The commuting variety of $\mathfrak{pgl}_n$
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915763851362304 |
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| author | Roman, Vlad |
| author_facet | Roman, Vlad |
| contents | We are considering the commuting variety of the Lie algebra $\mathfrak{pgl}_n$ over an algebraically closed field of characteristic $p >0$, namely the set of pairs $ \{ (A,B) \in \mathfrak{pgl}_n \times \mathfrak{pgl}_n \mid [A,B]=0 \} $. We prove that if $n=pr$, then there are precisely two irreducible components, of dimensions $n^2+r-1$ and $n^2+n-2$. We also prove that the variety $\{ (x,y) \in GL_n(k) \times GL_n(k) \mid [x,y]=ζI \}$ is irreducible of dimension $n^2 +n/d$, where $ζ$ is a root of unity of order $d$ with $d$ dividing $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11106 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The commuting variety of $\mathfrak{pgl}_n$ Roman, Vlad Algebraic Geometry We are considering the commuting variety of the Lie algebra $\mathfrak{pgl}_n$ over an algebraically closed field of characteristic $p >0$, namely the set of pairs $ \{ (A,B) \in \mathfrak{pgl}_n \times \mathfrak{pgl}_n \mid [A,B]=0 \} $. We prove that if $n=pr$, then there are precisely two irreducible components, of dimensions $n^2+r-1$ and $n^2+n-2$. We also prove that the variety $\{ (x,y) \in GL_n(k) \times GL_n(k) \mid [x,y]=ζI \}$ is irreducible of dimension $n^2 +n/d$, where $ζ$ is a root of unity of order $d$ with $d$ dividing $n$. |
| title | The commuting variety of $\mathfrak{pgl}_n$ |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2402.11106 |