# Invariant theory in universal algebra

Let $$\mathcal{L}$$ be a finite first-order language with no relation symbols, and $$\mathcal{K}:=\mathcal{V}(\Theta)$$ a variety in this language definited by a set of identities $$\Theta$$.

My questions are motivated, of course, by Noether's Theorem in invariant theory, and Chevalley-Shephard-Todd theorem (which can be seem as statements in the variety of commutative associative algebras) for algebraic systems (using Malcev terminology).

Question 1 Let $$A$$ be a finitely generated algebraic system in $$\mathcal{K}$$, and $$G$$ a finite group acting by $$\mathcal{K}$$-automorphisms on $$A$$. When is $$A^G:=\{a \in A| g.a=a, \forall g \in G\}$$, again finitely generated? A subcase of important interest is when $$A$$ is a relatively free algebraic system of $$\mathcal{K}$$.

Question 2 Let $$A$$ be a finitely generated and relatively free algebraic system in $$\mathcal{K}$$ and $$G$$ be a finite group of automorphisms as above. When is $$A^G$$ is a (possibly infinitely generated) relatively free algebraic system of $$\mathcal{K}$$ itself?

For varieties of associative algebras the work on Questions 1 and 2 has reached a very mature form (cf. Formanek, Noncommutative invariant theory, MR0810646). There has been an intensive study of these questions for varieties of Lie algebras by a number of people (V. Drenksy, V. Petrogradskii, etc) but I know of no work for other important variaties of non-associative algebras (such as Jordan algebras, alternative algebras, Malcev algebras, etc), and also nothing about general algebraic systems.

Refinement of Questions 1 and 2 Is there any general result in Universal algebra (or Model theory) that is relevant for the pourposes of these questions? Is something known about variaties of non-associative algebras (other than Lie)?

• Can you please explain what do you mean by the symbol $A^G$? – Jakub Opršal Jul 21 at 15:03
• @JakubOpršal $A^G = \{ a \in A| g a = a, \forall g \in G \}$. Thanks, I will edit the question for clarity – jg1896 Jul 21 at 15:33