The Universal Multiplicative Envelope of the Free Malcev Superalgebra on One Odd Generator

Shestakov, I. P.; Zhukavets, Natalia

doi:10.1080/00927870500454570

Cited by 17 publications

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References 10 publications

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“…As a corollary, we conclude that there are no nontrivial skew-symmetric Malcev s-identities. Another corollary of our result is the speciality of the Malcev Grassmann algebra which was introduced in [9]. Finally, we consider the homomorphic images of M and prove that they are special as well.…”

Section: Introductionmentioning

confidence: 85%

“…The elements Skew z [4(k+1)] , Skew z [4k+1] , k > 0 (see [9,13]), are nonzero central skew-symmetric elements in the free alternative algebra of countable rank.…”

Section: The Pre-base Is a Basementioning

confidence: 99%

“…It seems natural to ask first whether there could exist Malcev s-identities of special type, for instance, skew-symmetric s-identities. In this case, due to [6,9,11], the problem is reduced to free Malcev and alternative superalgebras on one odd generator, which are easier to deal with.…”

Section: Introductionmentioning

confidence: 98%

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Speciality of Malcev superalgebras on one odd generator

Shestakov

Zhukavets

2006

Journal of Algebra

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It is proved that every Malcev superalgebra generated by an odd element is special, that is, isomorphic to a subsuperalgebra of the commutator Malcev superalgebra A − for a certain alternative superalgebra A. As a corollary, it is shown that the kernel of the natural homomorphism of the free Malcev algebra Malc[X] of countable rank into the commutator Malcev algebra Alt[X] − of the corresponding free alternative algebra Alt[X], does not contain skew-symmetric multilinear elements. In other words, there are no skew-symmetric Malcev s-identities. Another corollary is speciality of the Malcev Grassmann algebra.

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Section: Introductionmentioning

confidence: 85%

“…The elements Skew z [4(k+1)] , Skew z [4k+1] , k > 0 (see [9,13]), are nonzero central skew-symmetric elements in the free alternative algebra of countable rank.…”

Section: The Pre-base Is a Basementioning

confidence: 99%

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Speciality of Malcev superalgebras on one odd generator

Shestakov

Zhukavets

2006

Journal of Algebra

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“…In particular, a base of this superalgebra was constructed. In [14] a base of the universal multiplicative envelope R(M) was constructed. The results of [11], [14] were used to describe the structure of the space of skew-symmetric elements in the free Malcev algebra of countable range.…”

Section: Introductionmentioning

confidence: 99%

The Malcev Poisson Superalgebra of the Free Malcev Superalgebra on One Odd Generator

Shestakov¹,

Zhukavets²

2006

J. Algebra Appl.

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“…An alternative way to construct R(M ) is the following one. Consider the free product M = M * F x of the algebra M and the one-dimensional Malcev algebra F x in the variety of Malcev algebras (see [1]); then the algebra R(M ) is isomorphic to the subalgebra of the algebra End F (M ) generated by all the multiplication operators R m , m ∈ M (see [14]). …”

Section: Corollary 1 Let O Be An Octonion Division Algebra Over a Fimentioning

confidence: 99%

On the radical of a free Malcev algebra

Shestakov

Kornev

2012

Proc. Amer. Math. Soc.

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Abstract. We prove that the prime radical rad M of the free Malcev algebra M of rank more than two over a field of characteristic = 2 coincides with the set of all universally Engelian elements of M. Moreover, let T (M) be the ideal of M consisting of all stable identities of the split simple 7-dimensional Malcev algebra M over F . It is proved that rad M = J(M) ∩ T (M), where J(M) is the Jacobian ideal of M. Similar results were proved by I. Shestakov and E. Zelmanov for free alternative and free Jordan algebras. An algebra M is called a Malcev algebra if it satisfies the identitieswhere J(x, y, z) = (xy)z + (zx)y + (yz)x is the Jacobian of the elements x, y, z [7,9,5]. Since for a Lie algebra the Jacobian of any three elements vanishes, Lie algebras fall into the variety of Malcev algebras. Among the non-Lie Malcev algebras, the traceless elements of the octonion algebra with the product given by the commutator [x, y] = xy − yx is one of the most important examples [9,5,6]. In 1977 I. P. Shestakov [11] proved that the free Malcev algebra M n on n ≥ 9 free generators is not semiprime; that is, M n contains nonzero nilpotent ideals. In 1979, V. T. Filippov [3] extended this result to free Malcev algebras with more than four generators. Therefore, the prime radical rad M n = 0 for n > 4, and a natural question on the description of this radical arises.For free alternative algebras, it was proved by Shestakov in [10] that the prime radical coincides with the set of nilpotent elements. A similar fact was established by E. Zel manov [15] for free Jordan algebras.In anticommutative algebras, the role of nilpotent elements is played by engelian elements. An element a of an algebra M is called engelian if the operator of right multiplication R a : x → xa is nilpotent. We will call an element a ∈ M universally engelian if, for every algebra M ⊇ M , the element a is engelian in M . In other words, the image R a of the element a in the (associative) universal multiplicative enveloping algebra R(M ) of M is nilpotent. In the present paper, we prove that the

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The Universal Multiplicative Envelope of the Free Malcev Superalgebra on One Odd Generator

Cited by 17 publications

References 10 publications

Speciality of Malcev superalgebras on one odd generator

Speciality of Malcev superalgebras on one odd generator

The Malcev Poisson Superalgebra of the Free Malcev Superalgebra on One Odd Generator

On the radical of a free Malcev algebra

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