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

TL;DR

This paper introduces a novel deformation method for the Lie algebra sl_2( extbf{F}) that produces various non-classical three-dimensional Lie algebras, including the Heisenberg algebra, with potential links to non-commutative geometry.

Contribution

It applies a twisted derivation-based deformation scheme to sl_2( extbf{F}), enabling the construction of new algebraic structures beyond classical rigidity.

Findings

Deformation of sl_2( extbf{F}) into the Heisenberg algebra and other types.

Resulting algebras are quadratic and relate to geometric quadratic algebras.

Contrasts with classical deformation schemes showing flexibility of sl_2( extbf{F}).

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.