Vladimir Retakh
Rutgers, The State University of New Jersey, Mathematics, Faculty Member
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In this paper we study geometry of symmetric torsion-free connections which preserve a given symplectic form
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Dedicated to the memory of Chih-Han Sah, this volume covers a number of topics, including: combinatorial geometry, connections between logic and geometry, Lie groups, algebras and their representations, and non-communcative algebra and... more
Dedicated to the memory of Chih-Han Sah, this volume covers a number of topics, including: combinatorial geometry, connections between logic and geometry, Lie groups, algebras and their representations, and non-communcative algebra and its relations to physics.
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We study a fully noncommutative generalization of the commutative fourth Painlevé equation that possesses solutions in terms of an infinite Toda system over an associative unital division ring equipped by a derivation.
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Page 1. DETERMINANTS OF MATRICES OVER NONCOMMUTATIVE RINGS IM Gel'fand and VS Retakh UDC 512.55 In this paper we construct (quasi)determinants of matrices over noncommutative rings. Unlike the well-known ...
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The aim of this paper is to introduce and study Lie algebras over noncommutative rings. For any Lie algebra $g$ sitting inside an associative algebra $A$ and any associative algebra $F$ we introduce and study the $F$-loop algebra... more
The aim of this paper is to introduce and study Lie algebras over noncommutative rings. For any Lie algebra $g$ sitting inside an associative algebra $A$ and any associative algebra $F$ we introduce and study the $F$-loop algebra $(g,A)(F)$, which is the Lie subalgebra of $F\otimes A$ generated by $F\otimes g$. In most examples $A$ is the universal enveloping algebra of $\gg$. Our description of the loop algebra has a striking resemblance to the commutator expansions of $F$ used by M. Kapranov in his approach to noncommutative geometry. We also associate with each $F$-loops algebras $(g,A)(F)$ a "noncommutative algebraic" group which naturally acts on $(g,A)(F)$ by conjugations and conclude the paper with a number of examples of such groups.
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We construct and study a class of algebras associated to generalized layered graphs, i.e. directed graphs with a ranking function on their vertices. Each finite directed acyclic graph admits a structure of a generalized layered graph. We... more
We construct and study a class of algebras associated to generalized layered graphs, i.e. directed graphs with a ranking function on their vertices. Each finite directed acyclic graph admits a structure of a generalized layered graph. We construct linear bases in such algebras and compute their Hilbert series. Our interest to generalized layered graphs and algebras associated to those graphs is motivated by their relations to factorizations of polynomials over noncommutative rings.
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Page 1. Zh ~, bh, '~) ~ 0P~ 1 [exp {i (~h ~- . . t-~h)}], b~ ~ ~ C~(~I). ~ a quantum ~mical system defined by a Hamilton~n H ~ ~ (a ) we mean a group of cont~uous ope~tors ~Uh(t), -~ < t <... more
Page 1. Zh ~, bh, '~) ~ 0P~ 1 [exp {i (~h ~- . . t-~h)}], b~ ~ ~ C~(~I). ~ a quantum ~mical system defined by a Hamilton~n H ~ ~ (a ) we mean a group of cont~uous ope~tors ~Uh(t), -~ < t < ~} in Sh~ ~n) such t~t for ~e~ b ~ Sh ...
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Page 1. A Theory of Noncommutative Determinants and Characteristic Functions of Graphs IM Gel~fand and VS Retakh UDC 512.55 Introduction A. Cayley was the first, in 1845 [3], who define the determinant of a matrix with noncommutative... more
Page 1. A Theory of Noncommutative Determinants and Characteristic Functions of Graphs IM Gel~fand and VS Retakh UDC 512.55 Introduction A. Cayley was the first, in 1845 [3], who define the determinant of a matrix with noncommutative entries. ...
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Page 1. ON THE CONJUGATE HOMOMORPHISM OF LOCALLY CONVEX SPACES VS Retakh Let q~ : E ~ F be a homomorphism of locally convex spaces.* If the conjugate spaces are equipped with strong topologies, the conjugate ...
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Page 1. It would be of interest to determine whether the product UT of the operators U and T in 2~ of Corollary 3 belongs to the closure of the set of finite-dimensional mappings in L(X, Z), if only under the assumption that Y satisfies ...
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Discriminants and Local Invariants of Planar Fronts.- Crofton Densities, Symplectic Geometry and Hilbert's Fourth Problem.- Projective Convex Curves.- Topological Classification of Real Trigonometric Polynomials and Cyclic Serpents... more
Discriminants and Local Invariants of Planar Fronts.- Crofton Densities, Symplectic Geometry and Hilbert's Fourth Problem.- Projective Convex Curves.- Topological Classification of Real Trigonometric Polynomials and Cyclic Serpents Polyhedron.- Singularities of Short Linear Waves on the Plane.- New Generalizations of Poincare's Geometric Theorem.- Explicit Formulas for Arnold's Generic Curve Invariants.- Nonlinear Integrable Equations and Nonlinear Fourier Transform.- Elliptic Solutions of the Yang-Baxter Equation and Modular Hypergeometric Functions.- Combinatorics of Hypergeometric Functions Associated with Positive Roots.- Local Invariants of Mappings of Surfaces into Three-Space.- Theorem on Six Vertices of a Plane Curve Via Sturm Theory.- The Arf-Invariant and the Arnold Invariants of Plane Curves.- Produit cyclique d'espaces et operations de Steenrod.- Characteristic Classes of Singularity Theory.- Value of Generalized Hypergeometric Function at Unity.- Harish-Chandra Decomposition for Zonal Spherical Function of type An.- Positive Paths in the Linear Symplectic Group.- Invariants of Submanifolds in Euclidean Space.- On Combinatorics and Topology of Pairwise Intersections of Schubert Cells in SLn/B.