It is shown that exceptional algebras of a vector cross product are ternary central simple maltsev algebras, which are not 3lie algebras if the characteristic of a ground field is distinct from 2 and 3. Maltsev algebras for which every lie algebra is a maltsev algebra. The paper main concerns structures and representations of n hom lie algebras. Then one readily veri es that the quotient vector space ga has the natural structure of lie algebra.
We provide examples of the new structures and present some properties and. Also the minimality of l is characterized in terms of the connections and it is shown that the above direct sum is by means of the family of its minimal color gltideals, admitting each one a quasimultiplicative basis inherited by the one of l. We use this formalism to study nary symmetric superalgebras with invariant skewsymmetric forms. These will be discussed as well, since they underlie the cohomological properties of n lie algebras. Cohomology complex for deformations of n lie algebras 70 12. Homalgebras and homcoalgebras journal of algebra and its. It is proven that the canonical nvector on the dual of an nlie algebra g is npoisson iff dim g. The aim of this paper is to introduce n ary hom algebra structures generalizing the n ary algebras of lie type including n ary nambu algebras, n ary nambu lie algebras and n ary lie algebras. Induced 3lie algebras, superalgebras and induced representations. When the skewsymmetry of the n lie algebra is relaxed, one is led the n leibniz algebras. These notes are devoted to the multiple generalization of a lie algebra introduced by a. Generically, they describe the dynamics of a physical system, and there is a need of understanding their quantization. The class of generalized lietype color algebras contains the ones of generalized lietype algebras, of nlie algebras and superalgebras, commutative leibniz nary algebras and superalgebras, among others.
Secondly, we study the universal associative enveloping algebras of nlie algebras. For this reason, we often deal with a more wieldy object, namely the lie algebra corresponding to the group. Pdf the classification of nlie algebras researchgate. Oct 23, 2004 a class of nary maltsev algebras is also characterized by the following interesting properties. The standard poisson structure may also be extended to the nary case. The notion of nlie filippov algebra was introduced in 1985 by filippov 1985. Representations and cohomology of nary multiplicative hom. There are two main ways of achieving this, depending on how the jacobi identity ji of the ordinary lie algebras is. M can be considered as a representation of the leibniz algebra dn 1l.
Also we show that a cohomology of nary homnambulie algebras could be derived from the cohomology of. Polynomial identities and enveloping algebras for nary. An nlie algebra is a vector space l with a multilinear product ln. The paper main concerns structures and representations of nhom lie algebras. Tensor models and hierarchy of nary algebras international. Homnambu lie algebras provide a framework that might be an appropriate setting in which n lie algebras n ary nambu lie algebras can be deformed, and their quantization studied. Lie theory, universal enveloping algebras, and the poincar. Also various quantum nary extensions of numbu mechanics and related nary extensions of differential structures and of lie algebras jacobi identities were considered. These generalizations are algebraic structures in which the two entries lie bracket has been replaced by a bracket with n entries. In the particular case n 3 a lie superalgebra with ternary graded lie bracket is usually called 3lie superalgebra. An nary quasigroup is a set with an nary operation, q, f with f. The aim of this paper is to provide cohomologies of nary homnambulie algebras governing central extensions and one parameter formal deformations.
On nary homnambu and homnambulie algebras sciencedirect. We rst introduce a notion of representation of a leibniz n algebra l. The condition for the leibnitz rules of the nary algebras is discussed from the perspective of the invariance of the underlying algebra under the nary transformations. Representations of nlie algebras and twisted al gebraic. In the particular case n 3 a lie super algebra with ternary graded lie bracket is usually called 3 lie super algebra. Jul 23, 2010 when the skewsymmetry of the lie or n lie algebra bracket is relaxed, one is led to a more general type of n algebras, the n leibniz algebras. A filippov algebra over f is an fvector space a, together with nary multilinear and skewsymmetric operation x 1, x n, which satisfies the following generalized jacobi identity x 1. Leibniz n algebras a leibniz algebra of order n, or simply a leibniz n algebra, is a vec. Vinogradov, nary lie and associative algebras, math. We compare definitions of such algebras in the usual and invariant case. A leibniz n algebra is a vector space equipped with an n ary operation which has the property of being a derivation for itself.
An nlie algebra l is a vector space with an n ary skewsymmetric operation satisfying x. When the skewsymmetry of the lie or nlie algebra bracket is relaxed, one is led to a more general type of nalgebras, the nleibniz algebras. In terms of the magnetic translation operators, an explicit physical realization of the cosine n algebra is given. Representations of nlie algebras and twisted algebraic structures abdenacer makhlouf 1, 1 university of haute alsace, mulhouse, france the notion of nlie algebra was introduced by filippov in 2. Aug 21, 2018 the class of generalized lie type color algebras contains the ones of generalized lie type algebras, of n lie algebras and superalgebras, commutative leibniz n ary algebras and superalgebras, among others. The problem of compatibility of two nlie algebra structures is analyzed and the.
