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| Format: | Preprint |
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2021
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| Online Access: | https://arxiv.org/abs/2105.08637 |
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| _version_ | 1866910091631919104 |
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| author | Cuypers, Hans |
| author_facet | Cuypers, Hans |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_08637 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Quasi-Clifford algebras, Quadratic forms over $\mathbb{F}_2$, and Lie Algebras Cuypers, Hans Rings and Algebras 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. |
| title | Quasi-Clifford algebras, Quadratic forms over $\mathbb{F}_2$, and Lie Algebras |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2105.08637 |