Simple Special Jordan Superalgebras with Associative Even Part

Zhelyabin, V. N.; Shestakov, I. P.

doi:10.1023/b:simj.0000042476.85436.a3

Cited by 25 publications

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References 10 publications

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“…By (5) we obtain D ik (a)x j = D jk (a)x i . Therefore, J(A, Δ) is a Jordan superalgebra by Proposition 3 from [9]. Let L = AD 11 + · · · + AD ij + · · · + AD nn .…”

Section: Propositionmentioning

confidence: 90%

“…The speciality of this Jordan superalgebra was proved in [6]. In [7] the universal associative enveloping algebra was constructed for a simple Jordan superalgebra of vector type with one derivation.In [8,9], the unital simple special Jordan superalgebras with associative even part A were described such that the odd part M is an associative A-module. If such superalgebra is not a superalgebra of nondegenerate bilinear superform then its even part A is a differentiably simple algebra with respect to some set of derivations, and the odd part M is a finitely generated projective A-module of rank 1.…”

mentioning

confidence: 99%

“…In [8,9], the unital simple special Jordan superalgebras with associative even part A were described such that the odd part M is an associative A-module. If such superalgebra is not a superalgebra of nondegenerate bilinear superform then its even part A is a differentiably simple algebra with respect to some set of derivations, and the odd part M is a finitely generated projective A-module of rank 1.…”

mentioning

confidence: 99%

“…This example may be found in [9]. Given a field F of characteristic zero in which the equation t 2 + 1 = 0 is unsolvable, some example of a similar superalgebra over F was given in [12][13].…”

mentioning

confidence: 99%

“…The module M is A-torsion free. As is known (for example, see [9]), the mapping D x,y : A → A given by the rule D x,y (a) = (a, x, y) is a derivation when x, y ∈ M . We have…”

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confidence: 99%

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Jordan superalgebras of vector type and projective modules

Zhelyabin

2012

Sib Math J

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We study connection between the Jordan superalgebras of vector type and the finitely generated projective modules of rank 1 over an integral domain.Keywords: Jordan superalgebra of vector type, differentiably simple algebra, integral domain, field of quotients, polynomial algebra, projective module, Picard groupThe well-known Kantor construction [1, 2] allows us to construct a Jordan superalgebra that is based on an associative commutative algebra with derivation. The so-constructed Jordan superalgebra belongs to the class of superalgebras of vector type. The superalgebras of vector type play an important role in exhibition of some counterexamples. In [3], some examples are given of prime (−1, 1)-algebras and Jordan algebras with absolute zero-divisors, the so-called Pchelintsev's monsters. Other constructions of Pchelintsev's monsters were given in [4] by using the Jordan superalgebras of vector type. Some examples of prime algebras with absolute zero-divisors in the varieties of alternative, Jordan, and (−1, 1)-algebras were obtained in [5] with the help of the Jordan superalgebras of vector type. The Jordan superalgebras of vector type with one derivation were studied in [2,6]. Thus, the simplicity criterium for a Jordan superalgebra was found in [2]. The speciality of this Jordan superalgebra was proved in [6]. In [7] the universal associative enveloping algebra was constructed for a simple Jordan superalgebra of vector type with one derivation.In [8,9], the unital simple special Jordan superalgebras with associative even part A were described such that the odd part M is an associative A-module. If such superalgebra is not a superalgebra of nondegenerate bilinear superform then its even part A is a differentiably simple algebra with respect to some set of derivations, and the odd part M is a finitely generated projective A-module of rank 1. The multiplication on M is given by some fixed finite sets of derivations and elements of A. Furthermore, each of such Jordan superalgebras is a subsuperalgebra of vector type J(Γ, D) with derivation D. If the number of generators of the A-module M is equal to 1 then the initial Jordan superalgebra is a superalgebra of vector type J(Γ, D). Under some restriction on A, the odd part M is a one-generated A-module. For example, if A is a local algebra then M is a free (and, consequently, one-generated) A-module by the well-known Kaplansky theorem. If the ground field is of characteristic p > 2 then A is a local algebra by [10]; therefore, M is a one-generated A-module. If A is a unique factorization ring then it is known that M is a free (and, consequently, one-generated) A-module (for example, see [11]).The first example of a simple superalgebra of vector type over the reals with several derivations which is not isomorphic to J(Γ, D) was constructed by I. P. Shestakov. This example may be found in [9]. Given a field F of characteristic zero in which the equation t 2 + 1 = 0 is unsolvable, some example of a similar superalgebra over F was given in [12][13]. Finally, the exampl...

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“…By (5) we obtain D ik (a)x j = D jk (a)x i . Therefore, J(A, Δ) is a Jordan superalgebra by Proposition 3 from [9]. Let L = AD 11 + · · · + AD ij + · · · + AD nn .…”

Section: Propositionmentioning

confidence: 90%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…The module M is A-torsion free. As is known (for example, see [9]), the mapping D x,y : A → A given by the rule D x,y (a) = (a, x, y) is a derivation when x, y ∈ M . We have…”

mentioning

confidence: 99%

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Jordan superalgebras of vector type and projective modules

Zhelyabin

2012

Sib Math J

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Novikov-Poisson algebras and associative commutative derivation algebras

Zhelyabin

Tikhov

2008

Algebra Logic

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We describe Novikov-Poisson algebras in which a Novikov algebra is not simple while its corresponding associative commutative derivation algebra is differentially simple. In particular, it is proved that a Novikov algebra is simple over a field of characteristic not 2 iff its associative commutative derivation algebra is differentially simple. The relationship is established between Novikov-Poisson algebras and Jordan superalgebras.

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Embedding Novikov–Poisson algebras in Novikov–Poisson algebras of vector type

Zakharov

2013

Algebra Logic

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Simple Special Jordan Superalgebras with Associative Even Part

Cited by 25 publications

References 10 publications

Jordan superalgebras of vector type and projective modules

Jordan superalgebras of vector type and projective modules

Novikov-Poisson algebras and associative commutative derivation algebras

Embedding Novikov–Poisson algebras in Novikov–Poisson algebras of vector type

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