Combinatorial aspects of the theory of Lie superalgebras

Михалев, А. В.; Fong, Y.; Knauer, Ulrich; Mikhalev, A. V.

doi:10.1515/9783110883220-003

Cited by 26 publications

(40 citation statements)

References 28 publications

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“…It is known that a suitable basis of a free Lie (super)algebra can be formed by regular (ordinary Lie algebra), s-regular (Lie superalgebra in characteristic 0) and ps-regular (Lie superalgebra in characteristic p) Lie monomials (Bahturin et al, 1992;Mikhalev and Zolotykh, 1995).…”

Section: Algorithmmentioning

confidence: 99%

“…The lefthand sides of the latter form a set of Lie polynomials which is often called canonical or Gröbner basis. In fact, canonical bases for ideals of free Lie algebras were introduced in (Shirshov, 1962), and canonical bases for ideals of free Lie superalgebras were introduced in (Mikhalev, 1989) (see also Ufnarovsky, 1990;Bahturin et al, 1992;Mikhalev and Zolotykh, 1995).…”

Section: Algorithmmentioning

confidence: 99%

“…These definitions can be generalized in the following way (Bahturin et al, 1992;Mikhalev and Zolotykh, 1995). Let G be an abelian additive group grading a certain algebra L = ⊕ g∈G L g .…”

Section: Introductionmentioning

confidence: 99%

“…Characteristic 2 requires also the existence of a quadratic operator q mapping odd elements of the algebra into even ones (Ufnarovsky, 1990;Bahturin et al, 1992;Mikhalev and Zolotykh, 1995) such that q(αu) = α 2 q(α), [u, v] = q(u + v) − q(u) − q(v), [u, [u, v]…”

Section: Introductionmentioning

confidence: 99%

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Construction of Finitely Presented Lie Algebras and Superalgebras

Gerdt

Kornyak

1996

Journal of Symbolic Computation

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We consider the following problem: what is the most general Lie algebra or superalgebra satisfying a given set of Lie polynomial equations?The presentation of Lie (super)algebras by a finite set of generators and defining relations is one of the most general mathematical and algorithmic schemes of their analysis. That problem is of great practical importance covering applications ranging from mathematical physics to combinatorial algebra. Some particular applications are construction of prolongation algebras in the Wahlquist-Estabrook method for integrability analysis of nonlinear partial differential equations and investigation of Lie (super)algebras arising in different (super)symmetrical physical models. The finite presentations also indicate a way to q-quantize Lie (super)algebras.To solve this problem, one should perform a large volume of algebraic transformations which is sharply increased with growth of the number of generators and relations. For this reason one needs, in practice, to use a computer algebra tool. We describe here an algorithm for constructing the basis of a finitely presented Lie (super)algebra and its commutator table, and its implementation in C. Some computer results illustrating our algorithm and its actual implementation are also presented.

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Section: Algorithmmentioning

confidence: 99%

Section: Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Construction of Finitely Presented Lie Algebras and Superalgebras

Gerdt

Kornyak

1996

Journal of Symbolic Computation

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“…We are deeply indebted to the pioneering work of Drinfel'd [15]; Grosshans, Rota, and Stein [16]; Gurevich [17]; Hashimoto and Hayashi [19]; Joyal and Street [23,24]; Majid [29,30]; Mikhalev and Zolotykh [32]; Pareigis [34]; and Scheunert [37,38].…”

Section: Introductionmentioning

confidence: 99%