q = 0. Obviously this denition also makes sense for mappings M! N between G-graded A-bimodules, where now A l is left multiplication of A 2 A on any G-graded A-bimodule, etc Example. A = L(V; V ) Let V be a nite dimensional vector space, ungraded for simplicity's sake, and let us consider the associative algebra A = L(V; V ). Proposition. If : L(V; V )! L(V; V ) is a dierential operator of order (p; q) with (p; q > 0); then = where P and Q are in L(V; V ). 8 >< P r ; if l p = 0 Q >: l ; if r q = 0 P r + Q l ; if l p r q = 0 Proof. We shall use the notation l Y := [Y l ; ] and similarly r Y = [Y r ; ], for Y 2 L(V; V ). We start with the following Claim. If l Y = P l Y + Qr Y for each Y 2 L(V; V ) and suitable P = P Y ; Q = Q Y : L(V; V )! L(V; V ), then we have = A l + B r where A = 0 if P = 0. If on the other hand r Y = PY l + Qr Y for each Y then we have = Al + B r where B = 0 if Q = 0. Let us assume that l Y = PY l + Qr Y for each Y. By replacing by? (1)r we may assume without loss that (1) = 0. We have (l Y )(X) = P X + XQ = (P + Q)X? [Q; X] =: [R; X] + SX; if we assume that R is traceless then R =?Q and S = P + Q are uniquely determined, thus linear in Y. Thus Y (X)? (Y X) = [R Y ; X] + S Y X

16 n-ary Lie and associative algebras 15 Insert X = 1 and use (1) = 0 to obtain (Y ) =?S Y, hence (1) [R Y ; X] = Y (X) + (Y )X? (Y X) Replacing Y by Y Z and applying the equation (1) repeatedly we obtain [R Y Z ; X] = Y Z(X) + (Y Z)X? (Y ZX) = Y Z(X) + Y (Z)X + (Y )ZX? [R Y ; Z]X? Y (ZX)? (Y )ZX + [R Y ; ZX] = Y Z(X) + Y (Z)X? Y Z(X)? Y (Z)X + Y [R Z ; X] + Z[R Y ; X] = Y [R Z ; X] + Z[R Y ; X]: The right hand side is symmetric in Y and Z, thus [R [Y;Z] ; X] = 0; inserting Y = Z = 1 we get also [R 1 ; X] = 0, hence R = 0. From (1) we see that : L(V; V )! L(V; V ) is a derivation, thus of the form (X) = [A; X] = (A l? A r )(X). If P = 0 then =?S = R? P = 0. So the rst part of the claim follows since we already substracted (1) r from the original. The second part of the claim follows by mirroring the above proof. Now we prove the proposition itself. If l p = 0 then by induction using the rst part of the claim with P = 0 we have = B r. Similarly for r q = 0 we get = A l. If l p r q = 0 with p; q > 0, by induction on p + q 2, using the claim, the result follows. The obtained result is parallel to the obvious fact that dierential operators over 0{dimensional manifolds are of zero order. 8. Remarks on Filipov's n-ary Lie algebras Here we show how Filoppov's concept of an n{lie algabra is related with that of 5.1 and sketch a similar framework for it. For simplicity's sake no grading on the vector space is assumed Let V be a vector space. According to [3], an n-linear skew symmetric mapping : V : : : V! V is called an F-Lie algebra structure if we have (1) ((Y 1 ; : : :; Y n ); X 2 ; : : :; X n ) = nx i=1 (Y 1 : : :; Y i?1 ; (Y i ; X 2 ; : : :; X n ); Y i+1 ; : : :; Y n ) The idea is that ( ; X 2 ; : : :; X n ) should act as derivation with respect to the `multiplication' (Y 1 ; : : :; Y n ) The dot product. For P 2 L p (V ; L(V; V )) and Q 2 L q (V ; L(V; V )) let us consider the rst entry as the distinguished one (belonging to L(V; V ), so that P ( ; X 1 ; : : :; X p ) 2 L(V; V )) and then let us dene P Q 2 L p+q (V ; L(V; V )) by (P Q)(Z; Y 1 ; : : :; Y q ; X 1 ; : : :; X p ) := = P (Q(Z; Y 1 ; : : :; Y q ); X 1 ; : : :; X p )? Q(P (Z; X 1 ; : : :; X p ); Y 1 ; : : :; Y q )?? qx i=1 Q(Z; Y 1 ; : : :; P (Y i ; X 1 ; : : :; X p ); : : :; Y q )

