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

Hans Cuypers

Abstract

Let $Γ=(\mathcal{V},\mathcal{E})$ be a graph, whose vertices $v\in \mathcal{V}$ are colored black and white and labeled with invertible elements $λ_v$ from a commutative and associative ring $R$ containing $\pm 1$. Then we consider the associative algebra $\mathfrak{C}(Γ)$ with identity element $\mathbf{1}$ generated by the elements of $\mathcal{V}$ such that for all $v,w\in \mathcal{V}$ we have \[\begin{array}{lll}v^2 &=λ_v\mathbf{1}&\textrm{if } v \textrm{ is white}, v^2 &=-λ_v\mathbf{1}&\textrm{if } v \textrm{ is black}, vw+wv&=0&\textrm{if } \{v,w\}\in \mathcal{E}, vw-wv&=0&\textrm{if } \{v,w\}\not\in \mathcal{E}.\\ \end{array}\] If $Γ$ is the complete graph, $\mathfrak{C}(Γ)$ 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 $\mathfrak{C}(Γ)$ in several interesting examples. As the algebra $\mathfrak{C}(Γ)$ is associative, we can also consider the corresponding Lie algebra and some of its subalgebras. In case $λ_v=1$ 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.

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

Abstract

Let be a graph, whose vertices are colored black and white and labeled with invertible elements from a commutative and associative ring containing . Then we consider the associative algebra with identity element generated by the elements of such that for all we have If is the complete graph, 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 over the field . Using this description, we determine the isomorphism type of in several interesting examples. As the algebra is associative, we can also consider the corresponding Lie algebra and some of its subalgebras. In case for all , and all vertices are black, we find that the elements satisfy the following relations In case is a field of characteristic , 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.

Paper Structure

This paper contains 12 sections, 32 theorems, 89 equations, 4 figures, 6 tables.

Table of Contents

  1. Introduction
  2. A class of algebras obtained from bilinear forms over $\mathbb{F}_2$
  3. From relations to algebra
  4. Special quasi-Clifford algebras
  5. The structure of the algebras
  6. Special quasi-Clifford algebras obtained from graphs of type $A$, $D$ and $E$
  7. Involutions
  8. Lie and Jordan algebras
  9. Lie algebras obtained from graphs
  10. A characterization of Lie algebras associated to a graph
  11. Spin representations and compact subalgebras of Kac-Moody algebras
  12. Jordan Algebras

Key Result

Theorem 1.1

Let $\Gamma=(\mathcal{V},\mathcal{E},\lambda)$ be a labeled graph. Then, for every bilinear form $g$ on $V$ with $g(v,w)+g(w,v)=f_\Gamma(v,w)$ for $v, w\in \mathcal{V}$, we find $\mathfrak{C}(\Gamma)$ to be isomorphic to the $R$-algebra $\mathfrak{C}(V,g,\lambda)$. $\blacktriangleleft$$\blacktriangl

Figures (4)

  • Figure 1: Graph of type $A_n$ obtained by changing the generators.
  • Figure 2: Graphs of type $D_n$.
  • Figure 3: Graphs of type $E_n$.
  • Figure 4: The nine forbidden subgraphs for a line graph.

Theorems & Definitions (66)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Corollary 3.3
  • Proposition 4.1
  • ...and 56 more