[Paper Review] Semisimple and $G$-equivariant simple algebras over operads

Pavel Etingof|arXiv (Cornell University)|Dec 23, 2015

Advanced Topics in Algebra1 references4 citations

TL;DR

This paper generalizes the classification of simple $G$-equivariant algebras—previously known for commutative, associative, and Lie algebras—to arbitrary linear operads. It proves that any simple $G$-equivariant $C$-algebra arises as $\mathrm{Fun}_H(G,B)$, where $H \leq G$ and $B$ is a simple $C$-algebra with an $H$-action, extending results in Deligne categories to finitely generated operads over $\mathbb{C}$.

ABSTRACT

Let $G$ be a finite group. There is a standard theorem on the classification of $G$-equivariant finite dimensional simple commutative, associative, and Lie algebras (i.e., simple algebras of these types in the category of representations of $G$). Namely, such an algebra is of the form $A={ m Fun}_H(G,B)$, where $H$ is a subgroup of $G$, and $B$ is a simple algebra of the corresponding type with an $H$-action. We explain that such a result holds in the generality of algebras over a linear operad. This allows one to extend Theorem 5.5 of arXiv:1506.07565 on the classification of simple commutative algebras in the Deligne category ${ m Rep}(S_t)$ to algebras over any finitely generated linear operad.

Motivation & Objective

• To extend the classification of $G$-equivariant simple algebras—previously known for specific types like commutative, associative, and Lie algebras—to arbitrary linear operads.
• To provide a general framework for classifying simple $C$-algebras in the Deligne category $\mathrm{Rep}(S_t)$ for transcendental $t$, using operadic structure.
• To generalize Theorem 5.5 of [S] on simple commutative algebras in $\mathrm{Rep}(S_t)$ to algebras over any finitely generated linear operad.
• To establish that such algebras are induced from simple $C$-algebras with group actions, using Frobenius reciprocity and representation-theoretic techniques.

Proposed method

• Define $C$-algebras over a linear operad $C$ over a field $F$, with structure maps $\alpha_n: C(n) \to \mathrm{Hom}_F(A^{igotimes n}, A)$ compatible with operadic composition.
• Introduce the algebra $R_A = L_A + E_A$, where $L_A$ is the image of $C(1)$ and $E_A$ is the span of all partial multiplication operators; $E_A$ is an ideal in $R_A$.
• Define a $C$-algebra as simple if it has no nontrivial $G$-invariant ideals and $E_A \neq 0$, and semisimple if it is a finite direct sum of simple $C$-algebras.
• Construct the radical $\mathrm{Rad}(A)$ as the kernel of the projection onto the maximal semisimple quotient as an $R_A$-module, showing $A$ is semisimple iff $\mathrm{Rad}(A) = 0$.
• Use Frobenius reciprocity to show that any simple $G$-equivariant $C$-algebra is isomorphic to $\mathrm{Fun}_H(G,B)$, where $H \leq G$ is the stabilizer of a minimal ideal and $B$ is a simple $C$-algebra with $H$-action.
• Prove that $H$ is unique up to conjugation and the action $\phi: H \to \mathrm{Aut}(B)$ is unique up to conjugation in $\mathrm{Aut}(B)$, establishing the classification.

Experimental results

Research questions

• RQ1 Can the classification of $G$-equivariant simple algebras be extended beyond commutative, associative, and Lie algebras to algebras over arbitrary linear operads?
• RQ2 How does the structure of $R_A = L_A + E_A$ and the ideal $E_A$ constrain the classification of simple $G$-equivariant $C$-algebras?
• RQ3 What is the role of the radical $\mathrm{Rad}(A)$ in determining semisimplicity and in the quotient construction of $\overline{A}$?
• RQ4 To what extent does the Frobenius reciprocity argument generalize to operadic algebras, particularly in the context of Deligne categories?
• RQ5 Can Theorem 5.5 of [S] on simple commutative algebras in $\mathrm{Rep}(S_t)$ be extended to algebras over finitely generated linear operads?

Key findings

• Any simple $G$-equivariant $C$-algebra is isomorphic to $\mathrm{Fun}_H(G,B)$ for some subgroup $H \leq G$ and a simple $C$-algebra $B$ with an $H$-action.
• The subgroup $H$ is uniquely determined up to conjugation in $G$, and the homomorphism $\phi: H \to \mathrm{Aut}(B)$ is unique up to conjugation in $\mathrm{Aut}(B)$.
• The radical $\mathrm{Rad}(A)$ vanishes if and only if $A$ is semisimple, and $A/\mathrm{Rad}(A)$ is always a semisimple $C$-algebra.
• The classification holds verbatim for infinite groups or affine algebraic groups, with the same construction via finite index subgroups and homomorphisms to $\mathrm{Aut}(B)$.
• The results extend Theorem 5.5 of [S] to algebras over any finitely generated linear operad over $\mathbb{C}$, with $B$ being an interpolation of $G \times S_{n-k}$-equivariant simple algebras.
• In the case of associative unital algebras, $B = \mathrm{End}(V)$ for an object $V$ in $\mathrm{Rep}(S_t)$, and for Lie algebras, $B = \mathfrak{sl}(V), \mathfrak{o}(V)$, or $\mathfrak{sp}(V)$, depending on the form on $V$.