Quasi-Clifford algebras, Quadratic forms over $\mathbb{F}_2$, and Lie Algebras

Abstract

Let $\Gamma=(\mathcal{V},\mathcal{E})$ be a graph, whose vertices $v\in \mathcal{V}$ are colored black and white and labeled with invertible elements $\lambda_v$ from a commutative and associative ring $R$ containing $\pm 1$. Then we consider the associative algebra $\mathfrak{C}(\Gamma)$ with identity element $\mathbf{1}$ generated by the elements of $\mathcal{V}$ such that for all $v,w\in \mathcal{V}$ we have \[\begin{array}{lll}v^2 &=\lambda_v\mathbf{1}&\textrm{if } v \textrm{ is white}, v^2 &=-\lambda_v\mathbf{1}&\textrm{if } v \textrm{ is black}, vw+wv&=0&\textrm{if } \{v,w\}\in \mathcal{E}, vw-wv&=0&\textrm{if } \{v,w\}\not\in \mathcal{E}.\\ \end{array}\] If $\Gamma$ is the complete graph, $\mathfrak{C}(\Gamma)$ is a Clifford algebra, otherwise it is a so-called quasi-Clifford algebra. We describe this algebra as a twisted group algebra with the help of a quadratic space $(V,Q)$ over the field $\mathbb{F}_2$. Using this description, we determine the isomorphism type of $\mathfrak{C}(\Gamma)$ in several interesting examples. As the algebra $\mathfrak{C}(\Gamma)$ is associative, we can also consider the corresponding Lie algebra and some of its subalgebras. In case $\lambda_v=1$ for all $v\in \mathcal{V}$, and all vertices are black, we find that the elements $v,w\in \mathcal{V}$ satisfy the following relations $$\begin{array}{lll} [v,w]&=0&\textrm{if } \{v,w\}\not\in \mathcal{E}, {[v,[v,w]]}&=-w&\textrm{if } \{v,w\}\in \mathcal{E}.\\ \end{array}$$ In case $R$ is a field of characteristic $0$, we identify these algebras as quotients of the compact subalgebras of Kac-Moody Lie algebras and prove that they admit a so-called generalized spin representation.