[Paper Review] Constructive noncommutative invariant theory

M. Domokos, Vesselin Drensky|arXiv (Cornell University)|Nov 15, 2018

Advanced Topics in Algebra16 references3 citations

TL;DR

This paper establishes a constructive method to generate invariants in the universal enveloping algebra of a finite-dimensional Lie algebra under a group action, by lifting homogeneous generators from the symmetric algebra via the canonical bijection. It proves a constructive Hilbert-Nagata theorem with explicit degree bounds for invariants in Lie nilpotent relatively free associative algebras with reductive group actions.

ABSTRACT

The problem of finding generators of the subalgebra of invariants under the action of a group of automorphisms of a finite dimensional Lie algebra on its universal enveloping algebra is reduced to finding homogeneous generators of the same group acting on the symmetric tensor algebra of the Lie algebra. This process is applied to prove a constructive Hilbert-Nagata Theorem (including degree bounds) for the algebra of invariants in a Lie nilpotent relatively free associative algebra endowed with an action induced by a representation of a reductive group.

Motivation & Objective

• To provide a constructive method for generating invariants in the universal enveloping algebra $U(L)^G$ of a finite-dimensional Lie algebra $L$ under a group $G$ of automorphisms.
• To reduce the problem of finding $G$-invariant generators in $U(L)$ to the commutative case via the canonical bijection $\omega: S(L) \to U(L)$.
• To establish a constructive Hilbert-Nagata theorem with explicit degree bounds for invariants in relatively free associative algebras satisfying Lie nilpotency identities.
• To extend classical invariant theory to noncommutative settings by leveraging representation theory and Schur functors in the context of reductive group actions.

Proposed method

• Use the canonical bijection $\omega: S(L) \to U(L)$, a $K$-vector space isomorphism, to transfer homogeneous generators from the symmetric algebra $S(L)^G$ to the universal enveloping algebra $U(L)^G$.
• Leverage the $GL(V)$-equivariance of the canonical surjections $T(V) \twoheadrightarrow F(\mathfrak{R},V) \twoheadrightarrow S(V)$ to relate invariants across tensor, relatively free, and symmetric algebras.
• Apply Pieri’s rules and the Cauchy identity to bound the height of irreducible $GL(E)$-modules in the decomposition of $F(\mathfrak{R},U + W \otimes E)_d$, ensuring that only partitions $\mu$ with $\mathrm{ht}(\mu) \leq nh$ appear.
• Use the fact that $G$-invariants in $F(\mathfrak{R},U + W \otimes E)$ are $GL(E)$-submodules, and that highest weight vectors for $S^\mu(E)$ with $\mathrm{ht}(\mu) \leq nh$ must lie in $F(\mathfrak{R},U + nhW)^G$.
• Prove that $F(\mathfrak{R},U + W \otimes E)^G$ is generated by $F(\mathfrak{R},U + nhW)^G$ as a $GL(E)$-module, using the action of $GL(K^m)$ on these invariants.
• Establish that the subalgebra of invariants $F(\mathfrak{R},V)^G$ is finitely generated with explicit degree bounds via the height bound $h = h(\mathfrak{R})$ of the variety $\mathfrak{R}$.

Experimental results

Research questions

• RQ1 Can the generators of the invariant subalgebra $U(L)^G$ in the universal enveloping algebra be constructed explicitly from generators of $S(L)^G$?
• RQ2 What is the relationship between the invariant theory of $U(L)$ and that of the symmetric algebra $S(L)$ under a group action?
• RQ3 What degree bounds can be established for generators of $F(\mathfrak{R},V)^G$ in relatively free associative algebras satisfying Lie nilpotency identities?
• RQ4 How does the structure of $GL(V)$-representations constrain the form of $G$-invariant elements in $F(\mathfrak{R},V)$ for reductive $G$?
• RQ5 To what extent can the classical Hilbert-Nagata theorem be made constructive in the noncommutative setting?

Key findings

• Theorem 1.1 establishes that if $\{f_\lambda\}$ is a homogeneous generating system of $S(L)^G$, then $\{\omega(f_\lambda)\}$ generates $U(L)^G$, providing a constructive lift from the commutative to the noncommutative setting.
• The paper proves a constructive Hilbert-Nagata theorem for $F(\mathfrak{R},V)^G$ when $\mathfrak{R} \subseteq \mathfrak{N}_p$, showing that the algebra of invariants is finitely generated with explicit degree bounds.
• The degree of any homogeneous generator of $F(\mathfrak{R},V)^G$ is bounded by $d \leq nh$, where $n = \dim(W)$ and $h = h(\mathfrak{R})$ is the height of the variety $\mathfrak{R}$, under the assumption that $\mathfrak{R}$ satisfies a Lie nilpotency identity.
• The proof relies on representation-theoretic techniques: the decomposition of $F(\mathfrak{R},V)_d$ into Schur functors $S^\lambda(V)$, with $\mathrm{ht}(\lambda) \leq h$, and the subsequent bound $\mathrm{ht}(\mu) \leq nh$ on the irreducible components in the $GL(E)$-decomposition.
• The invariant subalgebra $F(\mathfrak{R},U + W \otimes E)^G$ is contained in the $GL(E)$-submodule generated by $F(\mathfrak{R},U + nhW)^G$, showing that invariants over a larger space are generated from those over a finite-dimensional base.
• The result generalizes Weyl’s polarization theorem (when $\mathfrak{R}$ is commutative, so $h=1$) to noncommutative settings, providing a noncommutative analog of classical invariant theory with effective bounds.