On the Linear Algebraic Monoids Associated to Congruence of Matrices
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866910317306445824 |
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| author | Mukherjee, Himadri Sachdeva, Gunja |
| author_facet | Mukherjee, Himadri Sachdeva, Gunja |
| contents | This paper discusses the generalized congruence equation $X^tAX=B$, for $X \in M_n(k)$ over any field $k$, through the action of monoid $Sol_A \times Sol_B := \{X \ | \ X^tAX = A\} \times \{X \ | \ X^tBX = B\}$. We have completely characterized for what matrices $A$, the monoid $Sol_A$ is a Lie group. We have given the structure of the Lie group $Sol_A$ and $Sol_{A^2}$, and their Lie algebras when $A$ is $n \times n$ nilpotent matrix of nilpotency $n$. In this case, we have also proved that the invariants of $Sol_A$ for any $n$, and $Sol_{A^2}$ for $n$ even, are finitely generated. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_13958 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Linear Algebraic Monoids Associated to Congruence of Matrices Mukherjee, Himadri Sachdeva, Gunja Rings and Algebras Algebraic Geometry Group Theory Representation Theory 22E20, 22E25, 11E99, 14L10 This paper discusses the generalized congruence equation $X^tAX=B$, for $X \in M_n(k)$ over any field $k$, through the action of monoid $Sol_A \times Sol_B := \{X \ | \ X^tAX = A\} \times \{X \ | \ X^tBX = B\}$. We have completely characterized for what matrices $A$, the monoid $Sol_A$ is a Lie group. We have given the structure of the Lie group $Sol_A$ and $Sol_{A^2}$, and their Lie algebras when $A$ is $n \times n$ nilpotent matrix of nilpotency $n$. In this case, we have also proved that the invariants of $Sol_A$ for any $n$, and $Sol_{A^2}$ for $n$ even, are finitely generated. |
| title | On the Linear Algebraic Monoids Associated to Congruence of Matrices |
| topic | Rings and Algebras Algebraic Geometry Group Theory Representation Theory 22E20, 22E25, 11E99, 14L10 |
| url | https://arxiv.org/abs/2305.13958 |