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Problems in algebra inspired by universal algebraic geometry. (Russian, English) Zbl 1072.08002
Fundam. Prikl. Mat. 10, No. 3, 181-197 (2004); translation in J. Math. Sci., New York 139, No. 4, 6780-6791 (2006).
For a given variety of algebras $$V$$ and an arbitrary algebra $$A\in V$$ it is natural to consider the algebraic geometry of $$V$$ over $$A$$. From this point of view classical algebraic geometry deals with the variety of all commutative associative algebras. Also geometry of groups has a natural interpretation in this language. Many interesting problems, such as Tarski’s well-known problem concerning the logic of a free group, appear naturally in this context. Moreover, a general consideration of algebraic geometry in different varieties of algebras gives rise to many new problems in algebra and algebraic geometry. The author considers a special category invariant $$K_V(A)$$ of the constructed geometry. One of the natural problems that arise in this context is the following: What are the conditions on algebras $$A_1$$, $$A_2$$ from a variety $$V$$ that guarantee the coincidence of corresponding algebraic geometries? The author considers two variants of such a coincidence: 1) $$K_V(A_1)$$ is isomorphic to $$K_V(A_2)$$; 2) corresponding categories are equivalent. This problem is closely related to the problem of the description of automorphism and autoequivalence groups of categories of free algebras of given varieties. The paper contains a big amount of interesting open problems. Among them there are: 1) To find conditions under which a wreath product is geometrically Noetherian. 2) To find conditions under which a wreath product is logically Noetherian but not geometrically Noetherian. 3) Is it possible that a wreath product is not logically Noetherian? 4) Is it true that any free associative algebra (free Lie algebra) is geometrically (logically) Noetherian? 5) Is it true that there exists a continuum of different $$k$$-generated simple Lie algebras for a fixed $$k$$? 6) To investigate the group $$\text{Aut(Eng}(W(X)))$$, where $$W(X)$$ is a free Lie algebra on the finite set $$X$$. 7) To investigate automorphism and autoequivalence groups of categories of free algebras of certain special subvarieties of the variety of all Lie algebras.

##### MSC:
 08C05 Categories of algebras 14A99 Foundations of algebraic geometry 08B99 Varieties 17B01 Identities, free Lie (super)algebras