Jordan superalgebras of type \(\mathcal{M}_{n\mid m}(\mathbb{F})^{(+)}\) and the Wedderburn principal theorem (WPT). (English) Zbl 1402.17037
Summary: An analogue of the Wedderburn Principal Theorem (WPT) is considered for finite-dimensional Jordan superalgebras \(\mathcal{F}\) with solvable radical \(\mathcal{N}\), \(\mathcal{N}^{2}=0\), and such that \(\mathcal{F}/\mathcal{N}\cong\mathcal{M}_{n\mid m}(\mathbb{F})^{(+)}\), where \(\mathbb{F}\) is a field of characteristic zero. It is proved that the WPT is valid under some restrictions over the irreducible \(\mathcal{M}_{n\mid m}(\mathbb{F})^{(+)}\)-bimodules contained in \(\mathcal{N}\), and it is shown with counterexamples that these restrictions cannot be weakened.
MSC:
17C70 | Super structures |
17C10 | Structure theory for Jordan algebras |
17C17 | Radicals in Jordan algebras |
References:
[1] | DOI: 10.2307/1969210 · Zbl 0029.01002 · doi:10.2307/1969210 |
[2] | Askinuze V., Ukrain Math. Z 3 pp 381– (1951) |
[3] | Taft E. J., Illinois J. Math 1 pp 565– (1957) |
[4] | Jacobson N., Structure and Representation of Jordan Algebras (1968) · Zbl 0218.17010 · doi:10.1090/coll/039 |
[5] | DOI: 10.1080/00927877708822224 · Zbl 0367.17007 · doi:10.1080/00927877708822224 |
[6] | Kantor I. L., Amer. Math. Soc. Transl. Ser 2 pp 151– (1992) |
[7] | Martinez, C., Zelmanov, E. I. (2010). Representation theory of Jordan superalgebras with unity I.Trans. Amer. Math. Soc.362(2)815–846. · Zbl 1220.17024 |
[8] | DOI: 10.1090/S0002-9947-1951-0041120-7 · doi:10.1090/S0002-9947-1951-0041120-7 |
[9] | DOI: 10.1515/crll.1964.213.187 · Zbl 0125.01904 · doi:10.1515/crll.1964.213.187 |
[10] | Zhevlakov, K. A., Slin’ko, A. M., Shestakov, I. P., Shirshov, A. I. (1982).Rings That Are Nearly Associative.Academic Press. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.