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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)

- Vladimir Retakh, Christophe Reutenauer, Arkady Vaintrob
- 2000

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)

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. Our… (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)

There are two ways to generalize basic constructions of commutative algebra for a noncommutative case. More traditional way is to define commutative functions like trace or determinant over noncommuting variables. Beginning with [6] this approach was widely used by different authors, see for example [5], [15], [14], [12], [11], [7]. However, there is… (More)

Quasideterminants of noncommutative matrices introduced in [GR, GR1] have proved to be a powerfull tool in basic problems of noncommutative algebra and geometry (see [GR, GR1-GR4, GKLLRT, GV, EGR, EGR1, ER,KL, KLT, LST, Mo, Mo1, P, RS, RRV, Rsh, Sch]). In general, the quasideterminants of matrix A = (aij) are rational functions in (aij)’s. The minimal… (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)

In [3] we introduced a new class of algebras A(Γ) associated to layered directed graphs Γ. These algebras arose as generalizations of the algebras Qn (which are related to factorizations of noncommutative polynomials, see [2, 5, 9]), but the new class of algebras seems to be interesting by itself. Various results have been proven for algebras A(Γ). In [3]… (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)