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Second cohomology group of the finite-dimensional simple Jordan superalgebra $𝒟_{t}$ , $t \neq 0$

Corrections for this article
- Corrigendum: Second cohomology group of the finite-dimensional simple Jordan superalgebra $𝒟_{t}$ , $t \neq 0$

F. A. Gómez González

Institute of Mathematics, University of Antioquia, Medellín, Colombia

E-mail Address: faber.gomez@udea.edu.co

Corresponding author. The first author was partially supported by Universidad de Antioquia, SIIU2019-26870.

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and

J. A. Ramírez Bermúdez

Institute of Mathematics, University of Antioquia, Medellín, Colombia

E-mail Address: jalexander.ramirez@udea.edu.co

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https://doi.org/10.1142/S0219498822500918Cited by:2 (Source: Crossref)

Abstract

The second cohomology group (SCG) of the Jordan superalgebra $𝒟_{t}$ , $t \neq 0$ , over an algebraically closed field $𝔽$ of characteristic zero is calculated by using the coefficients which appear in the regular superbimodule $Reg 𝒟_{t}$ . Contrary to the case of algebras, this group is nontrivial thanks to the non-splitting caused by the Wedderburn Decomposition Theorem [F. A. Gómez-González, Wedderburn principal theorem for Jordan superalgebras I, J. Algebra505 (2018) 1–32]. First, to calculate the SCG of a Jordan superalgebra we use split-null extension of the Jordan superalgebra and the Jordan superalgebra representation. We prove conditions that satisfy the bilinear forms $h$ that determine the SCG in Jordan superalgebras. We use these to calculate the SCG for the Jordan superalgebra $𝒟_{t}$ , $t \neq 0$ . Finally, we prove that $ℋ^{2} (𝒟_{t}, Reg 𝒟_{t}) = 0 \dot{+} 𝔽^{2}$ , $t \neq 0$ .

Communicated by V. Futorny

Keywords:

AMSC: 17A70, 17C70, 17A60

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