Abstract
In this paper we construct a minimal faithful representation of the \((2m+2)\)-dimensional complex general Diamond Lie algebra, \(\mathfrak {D}_m(\mathbb {C})\), which is isomorphic to a subalgebra of the special linear Lie algebra \(\mathfrak {sl}(m+2,\mathbb {C})\). We also construct a faithful representation of the real general Diamond Lie algebra \(\mathfrak {D}_m\) which is isomorphic to a subalgebra of the special symplectic Lie algebra \(\mathfrak {sp}(2m+2,\mathbb {R})\). Furthermore, we describe Leibniz algebras with corresponding \((2m+2)\)-dimensional real general Diamond Lie algebra \(\mathfrak {D}_m\) and such that the ideal generated by the squares of elements provides a faithful representation of \(\mathfrak {D}_m\).
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We thank the referees for the helpful comments and suggestions which made the paper more enhanced.
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Communicated by Miin Huey Ang.
The work was partially supported was supported by Agencia Estatal de Investigación (Spain), grant MTM2016-79661-P (European FEDER support included, UE) and by Xunta de Galicia, Grant GRC2013-045 (European FEDER support included). The work was also partially supported by Grant No. 0828/GF4 of Ministry of Education and Science of the Republic of Kazakhstan.
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Camacho, L.M., Karimjanov, I.A., Ladra, M. et al. Leibniz Algebras Constructed by Representations of General Diamond Lie Algebras. Bull. Malays. Math. Sci. Soc. 42, 1281–1293 (2019). https://doi.org/10.1007/s40840-017-0541-5
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DOI: https://doi.org/10.1007/s40840-017-0541-5