Invariant theory in universal algebra

Question

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

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

Question 1 Let $A$ be a finitely generated algebraic system in $\mathcal{K}$ and let $G$ be a finite group acting by $\mathcal{K}$-automorphisms on $A$. When is $A^G:=\{a \in A| g\cdot a=a, \forall g \in G\}$ 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 let $G$ be a finite group of automorphisms, as above. When is $A^G$ 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 varieties 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 purposes of these questions? Is something known about varieties of non-associative algebras (other than Lie)?

Answer 1

Refinement of Questions 1 and 2: Is there any general result in Universal algebra (or Model theory) that is relevant for the purposes of these questions?

I think that this is too general of a question to expect a complete and satisfying answer. Let me make a few remarks anyway.

Remark 1. The Math Subject Classification numbers for General Algebraic Systems all have the form 08XXX. The Math Subject Classification numbers for invariant theory (in the sense of this question) are 13A50 (actions of groups on commutative rings), 14L24 (geometric invariant theory), 16R30 (trace rings and invariant theory), and 16W22 (actions of groups and semigroups; invariant theory (associative rings and algebras)). If you search the MathSciNet database for papers with Primary or Secondary MSC number 08XXX and at least one Primary or Secondary MSC number from 13A50, 14L24, 16R30, or 16W22, you get exactly three papers. None of them seems to be related to this problem. You can examine the papers yourself, if you want to check this, they are:

Khrypchenko, Mykola; Novikov, Boris
Reflectors and globalizations of partial actions of groups.
J. Aust. Math. Soc. 104 (2018), no. 3, 358-379.
MSC: 18A40 (08A02 08A55 08B25 08C05 16W22)

Dokuchaev, M.
Recent developments around partial actions.
São Paulo J. Math. Sci. 13 (2019), no. 1, 195-247.
MSC: 6W22 (08A02 16S10 16S35 20C25 46L55 54H15)

Brookes, Matthew D. G. K.
Congruences on the partial automorphism monoid of a free group action.
Internat. J. Algebra Comput. 31 (2021), no. 6, 1147-1176.
MSC: 08A30 (16W22 20M18 20M20)

So, it seems that no one has combined the types of ideas mentioned in the problem statement. (Perhaps in very old papers this has been done, and the results are not reflected well in the MSC system.)

Remark 2. The original question asks about more than just papers/results on this topic, but also which universal algebraic results might be relevant. As I wrote above, I think that this is far too general a question. The field of UA is approximately 90 years old. MathSciNet lists 13790 papers in MSC 08XXX stretching back to 1931. It is possible that something in one of those papers is relevant. Today there are 536 papers in the database which have MSC Primary or Secondary number 08B20, which is the number for Free Algebras. Any of those might be relevant.

Remark 3. I think that questions of this type could have interesting answers. I could imagine posing the following project to someone.

Let Property $P(G)$ be the property of a variety $\mathcal V$ that, for a finite group $G$, the subalgebra $\mathbf{F}_{\mathcal V}(n)^G$ of the $n$-generated $\mathcal V$-free algebra $\mathbf{F}_{\mathcal V}(n)$ is also $\mathcal V$-free algebra. The project is: examine all varieties generated by $2$-element algebras and determine which of them have property $P(G)$ for some/all finite groups $G$.

It is easy to see that the variety generated by the $2$-element semilattice has the property $P(G)$ for any finite $G$. It is not too hard to see that the variety generated by the $2$-element lattice does NOT have the property $P(G)$ for the $2$-element group $G$. Perhaps by examining all the varieties generated by $2$-element algebras, some patterns might emerge. I think the project is feasible, because varieties generated by $2$-element algebras have been classified.

In order to add some math to this answer, let me close by explaining why the variety generated by the $2$-element lattice does NOT have the property $P(G)$ for the $2$-element group $G$. The variety in question is the variety of distributive lattices. The sizes of the small free algebras are $|\mathbf{F}_{\mathcal V}(0)|=0$, $|\mathbf{F}_{\mathcal V}(1)|=1$, $|\mathbf{F}_{\mathcal V}(2)|=4$, $|\mathbf{F}_{\mathcal V}(3)|=18$. Let $G$ be the group of automorphisms generated by the automorphism $\alpha$ of $\mathbf{F}_{\mathcal V}(x,y,z)$ which fixes $x$ and swaps $y$ and $z$. It is not hard to see that $|\mathbf{F}_{\mathcal V}(3)^G| = 8$, so this subalgebra of $\mathbf{F}_{\mathcal V}(3)$ cannot be a relatively free algebra. In fact, the universe of $\mathbf{F}_{\mathcal V}(3)^G$ is generated by $\{x, y\wedge z, y\vee z\}$ and consists of $\{x\wedge y\wedge z, y\wedge z, x\wedge (y\vee z), (x\wedge (y\vee z))\vee(x\wedge y), x, y\vee z, x\vee (y\wedge z), x\vee y\vee z\}$.