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On the Linear Algebraic Monoids Associated to Congruence of Matrices

Himadri Mukherjee, Gunja Sachdeva

Abstract

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.

On the Linear Algebraic Monoids Associated to Congruence of Matrices

Abstract

This paper discusses the generalized congruence equation , for over any field , through the action of monoid . We have completely characterized for what matrices , the monoid is a Lie group. We have given the structure of the Lie group and , and their Lie algebras when is nilpotent matrix of nilpotency . In this case, we have also proved that the invariants of for any , and for even, are finitely generated.
Paper Structure (26 sections, 66 theorems, 66 equations)

This paper contains 26 sections, 66 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Notations and terminologies
  3. The Congruence Monoid
  4. Congruence matrices
  5. The structure of $Sol_{A_n}$
  6. Example.
  7. Lie algebra of $\text{Sol}_{A_n}$
  8. Lie group of $\text{Sol}_{A_n}$
  9. The case of $n \equiv 1(\text{mod}\ 2)$
  10. The case of $n \equiv 0(\text{mod } 2)$
  11. $\text{Sol}_{A_n}$ as a semi-direct product
  12. The structure of $Sol_{A^2_n}$ with $n \geq 4$
  13. Lie algebra of $\text{Sol}_{A^2_n}$
  14. Case 1: $\bf{n \equiv 0 (\text{mod } 4)}; \ n \geq 4$
  15. Case 2: $\bf{n \equiv 2 (\text{mod } 4)}; \ n \geq 6$
  16. ...and 11 more sections

Key Result

Lemma 3.1

Let $V$ be a finite dimensional vector space over $\mathbb{C}$, and $N$ be the matrix ${\footnotesize}.$ The following are true:

Theorems & Definitions (72)

  • Lemma 3.1
  • Theorem 3.2
  • Example 3.1
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Example 3.2
  • Lemma 3.7
  • Theorem 3.8
  • ...and 62 more