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On $R$-equivalence of Automorphism Groups of Associative Algebras

Dibyendu Das

TL;DR

This work investigates the rationality and R-equivalence properties of the automorphism group G_A = Aut_k(A)^0 of a finite-dimensional unital associative algebra A over a field k, with a focus on split finite-dimensional and split local algebras. The authors establish a key reduction: G_A is R-trivial exactly when the subgroup G_{A,A_s} fixing a semisimple subalgebra A_s is R-trivial, and they derive practical sufficient conditions (e.g., J^2 = 0, dim(J/J^2) ≤ 5, or rank(G_{A,A_s}) = dim(J/J^2)) that guarantee R-triviality; they also prove that Aut_k(A) nilpotent implies G_A is rational. By reducing to split local algebras, they relate G_A to Im(Φ_A): G_A → GL(J/J^2), and show that for such A, G_A is stably birational to Im(Φ_A), turning the problem into the study of stabilizers of radical-graded data via quiver-and-relations methods. In the commutative local case, they express Aut_k(A) as Im(Φ_A) ⋉ Ker(Φ_A) with Ker(Φ_A) unipotent and Im(Φ_A) often captured by stabilizers of polynomials, yielding criteria that connect rationality to the stabilizer of a defining polynomial f. Despite many positive results (including many cases where G_A is rational or R-trivial), the authors construct explicit counterexamples with large Jacobson radical quotients (dim(J/J^2) ≥ 6) where G_A is not R-trivial, and they demonstrate that R-triviality is not forced in general. The paper thus provides a comprehensive framework for analyzing R-equivalence in Aut_k(A) using Morita invariance, graded- by-radical structure, and stabilizer techniques, and highlights several open questions about necessary conditions for R-triviality and rationality in broader classes of algebras.

Abstract

Let $A$ be a finite-dimensional associative $k$-algebra with identity. The primary aim of this paper is to study the rationality properties of the group of all $k$-algebra automorphisms of $A$, as an affine algebraic group over an arbitrary field $k$. We investigate mainly the $R$-equivalence property of the identity component of $\mathrm{Aut}_{k}(A)$ over a perfect field $k$.

On $R$-equivalence of Automorphism Groups of Associative Algebras

TL;DR

This work investigates the rationality and R-equivalence properties of the automorphism group G_A = Aut_k(A)^0 of a finite-dimensional unital associative algebra A over a field k, with a focus on split finite-dimensional and split local algebras. The authors establish a key reduction: G_A is R-trivial exactly when the subgroup G_{A,A_s} fixing a semisimple subalgebra A_s is R-trivial, and they derive practical sufficient conditions (e.g., J^2 = 0, dim(J/J^2) ≤ 5, or rank(G_{A,A_s}) = dim(J/J^2)) that guarantee R-triviality; they also prove that Aut_k(A) nilpotent implies G_A is rational. By reducing to split local algebras, they relate G_A to Im(Φ_A): G_A → GL(J/J^2), and show that for such A, G_A is stably birational to Im(Φ_A), turning the problem into the study of stabilizers of radical-graded data via quiver-and-relations methods. In the commutative local case, they express Aut_k(A) as Im(Φ_A) ⋉ Ker(Φ_A) with Ker(Φ_A) unipotent and Im(Φ_A) often captured by stabilizers of polynomials, yielding criteria that connect rationality to the stabilizer of a defining polynomial f. Despite many positive results (including many cases where G_A is rational or R-trivial), the authors construct explicit counterexamples with large Jacobson radical quotients (dim(J/J^2) ≥ 6) where G_A is not R-trivial, and they demonstrate that R-triviality is not forced in general. The paper thus provides a comprehensive framework for analyzing R-equivalence in Aut_k(A) using Morita invariance, graded- by-radical structure, and stabilizer techniques, and highlights several open questions about necessary conditions for R-triviality and rationality in broader classes of algebras.

Abstract

Let be a finite-dimensional associative -algebra with identity. The primary aim of this paper is to study the rationality properties of the group of all -algebra automorphisms of , as an affine algebraic group over an arbitrary field . We investigate mainly the -equivalence property of the identity component of over a perfect field .
Paper Structure (6 sections, 63 theorems, 53 equations)

This paper contains 6 sections, 63 theorems, 53 equations.

Table of Contents

  1. Introduction and Main results
  2. Preliminaries
  3. Review of Pollack's Results over Perfect fields
  4. $R$-Equivalence and $R$-triviality
  5. Finite Dimensional Commutative Algebras
  6. Counter-Example to $R$-triviality of $G_A$

Key Result

Theorem A

$(=\text{Theorem }Theorem 4.20)$ Let $A$ be a split finite-dimensional associative algebra over a perfect field $k$ with $A_s$ a semisimple subalgebra of $A$ such that $A= A_s\oplus J$ (vector space decomposition) and $G_{A, A_s}\coloneqq\{\sigma\in G_A: \sigma(a)=a \quad \forall a\in A_s\}$. Then, Then, $\mathrm{Aut}_k(A)^{0}\coloneqq G_A$ is $R$-trivial.

Theorems & Definitions (133)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem F
  • Theorem G
  • Theorem H
  • Theorem 2.1
  • Corollary 2.2
  • ...and 123 more