Monomialn-Lie algebras

Pozhidaev, A. P.

doi:10.1007/bf02671633

Cited by 15 publications

(11 citation statements)

References 4 publications

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“…At present, these are often referred to as Filippov algebras. We also mention the article [4] which is close to our topic.…”

Section: Introductionmentioning

confidence: 89%

The n-lie property of the Jacobian as a condition for complete integrability

Dzhumadil’daev

2006

Sib Math J

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We prove that an associative commutative algebra U with derivations D 1 , . . . , D n ∈ Der U is an n-Lie algebra with respect to the n-multiplication D 1 ∧ · · · ∧ D n if the system {D 1 , . . . , D n } is in involution. In the case of pairwise commuting derivations this fact was established by V. T. Filippov. One more formulation of the Frobenius condition for complete integrability is obtained in terms of n-Lie multiplications. A differential system {D 1 , . . . , D n } of rank n on a manifold M m is in involution if and only if the space of smooth functions on M is an n-Lie algebra with respect to the Jacobian Det(D i u j ).

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“…At present, these are often referred to as Filippov algebras. We also mention the article [4] which is close to our topic.…”

Section: Introductionmentioning

confidence: 89%

The n-lie property of the Jacobian as a condition for complete integrability

Dzhumadil’daev

2006

Sib Math J

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“…Since L 7 is not solvable, it is not isomorphic to the algebras L i for i = 7. Since the algebras L 9 , L 11 are solvable with index 4, each of them is not isomorphic Periodic Algebras Generated by Groups 553 to each of the algebras L 1 -L 6 , L 8 , L 10 . Moreover, since L 9 is not a Lie algebra, but L 11 is a Lie algebra, they are not isomorphic.…”

Section: S Albeverio Ba Omirov Ua Rozikovmentioning

confidence: 99%

“…In this paper we consider algebras over a field K, with basis set {e a , a ∈ G}, which is indexed by elements of a group (G, •). The multiplication table is given as e a e b = f (a, b)e a•b•t , where t is a fixed element of G and f a map of a Cartesian square of G into the field K. A construction of this type for an n-ary algebra was first considered in [11]. For an infinite group G our construction gives an infinite dimensional algebra.…”

Section: Introductionmentioning

confidence: 99%

Periodic Algebras Generated by Groups

Albeverio

Omirov²,

Rozikov³

2015

Algebra Colloq.

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We consider algebras with basis numerated by elements of a group G. We fix a function f from G × G to a ground field and give a multiplication of the algebra which depends on f . We study the basic properties of such algebras. In particular, we find a condition on f under which the corresponding algebra is a Leibniz algebra. Moreover, for a given subgroup G of G we define a G-periodic algebra, which corresponds to a G-periodic function f, we establish a criterion for the right nilpotency of a G-periodic algebra. In addition, for G = Z we describe all 2Z-and 3Z-periodic algebras. Some properties of nZ-periodic algebras are obtained.

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“…In [13], Takhtajan studied the algebraic structures in Nambu mechanics, and indicated the relation between Nambu mechanics and n-Lie algebras. In 1985, Filippov introduced n-Lie algebras ( [14]), and then the structures are studied in [15,16,17,18,19,20]. These earlier studies demonstrate a strong and consistent association between n-Lie algebras and n-Lie Poisson algebras.…”

Section: Introductionmentioning

confidence: 99%

The infinite dimensional Unital 3-Lie Poisson algebra

Kang,

Bai,

2019

Preprint

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From a commutative associative algebra A, the infinite dimensional unital 3-Lie Poisson algebra L is constructed, which is also a canonical Nambu 3-Lie algebra, and the structure of L is discussed. It is proved that: (1) there is a minimal set of generators S consisting of six vectors; (2) the quotient algebra L/FL 0 0,0 is a simple 3-Lie Poisson algebra;(3) four important infinite dimensional 3-Lie algebras: 3-Virasoro-Witt algebra W 3 , A δ ω , A ω and the 3-W ∞ algebra can be embedded in L.

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Monomialn-Lie algebras

Cited by 15 publications

References 4 publications

The n-lie property of the Jacobian as a condition for complete integrability

The n-lie property of the Jacobian as a condition for complete integrability

Periodic Algebras Generated by Groups

The infinite dimensional Unital 3-Lie Poisson algebra

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