17 16 Michor, Vinogradov Then 2 L n?1 (V ; L(V; V )) which is skew symmetric in all arguments, is an F-Lie algebra structure if and only if = Lemma. We have Alt(P Q) = (p + 1)!(q + 1)!( 1 p+1 i Alt Q Alt P? (?1) pq i Alt P Alt Q); where Alt : L p (V; L(V; V ))! L p+1 skew (V ; V ) = Ap (V ) is the alternator in all appearing variables. In particular, if is an n-ary F-Lie algebra structure, then Alt is a Lie algebra structure in the sense of 5.1. Proof. An easy computation The grading operator. For a permutation 2 S p and a = (a 1 ; : : :; a p ) 2 N p 0 let the grading operator or (generalized) sign operator be given by S a : L a1++ap (V ; W )! L a1++ap (V ; W ); (S a P )(X 1 1; : : :; X 1 a 1 ; : : :; X p 1 ; : : :; Xp a p ) = P (X 1 1 ; : : :; X 1 which obviously satises S a = S (a) S a : a 1 ; : : :; X p 1 ; : : :; Xp a p ); We shall use the simplied version S a1;a2 a1 ;a2; = S for the permutation of the rst (12) two blocks of arguments of lenght a 1 and a 2. Note that also S a;b () =. If P is skew symmetric on V, then SP a = sign(; a)p, the sign from [7] or Lemma. For P 2 L p (V ; L(V; V )) and 2 L q (V; W ) let ((P ) )(X 1 ; : : :; X p ; Y 1 ; : : :; Y q ) :=? then we have for! 2 L (V ;R) qx i=1 (Y 1 ; : : :; P (Y i ; X 1 ; : : :; X p ); : : :; Y q ) (P )(!) = ((P ) )! + S q;p (P )!: Proof. A straightforward computation Lemma 8.5 suggests that (P ) behaves like a derivation with coecients in a trivial representation of gl(v ) with respect to the sign operators from 8.4. The corresponding derivation with coecients in the adjoint representation of gl(v ) then is given by the formula which follows directly from the denitions: where [P; Q] gl(v ) is the pointwise bracket P Q = [P; Q] gl(v ) + (P )Q; [P; Q] gl(v )(X 1 ; : : :) = [P (X 1 ; : : :); Q(X p+1 ; : : :)]: Moreover we have the following result

18 n-ary Lie and associative algebras Proposition. For P 2 L p (V ; L(V; V )) and Q 2 L q (V ; L(V; V )) we have where is a graded Lie bracket in the sense that P (Q R)? S q;p (Q (P R)) = [P; Q] R; [P; Q] S = [P; Q] gl(v ) + (P )Q? S q;p (Q)P [P; Q] S =?S q;p [Q; P ] S ; [P; [Q; R] S ] S = [[P; Q] S ; R] S + S q;p [Q; [P; R] S ] S : Also the derivation is well behaved with respect to this bracket, (P )(Q)? S q;p (Q)(P ) = ([P; Q] S ): Proof. For decomposable elements like in the proof of lemma 8.5 this is a long but straightforward computation. 9. Dynamical aspects It is natural to expect an eventual dynamical realization of algebraic constructions discussed above when the underlying vector space V is the algebra of observables of a mechanical or physical system. In the classical approach it should be an algebra of the form V = C 1 (M) with M being the space{time, conguration or phase space of a system, etc. The localizability principle forces us to limit the considerations to n{ary operations which are given by means of multi{erential operators. The following list of denitions is in conformity with these remarks. 9.1 Denition. An n{lie algebra structure (f 1 ; : : :; f n ) on C 1 (M) is called (1) local, if is a multi{dierential operator (2) n{ Jacobi, if is a rst{order dierential operator with respect to any its argument (3) n{poisson,if is an n{derivation. (M; ) is called an n{jacobi or n{poisson manifold if is an n{jacobi or, respectively, n{poisson structure on C 1 (M): It seemes plausible that Kirillov's theorem is still valid for the proposed n{ary generalization. It so, n{jacobi structures exhaust all local ones. 9.2 Examples. Any k{derivation on a manifold M is of the form (f 1 ; : : :; f k ) = P (df 1 ; : : :; df k ) where P = P is a k{vector eld on M and vice versa. If k is even, then is an n{poisson structure on M i [P ; P ] Schouten = 0: In particular, is a k{poisson structure in each of below listed cases: (1) P is of constant coecients on M = R m (2) P = X ^ Q where X is a vector eld on M such that L X (Q) = 0 (3) P = Q 1 ^ ^ Q r where all multi{vector elds Q i 's are of even degree and such that [Q i ; Q j ] Schouten = 0; 8 i; j: These examples are taken from [20] where the reader will nd a systematical exposition and further structural results.

