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There are two ways to generalize basic constructions of commutative algebra for a noncommutative case. More traditional way is to define com-mutative functions like trace or determinant over noncommuting variables. Beginning with  this approach was widely used by different authors, see However, there is another possibility to work with purely… (More)
Introduction A notion of quasideterminants for matrices over a noncommutative skew-field was introduced in [GR], [GR1], [GR2]. It proved its effectiveness for many areas including noncommutative symmetric functions [GKLLRT], noncommutative The main property of quasideterminants is a " heredity principle " : let A be a square matrix over a skew-field and (A… (More)
The aim of this paper is to introduce and study Lie algebras and Lie groups over noncommutative rings. For any Lie algebra g sitting inside an associative algebra A and any associative algebra F we introduce and study the algebra (g, A)(F), which is the Lie subalgebra of F ⊗ A generated by F ⊗ g. In most examples A is the universal enveloping algebra of g.… (More)
We integrate nonabelian Toda field equations [Kr] for root systems of types A, B, C, for functions with values in any associative algebra. The solution is expressed via quasideterminants introduced in [GR1],[GR2], [GR4]. In the appendix we review some results concerning noncommutative versions of other classical integrable equations.
M. Farber in  constructed invariants of µ-component boundary links with values in the algebra of noncommutative rational functions. In this paper we simplify his algebraic constructions and express them by using noncommutative generalizations of determinants introduced by Gelfand and Retakh. In particular, for every finite-dimensional module N over the… (More)
This is a part of a handwritten manuscript of the same authors. To read it is enough to be familiar with basic definitions, properties and notations of the theory of quasideterminants of noncommutative matrices. (See the paper by I.M. Gelfand and V.S. Retakh, " A theory of noncommutative determinants and characteristic functions of graphs, " Funct.
We find explicit (multisoliton) solutions for nonabelian integrable systems such as periodic Toda field equations, Langmuir equations, and Schrödinger equations for functions with values in any associative algebra. The solution for nonabelian Toda field equations for root systems of types A, B, C was expressed by the authors in [EGR] using quasideterminants… (More)
We study certain quadratic and quadratic linear algebras related to factorizations of noncommutative polynomials and differential polynomials. Such algebras possess a natural derivation and give us a new understanding of the nature of noncommutative symmetric functions.