Word equations in simple groups and polynomial equations in simple algebras

Abstract

To Kolya Vavilov, friend and colleague, on the occasion of the 60th anniversary Abstract. We give a brief survey of recent results on word maps on simple groups and polynomial maps on simple associative and Lie algebras. Our focus is on parallelism between these theories, allowing one to state many new open problems and giving new ways for solving older ones.

Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture

“…This conjecture is still widely open, there being only several partial results, see [6,10,45,46,47,65]. It is a special case of the conjecture in [57,Question 2]. Problem 1.…”

“…Different aspects of word maps are considered in a vast and extended literature; we refer to the papers [7,8,9,20,21,45,46,47,57,72,97,98,99,101] for details, surveys and further explanations. Waring type questions for rings were considered by Matei Brešar [17].…”

“…Unfortunately we saw such an example (Example 5.14, although its degree must be at least 5 byŠpenko [103, Lemma 7.4]). More general questions about surjectivity of word maps in groups and polynomials in algebras are considered in[57].…”

Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics

“…Some counterparts of Waring type properties discussed above can be formulated for maps of matrix algebras induced by associative noncommutative polynomials, see [61] for a survey.…”

Some New Parallels Between Groups and Lie Algebras, or What Can Be Simpler than Multiplication Table?

“…There are several cases where each element of G is a single commutator: G = A ∞ is the infinite alternating group (Ore [70]); G = G(k) is the group of k-points of a semisimple adjoint linear algebraic group G over an algebraically closed field k (Rimhak Ree [74]); and G is the automorphism group of some "nice" topological or combinatorial object (e.g., the Cantor set). Precise references and additional examples and generalisations can be found in our survey [48] (jointly with Alexey Kanel-Belov and Eugene Plotkin).…”