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Invariant theory and coefficient algebras of Lie algebras

Yin Chen, Runxuan Zhang

TL;DR

The paper investigates coefficient algebras B_g(V) arising from the characteristic polynomial varphi_g(V) of a Lie algebra g acting on a finite-dimensional representation V, revealing deep ties to classical invariant theory. For the standard symmetric powers S^d(C^n), it establishes: (a) B_{u_n(C)}(S^d(C^n)) is the ring of symmetric polynomials C[x_1,...,x_n]^{S_n}; (b) B_{gl_n(C)}(S^d(C^n)) coincides with the GL_n(C)-invariant ring C[gl_n(C)]^{GL_n(C)} ≅ C[s_1,...,s_n]; and (c) B_{sl_n(C)}(S^d(C^n)) is generated by the traces tr_2,...,tr_n, i.e., C[tr_2,...,tr_n] = C[SL_n(C) invariants]. Consequently, the SL_n(C)–standard representation’s characteristic polynomial is explicitly determined in terms of these trace invariants. The work illuminates how Lie-theoretic characteristic polynomials encode fundamental invariant-theoretic data and provides concrete computations and proofs for these foundational algebras, enabling applications to the structure of polynomial invariants and concrete characteristic polynomials.

Abstract

The coefficient algebra of a finite-dimensional Lie algebra on a finite-dimensional representation is defined as the subalgebra generated by all coefficients of the corresponding characteristic polynomial. We explore connections between classical invariant theory and the coefficient algebras of finite-dimensional complex Lie algebras on some representations. Specifically, we prove that with respect to any symmetric power of the standard representation: (a) the coefficient algebra of the upper triangular solvable complex Lie algebra is isomorphic to the ring of symmetric polynomials; (b) the coefficient algebra of the general linear complex Lie algebra is isomorphic to the invariant ring of the general linear group with the conjugacy action on the full space of matrices; and (c) the coefficient algebra of the special linear complex Lie algebra can be generated by classical trace functions. As an application, we determine the characteristic polynomial of the special linear complex Lie algebra on its standard representation.

Invariant theory and coefficient algebras of Lie algebras

TL;DR

The paper investigates coefficient algebras B_g(V) arising from the characteristic polynomial varphi_g(V) of a Lie algebra g acting on a finite-dimensional representation V, revealing deep ties to classical invariant theory. For the standard symmetric powers S^d(C^n), it establishes: (a) B_{u_n(C)}(S^d(C^n)) is the ring of symmetric polynomials C[x_1,...,x_n]^{S_n}; (b) B_{gl_n(C)}(S^d(C^n)) coincides with the GL_n(C)-invariant ring C[gl_n(C)]^{GL_n(C)} ≅ C[s_1,...,s_n]; and (c) B_{sl_n(C)}(S^d(C^n)) is generated by the traces tr_2,...,tr_n, i.e., C[tr_2,...,tr_n] = C[SL_n(C) invariants]. Consequently, the SL_n(C)–standard representation’s characteristic polynomial is explicitly determined in terms of these trace invariants. The work illuminates how Lie-theoretic characteristic polynomials encode fundamental invariant-theoretic data and provides concrete computations and proofs for these foundational algebras, enabling applications to the structure of polynomial invariants and concrete characteristic polynomials.

Abstract

The coefficient algebra of a finite-dimensional Lie algebra on a finite-dimensional representation is defined as the subalgebra generated by all coefficients of the corresponding characteristic polynomial. We explore connections between classical invariant theory and the coefficient algebras of finite-dimensional complex Lie algebras on some representations. Specifically, we prove that with respect to any symmetric power of the standard representation: (a) the coefficient algebra of the upper triangular solvable complex Lie algebra is isomorphic to the ring of symmetric polynomials; (b) the coefficient algebra of the general linear complex Lie algebra is isomorphic to the invariant ring of the general linear group with the conjugacy action on the full space of matrices; and (c) the coefficient algebra of the special linear complex Lie algebra can be generated by classical trace functions. As an application, we determine the characteristic polynomial of the special linear complex Lie algebra on its standard representation.

Paper Structure

This paper contains 14 sections, 14 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Characteristic polynomials and coefficient algebras
  4. Main results
  5. Layout and conventions
  6. $\germ{sl}_2(\mathbb{C})$ and Nilpotent Lie Algebras
  7. The Lie algebra $\germ{sl}_2(\mathbb{C})$
  8. Characteristic polynomials on self-dual representations
  9. Nilpotent Lie algebras
  10. Coefficient Algebras of $\germ{u}_n(\mathbb{C})$
  11. Actions on $S^d(V)$
  12. Proof of Theorem \ref{['thm1']}
  13. Examples
  14. Coefficient Algebras of $\germ{gl}_n(\mathbb{C})$ and $\germ{sl}_n(\mathbb{C})$

Key Result

Theorem 1.2

The coefficient algebra $B_{\germ{u}_n(\mathbb{C})}(S^d(\mathbb{C}^n))$ is isomorphic to the ring of symmetric polynomials $\mathbb{C}[x_1,x_2,\dots,x_n]^{S_n}$, where $S_n$ denotes the symmetric group of degree $n$ with the standard permutation action on $\{1,2,\dots,n\}$.

Theorems & Definitions (37)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Corollary 2.3
  • proof
  • Remark 2.4
  • ...and 27 more