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 Izv. RAN. Ser. Mat., 2016, Volume 80, Issue 6, Pages 247–257 (Mi izv8447)

Simple right alternative superalgebras of Abelian type whose even part is a field

S. V. Pchelintsev, O. V. Shashkov

Financial University under the Government of the Russian Federation, Moscow

Abstract: We study central simple unital right alternative superalgebras $B=\Gamma\oplus M$ of Abelian type of arbitrary dimension whose even part $\Gamma$ is a field. We prove that every such superalgebra $B=\Gamma\oplus M$, except for the superalgebra $B_{1|2}$, is a double, that is, the odd part can be represented in the form $M=\Gamma x$ for a suitable $x$.
If the generating element $x$ commutes with the even part $\Gamma$, then $B$ is isomorphic to a twisted superalgebra of vector type $B(\Gamma,D,\gamma)$ introduced by Shestakov [1], [2]. But if $x$ commutes with the odd part $M$, then $B$ is isomorphic to a superalgebra $B(\Gamma, ^*,R_\omega)$ introduced in [3] and called an $\omega$-double.
We prove that if the ground field is algebraically closed, then $B$ is isomorphic to one of the superalgebras $B_{1|2}$, $B(\Gamma,D,\gamma)$, $B(\Gamma, ^*,R_\omega)$.

Keywords: simple right alternative superalgebra, superalgebra of Abelian type.

 Funding Agency Grant Number Russian Foundation for Basic Research 14-01-00014 This research was supported by the Russian Foundation for Basic Research (grant no. 14-01-00014).

DOI: https://doi.org/10.4213/im8447

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English version:
Izvestiya: Mathematics, 2016, 80:6, 1231–1241

Bibliographic databases:

UDC: 512.554.5
MSC: 17A70, 17D15

Citation: S. V. Pchelintsev, O. V. Shashkov, “Simple right alternative superalgebras of Abelian type whose even part is a field”, Izv. RAN. Ser. Mat., 80:6 (2016), 247–257; Izv. Math., 80:6 (2016), 1231–1241

Citation in format AMSBIB
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• https://doi.org/10.4213/im8447
• http://mi.mathnet.ru/eng/izv/v80/i6/p247

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This publication is cited in the following articles:
1. S. V. Pchelintsev, O. V. Shashkov, “Simple finite-dimensional right-alternative superalgebras with semisimple strongly associative even part”, Sb. Math., 208:2 (2017), 223–236
2. S. V. Pchelintsev, O. V. Shashkov, “Simple finite-dimensional right-alternative unital superalgebras with strongly associative even part”, Sb. Math., 208:4 (2017), 531–545
3. S. V. Pchelintsev, O. V. Shashkov, “Singulyarnye 6-mernye superalgebry”, Sib. elektron. matem. izv., 15 (2018), 92–105
4. S. V. Pchelintsev, O. V. Shashkov, “Simple finite-dimensional right-alternative unital superalgebras with associative-commutative even part over a field of characteristic zero”, Izv. Math., 82:3 (2018), 578–595
5. V. N. Zhelyabin, “Structure of some unital simple Jordan superalgebras with associative even part”, Siberian Math. J., 59:6 (2018), 1051–1062
6. S. V. Pchelintsev, O. V. Shashkov, “Prostye unitalnye pravoalternativnye superalgebry nad algebroi matrits poryadka $2$”, Algebra i logika, 58:1 (2019), 108–131
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