19 18 Michor, Vinogradov References [1] Azcarraga, J.A. de; Perelomov, A.M.; Perez Bueno, J.C., New generalized Poisson structures, Preprint FTUV 96-1, IFIC [2] Alekseevsky, D. V.; Guha, P., Darboux' theorem for Nambu... [3] Filippov, V. T., n-ary Lie algebras, Sibirskii Math. J. 24, 6 (1985), 126{140. (Russian) [4] Gnedbaye, A. V., Les algebres k-aires et leurs operades, C. R. Acad. Sci. Paris, Serie I 321 (1995), 147{142. [5] Iba~nez, R.; Leon, M. de; Marrero, J.C.; Martin de Diego, D., Dynamics of generalized Poisson and Nambu-Poisson brackets, Preprint July 26, [6] Krasil'shchik, I. S.; Lychagin, V. V.; Vinogradov, A. M., Geometry of jet spaces and nonlinear partial dierential equations, Gordon and Breach, New York, [7] Lecomte, Pierre; Michor, Peter W.; Schicketanz, Hubert, The multigraded Nijenhuis-Richardson Algebra, its universal property and application, J. Pure Applied Algebra 77 (1992), 87{102. [8] Lecomte, P. B. A.; Roger, C., Modules et cohomologies des bigebres de Lie, C. R. Acad. Sci. Paris 310 (1990), 405{410; (Note recticative), C. R. Acad. Sci. Paris 311 (1990), 893{894. [9] Marmo, G.; Vilasi, G.; Vinogradov, A., The local structure of n- Poisson and n-jacobi manifolds and some applications, submitted to J.Geom.Phys.. [10] Michor, Peter W., Knit products of graded Lie algebras and groups, Suppl. Rendiconti Circolo Matematico di Palermo, Ser. II 22 (1989), 171{175. [11] Michor, Peter W.; Ruppert, Wolfgang; Wegenkittl, Klaus, A connection between Lie algebras and general algebras, Suppl. Rendiconti Circolo Matematico di Palermo, Serie II, 21 (1989), 265{274. [12] Myung, H. C., Malcev-admissible algebras, Progress in Mathematics Vol. 64, Birkhauser, Basel { Boston, [13] Nambu, Y., Generalized Hamiltonian dynamics, Phys. Rev. D7 (1973), 2405{2412. [14] Nijenhuis, A.; Richardson, R., Deformation of Lie algebra structures, J. Math. Mech. 17 (1967), 89{105. [15] Nijenhuis, A., On a class of common properties of some dierent types of algebras I, II, Nieuw Archief voor Wiskunde (3) 17 (1969), 17{46, 87{108. [16] Roger, C., Algebres de Lie graduees et quantication, Symplectic Geometry and Mathematical Physics (P. Donato et al., eds.), Progress in Math. 99, Birkhauser, [17] Takhtajan, Leon, On foundation of generalized Nambu mechanics, Comm. Math. Physics 160 (1994), 295{315. [18] Vinogradov, A. M., The C-spectral sequence, Lagrangian formalism and conservation laws; I. The linear theory; II. The non-linear theory, J. Math. Anal. and Appl. 100 (1984), 1{40, 41{129. [19] Vinogradov, A.M., The logic algebra for the theory of linear dierential operators, Sov. Math. Dokl. 13 (1972), 1058{1062. [20] Vinogradov, A.M.; Vinogradov, M., Alternative n-poisson manifolds, in progress. P. Michor: Institut fur Mathematik, Universitat Wien, Strudlhofgasse 4, A-1090 Wien, Austria address: peter.michor@esi.ac.at A. M. Vinogradov: Dip. Ing. Inf. e Mat. Universita di Salerno, Via S. Allende, Baronissi, Salerno, Italy address: vinograd@ponza.dia.unisa.it