Computational Aspects of Polynomial Identities

“…This development is mainly associated with the results obtained by Kemer in the 1980-ies. Details concerning Kemer's fundamental contributions to this theory can be found in several monographs [9,7,8].…”

Section: Introductionmentioning

confidence: 99%

Z-graded identities of the Lie algebra W1

2015

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“…In particular, the dimensions dim P n of an associative operad P are usually referred to as codimensions of the corresponding variety of algebras with polynomial identities; the exponential and the usual generating series of the operad are referred as the codimension series and the exponential codimension series of the corresponding variety. (See, e.g., [16,18]. ) We do not know whether each symmetric quotient operad of Assoc has a finite Gröbner basis.…”

Section: Varieties Of Associative Algebrasmentioning

confidence: 98%

On generating series of finitely presented operads

Khoroshkin

Piontkovski

2015

Journal of Algebra

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Given an operad P with a finite Groebner basis of relations, we study the generating functions for the dimensions of its graded components P(n). Under moderate assumptions on the relations we prove that the exponential generating function for the sequence {dim P(n)} is differential algebraic, and in fact algebraic if P is a symmetrization of a non-symmetric operad. If, in addition, the growth of the dimensions of P(n) is bounded by an exponent of n (or a polynomial of n, in the non-symmetric case) then, moreover, the ordinary generating function for the above sequence {dim P(n)} is rational. We give a number of examples of calculations and discuss conjectures about the above generating functions for more general classes of operads.Comment: Minor changes; references to recent articles by Berele and by Belov, Bokut, Rowen, and Yu are adde

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“…We shall conclude this section with some comments on the solution of different versions of the Specht problem for associative and Lie algebras. For further reading we refer to the book by Kanel-Belov, Karasik and Rowen [76].…”

Section: Both Identities Follow From the Cayleymentioning

confidence: 99%

“…For a background and further reading we refer e.g. to Drensky [32], Drensky and Formanek [34], Giambruno and Zaicev [56], Kanel-Belov, Karasik and Rowen [76] for associative algebras, Bahturin [6] for Lie algebras and Zhevlakov, Slinko, Shestakov and Shirshov [143] for Jordan algebras.…”

Section: Introductionmentioning

confidence: 99%

Weak polynomial identities and their applications

Drensky

2021

Communications in Mathematics

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Let R be an associative algebra over a field K generated by a vector subspace V. The polynomial f(x 1, . . . , xn ) of the free associative algebra K〈x 1, x 2, . . .〉 is a weak polynomial identity for the pair (R, V) if it vanishes in R when evaluated on V. We survey results on weak polynomial identities and on their applications to polynomial identities and central polynomials of associative and close to them nonassociative algebras and on the finite basis problem. We also present results on weak polynomial identities of degree three.

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Computational Aspects of Polynomial Identities

Cited by 9 publications

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Z-graded identities of the Lie algebra W1

Z-graded identities of the Lie algebra W1

On generating series of finitely presented operads

Weak polynomial identities and their applications

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