A 0ary, or nullary, quasigroup is just a constant element of q. We suppose there are three quantities tx, ty, and tz with a lie product indicated by, tx,ty itz, ty,tz itx, tz,tx ity. The aim of this paper is to introduce nary homalgebra structures generalizing the nary algebras of lie type including nary nambu algebras, nary nambulie algebras and nary lie algebras, and nary algebras of associative type including nary totally associative and nary partially associative algebras. The basic example of a lie algebra arises from using the commutator in an associative algebra, so x. It was noticed in particular that many of quantum algebras and qdeformed lie algebras obey certain qdeformed versions of jacobi identity generalizing lie algebras jacobi identity. One observes that if m is a representation of a leibniz n algebra l, then homl. Construction of n lie algebras and n ary homnambu lie algebras joakim arnlind, abdenacer makhlouf and sergei silvestrov 1 dec 2011 journal of mathematical physics, vol. It is shown that the nary transformations which keep the underlying algebraic structure invariant form closed finite n. Leibniz nalgebras a leibniz algebra of order n, or simply a leibniz nalgebra, is a vec. There are two main ways of achieving this, depending on how the jacobi identity ji of. An n nlie algebra is defined to be an algebraic structure which. The vector space together with this operation is a nonassociative algebra, meaning that the lie bracket is not necessarily associative lie algebras are closely related to lie groups. It is shown that the n ary transformations which keep the underlying algebraic structure invariant form closed finite n ary lie subalgebras. Please join the simons foundation and our generous member organizations in supporting arxiv during our giving campaign september 2327.
Classification of simple linearly compact nlie super. The special linear lie algebra sln,r consists of all n. When the skewsymmetry of the nlie algebra is relaxed, one is led the nleibniz algebras. An nary homnambulie algebra is an nary homnambu algebra in which the nary product, is antisymmetric. In 2, filippov formulated a theory of nlie algebras based on his proposed 2n. We compare definitions of such algebras in the usual. Each type of nary bracket satisfies a specific characteristic identity which plays the role of the jacobi identity for lie algebras. The basic result is that every n ary algebra of the vector cross product is an n ary central simple maltsev algebra. Commutative nary superalgebras with an invariant skewsymmetric.
It is shown that exceptional algebras of a vector cross product are ternary central simple maltsev algebras, which are not 3 lie algebras if the characteristic of a ground field is distinct from 2 and 3. These n ary algebras are a generalization of n ary lie algebras of type 1 and of type 2 for even n. These generalizations are algebraic structures in which the twoentry lie bracket has been replaced by a bracket with n entries. We introduce an alternative concept of bihomlie algebra called. The standard poisson structure may also be extended to the n ary case. We find that they are the sh nlie algebras for the n even case. A leibniz nalgebra is a vector space equipped with an nary operation which has the property of being a derivation for itself. In these, the standard lie bracket is replaced by a linear nary bracket with n 2 entries, the algebra structure being defined by the characteristic identity satisfied by the nary bracket. Lie algebras and ados theorem princeton university. If a m nk, then the space of n nmatrices has the structure of a lie algebra with lie bracket x.
The condition for the leibnitz rules of the n ary algebras is discussed from the perspective of the invariance of the underlying algebra under the n ary transformations. The theory of such algebras, being an extension of the theory of. An introduction to some novel applications of lie algebra. In these, the standard lie bracket is replaced by a linear n ary bracket with n 2 entries, the algebra structure being defined by the characteristic identity satisfied by the n ary bracket. Filippov in 8 proposed a notion of nlie algebra, which can be considered as an extension of the concept of binary lie bracket to nary. Matrix lie algebras edit a complex matrix group is a lie group consisting of matrices, g.
A non abelian lie algebra g is called simple if it has no non trivial ideals. In the particular case n 3 a lie superalgebra with ternary graded lie bracket is usually. As nary operations, generalizing lie and poisson algebras, arise in many different physical contexts, it is interesting to study general ways of constructing explicit realizations of such multilinear structures. There are two main ways of achieving this, depending on how the jacobi identity ji of the ordinary lie algebras is looked at. The aim of this paper is to develop the theory of homcoalgebras and related structures. Polyadic or multiary means nary for some nonnegative integer n. When the skewsymmetry of the lie or n lie algebra bracket is relaxed, one is led to a more general type of n algebras, the n leibniz algebras. Journal of mathematical physics, issn 00222488, eissn 10897658, vol. In order to transfer information from the lie algebra to the lie group, we use a function called the exponential mapping.
Ling proved in 1993 that for n 3, there is only one simple nitedimensional nlie algebra. Homnambulie algebras provide a framework that might be an. Filipov 3 proposed a generalization of the concept of a lie algebra by replacing the binary operation by an nary one. These generalizations are algebraic structures in which the. Generalize lietype algebra, nlie algebra, nleibniz algebra, superalgebra, color. Prove that o2 and n2 are abelian 1dimensional lie algebras, hence they are isomorphic to kwith zero. L satisfying nary anticommutativity and the nary jacobi identity.
Request pdf nary lie and associative algebras with the help of the multigraded nijenhuisrichardson bracket and the multigraded gerstenhaber bracket from 7 for every n 2 we define nary. Pdf generalization of nary nambu algebras and beyond. In these, the standard lie bracket is replaced by a linear nary. Assume that v, is an n ary lie algebra of t yp e r, l and the algebra struc ture is in v a rian t with resp ect to. By analogy with nlie algebras, which are a natural generalization of lie algebras to the case of nary multiplication, we define the concept of an nary maltsev algerba. Permission to use in presenting this thesis in partial ful lment of the requirements for a postgraduate degree from the university of saskatchewan, i agree that the libraries of. These structures were introduced by filippov in 1985. A class of nary maltsev algebras is also characterized by the following interesting properties. After a selfcontained introduction to lie algebra cohomology, we present. Construction of nlie algebras and nary homnambulie. These will be discussed as well, since they underlie the cohomological properties of nlie algebras. We generalize to nary algebras the notions of derivation and representation introduced by sheng for homlie algebras.
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