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Quasi-Deformations of sl_2(\F) using twisted derivations

Daniel Larsson·Sergei D. Silvestrov
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Abstract

In this paper we apply a method devised in \cite{HartLarsSilv1D,LarsSilv1D} to the three-dimensional simple Lie algebra $\sll$. One of the main points of this deformation method is that the deformed algebra comes endowed with a canonical twisted Jacobi identity. We show in the present paper that when our deformation scheme is applied to $\sll$ we can, by choosing parameters suitably, deform $\sll$ into the Heisenberg Lie algebra and some other three-dimensional Lie algebras in addition to more exotic types of algebras, this being in stark contrast to the classical deformation schemes where $\sll$ is rigid. The resulting algebras are quadratic and we point out possible connections to ``geometric quadratic algebras'' such as the Artin--Schelter regular algebras, studied extensively since the beginning of the 90's in connection with non-commutative projective geometry.

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Daniel Larsson, Sergei D. Silvestrov. 2005-10-17. Quasi-Deformations of sl_2(\F) using twisted derivations. https://arxiv.org/abs/math/0506172

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