DIS

Abstract

We consider a class of matrices with noncommutative entries, first considered by Yu. I. Manin in 1988. They can be defined as ``noncommutative endomorphisms'' of a polynomial algebra. The main aim of the talk is twofold: the first is to show that quite a lot of properties and theorems of linear algebra (e.g., a natural definition of the determinant, the Cayley-Hamilton theorem, and so on and so forth) have a straightforward and natural counterpart in this case. The second, to show how these matrices appear in the theory of integrable quantum spin models, and present a few applications. (Joint work(s) with A. Chervov and V. Rubtsov).