On the Linear Algebraic Monoids Associated to Congruence of Matrices

Abstract

This paper discusses the generalized congruence equation $X^{t}AX=B$, for $X\in M_n(k)$ over any field $k$, through the action of monoid $So{l}_A\times So{l}_B:=\{X|X^{t}AX=A\}\times \{X|X^{t}BX=B\}$. We have completely characterized for what matrices $A$, the monoid $So{l}_A$ is a Lie group. We have given the structure of the Lie group $So{l}_A$ and $So{l}_{A^2}$, 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 $So{l}_A$ for any $n$, and $So{l}_{A^2}$ for $n$ even, are finitely generated.