A Left-Symmetric Structure on The Semi-Direct Sum Real Frobenius Lie Algebra of Dimension 8

Edi Kurniadi, Nurul Gusriani, Betty Subartini

Abstract


Let  be the Lie algebra of  the semi-direct sum of the real vector space   and the Lie algebra  of the sets of all  real matrices. In this paper, a Frobenius functional is constructed in order for the Lie algebra  to be the real Frobenius Lie algebra of dimension 8. Moreover,  a bilinear form corresponding to this Frobenius functional is symplectic. Then the obtained symplectic bilinear form induces the left-symmetric algebra structures on . In other words, the Lie algebra   is the left-symmetric algebra. In particular, we give the formulas of its left-symmetric algebra structure explicitely. The left-symmetric algebra structures for case of higher dimension of this Lie algebra type are still an open problem to be investigated.


Keywords


Left-symmetric algebra; Frobenius Lie algebra; Frobenius functional; Semi-direct sum; Symplectic form.

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DOI: https://doi.org/10.18860/ca.v7i2.13462

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