Abstract
Let \((g,~[-,-],~\omega )\) be a finite-dimensional complex \(\omega \)-Lie superalgebra. In this paper, we introduce the notions of derivation superalgebra \({\mathrm{Der}}(g)\) and the automorphism group \({\mathrm{Aut}}(g)\) of \((g,~[-,-],~\omega )\). We study \({\mathrm{Der}}^{\omega }(g)\) and \({\mathrm{Aut}}^{\omega }(g)\), which are superalgebra of \({\mathrm{Der}}(g)\) and subgroup of \({\mathrm{Aut}}(g)\), respectively. For any 3-dimensional or 4-dimensional complex \(\omega \)-Lie superalgebra g, we explicitly calculate \({\mathrm{Der}}(g)\) and \({\mathrm{Aut}}(g)\), and obtain Jordan standard forms of elements in the two sets. We also study representation theory of \(\omega \)-Lie superalgebras and give a conclusion that all nontrivial non-\(\omega \)-Lie 3-dimensional and 4-dimensional \(\omega \)-Lie superalgebras are multiplicative, as well as we show that any irreducible respresentation of the 4-dimensional \(\omega \)-Lie superalgebra \(P_{2,k}(k\ne 0,-1)\) is 1-dimensional.
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Acknowledgements
The authors would like to thank the referee for valuable comments and suggestions on this article. Supported by NNSF of China (nos. 11771069 and 12071405).
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Appendix
Appendix
The characterizations about \({\mathrm{Der}}(g)\), \({\mathrm{Aut}}(g)\) and Jordan standard forms of elements in \({\mathrm{Der}}(g)\) and \({\mathrm{Aut}}(g)\) for the remaining nontrival non-\(\omega \)-Lie 4-dimensional \(\omega \)-Lie superalgebras are summarized in the following Tables 1, 2, 3 and 4, without the detailed proofs.
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Zhou, J., Chen, L. On Low-Dimensional Complex \(\omega \)-Lie Superalgebras. Adv. Appl. Clifford Algebras 31, 54 (2021). https://doi.org/10.1007/s00006-021-01141-8
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DOI: https://doi.org/10.1007/s00006-021-01141-8