Vladimir Retakh

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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 integrable systems [RS],[EGR], quantum algebras [GR], [GR1], [GR2], [KL], [M], etc. The main property of(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. Introduction Let x1, . . . , xn be the roots of a generic polynomial P (x) = x n + a1x(More)
M. Farber in [2] 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)
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. Introduction Nonabelian(More)
This is a survey of recently published results. We introduce and study a wide class algebras associated to directed graphs and related to factorizations of noncommutative polynomials. In particular, we show that for many well-known graphs such algebras are Koszul and compute their Hilbert series. Let R be an associative ring with unit and P (t) = a0t +a1t(More)
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)