Vladimir Retakh
Rutgers University
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Communications in Mathematical Physics | 1986
M. I. Monastyrsky; Vladimir Retakh
The present paper contains a systematic study of several linked singularities in condensed matter. We introduce a hierarchy of conservation laws in terms of differential forms corresponding to a sequence of linking invariants so that we can distinguish nontrivial links possessing zero Gauss linking coefficients. We obtain a set of topological obstruction rules for links in nematics, cholesterics, and superfluid3He and4He.
arXiv: Quantum Algebra | 1996
Israel M. Gelfand; Vladimir Retakh
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].
Letters in Mathematical Physics | 2005
Israel M. Gelfand; Sergei Gelfand; Vladimir Retakh; Robert Lee Wilson
To directed graphs with unique sink and source we associate a noncommutative associative algebra and a polynomial over this algebra. Edges of the graph correspond to pseudo-roots of the polynomial. We give a sufficient condition when coefficients of the polynomial can be rationally expressed via elements of a given set of pseudo-roots (edges). Our results are based on a new theorem for directed graphs also proved in this paper.
Journal of Pure and Applied Algebra | 1993
Vladimir Retakh
The difinition of n-homotopically multiplicative maps of differential graded Lie algebras is given. It is shown that such maps conserve n-Lie-Massey brackets.
arXiv: Spectral Theory | 2011
Edinah K. Gnang; Ahmed M. Elgammal; Vladimir Retakh
In this paper we propose a general spectral theory for tensors. Our proposed factorization decomposes a tensor into a product of orthogonal and scaling tensors. At the same time, our factorization yields an expansion of a tensor as a summation of outer products of lower order tensors . Our proposed factorization shows the relationship between the eigen-objects and the generalised characteristic polynomials. Our framework is based on a consistent multilinear algebra which explains how to generalise the notion of matrix hermicity, matrix transpose, and most importantly the notion of orthogonality. Our proposed factorization for a tensor in terms of lower order tensors can be recursively applied so as to naturally induces a spectral hierarchy for tensors.
International Mathematics Research Notices | 2005
Arkady Berenstein; Vladimir Retakh
This paper is a first attempt to generalize results of A. Berenstein, S. Fomin, and A. Zelevinsky on total positivity of matrices over commutative rings to matrices over noncommutative rings. The classical theory of total positivity studies matrices whose minors are all nonnegative. Motivated by a surprising connection discovered by Lusztig [10, 11] between total positivity of matrices and canonical bases for quantum groups, Berenstein, Fomin, and Zelevinsky, in a series of papers [1, 2, 3, 4], systematically investigated the problem of total positivity from a representation-theoretic point of view. In particular, they showed that a natural framework for the study of totally positive matrices is provided by the decomposition of a reductive group G into the disjoint union of double Bruhat cells G = BuB∩B−vB−, where B and B− are two opposite Borel subgroups in G, and u and v belong to the Weyl group W of G. According to [1, 3, 4] there, exist families of birational parametrizations of G, one for each reduced expression of the element (u, v) in the Coxeter group W ×W. Every such parametrization can be thought of as a system of local coordinates in G. Such coordinates are called the factorization parameters associated to the reduced expression of (u, v). The coordinates are obtained by expressing a generic element x ∈ G as an element of the maximal torus H = B ∩ B− multiplied by the product of elements of various
Open Mathematics | 2010
Vladimir Retakh
This is an extended version of a talk given by the author at the conference “Algebra and Topology in Interaction” on the occasion of the 70th Anniversary of D.B. Fuchs at UC Davis in September 2009. It is a brief survey of an area originated around 1995 by I. Gelfand and the speaker.
International Mathematics Research Notices | 2004
Alexander Molev; Vladimir Retakh
We apply the techniques of quasideterminants to construct new families of Casimir elements for the general linear Lie superalgebra Gl(m|n) whose images under the Harish-Chandra isomorphism are the elementary, complete, and power sums supersymmetric functions, respectively.
Journal of Geometry and Physics | 2014
Vladimir Retakh
We present a definition and discuss basic properties of cross-ratios over noncommutative skew-fields.
Letters in Mathematical Physics | 1993
Mark Iosifovich Graev; Vladimir Retakh
We describe multivariable hypergeometric series in orthogonal polynomials. These series are solutions of special systems of Gelfand-type equations. The difference andq-analogs are also given.