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Aut-stable subspaces of Grassmann algebras

Mithat Konuralp Demir, Zahra Nazemian

TL;DR

The paper analyzes Aut-stable subspaces in algebras, focusing on Grassmann (exterior) algebras, and extends prior results to classify Aut-stable subspaces and subalgebras for finite generators. It leverages the automorphism group structure \mathrm{Aut}_{alg}(\mathcal{E}) = N_1 \rtimes F_0, the center Z(\mathcal{E}), and the commutator-generated subalgebra A_{com} to develop a framework for invariance under all alg-automorphisms. The main contribution is an explicit, complete classification: Aut-stable subspaces have forms (a)-(d) as stated, and Aut-stable subalgebras are precisely of the form \mathbb{k}+B' with B' Aut-stable; these results illuminate the interaction between grading, centers, and automorphisms in exterior algebras. The work connects invariant-theoretic considerations with concrete automorphism-driven structure in Grassmann algebras and raises questions for countably infinite generators.

Abstract

Recently, the concept of Aut-stable subspaces has played an important role in the characterization of polynomial rings, a topic that remains a challenging problem in algebraic geometry (see [8]). It turns out that polynomial rings with more than two variables do not have any Aut-stable subspaces over an algebraically closed field of characteristic zero [7]. In this work, we characterize all Aut-stable subspaces and Aut-stable subalgebras of Grassmann algebras.

Aut-stable subspaces of Grassmann algebras

TL;DR

The paper analyzes Aut-stable subspaces in algebras, focusing on Grassmann (exterior) algebras, and extends prior results to classify Aut-stable subspaces and subalgebras for finite generators. It leverages the automorphism group structure \mathrm{Aut}_{alg}(\mathcal{E}) = N_1 \rtimes F_0, the center Z(\mathcal{E}), and the commutator-generated subalgebra A_{com} to develop a framework for invariance under all alg-automorphisms. The main contribution is an explicit, complete classification: Aut-stable subspaces have forms (a)-(d) as stated, and Aut-stable subalgebras are precisely of the form \mathbb{k}+B' with B' Aut-stable; these results illuminate the interaction between grading, centers, and automorphisms in exterior algebras. The work connects invariant-theoretic considerations with concrete automorphism-driven structure in Grassmann algebras and raises questions for countably infinite generators.

Abstract

Recently, the concept of Aut-stable subspaces has played an important role in the characterization of polynomial rings, a topic that remains a challenging problem in algebraic geometry (see [8]). It turns out that polynomial rings with more than two variables do not have any Aut-stable subspaces over an algebraically closed field of characteristic zero [7]. In this work, we characterize all Aut-stable subspaces and Aut-stable subalgebras of Grassmann algebras.

Paper Structure

This paper contains 2 sections, 13 theorems, 61 equations.

Table of Contents

  1. Basic Notions and results
  2. $\mathop{\rm Aut}\nolimits$-stable subspaces (subalgebras) of $\mathcal{E}$

Key Result

Theorem 2

ZJLM Suppose $\Bbbk$ is an algebraically closed field of characteristic zero and $A$ is a ${\Bbbk}$-algebra. Then $A$ is isomorphic to $\Bbbk[z_1,\cdots,z_m]$ for some integer $m\geq 2$ if and only if the following two conditions hold.

Theorems & Definitions (23)

  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 1.1
  • Remark 1.2
  • Proposition 1.3
  • proof
  • Definition 1.4
  • Lemma 1.6
  • proof
  • ...and 13 more