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Vladimir Retakh

Publications and source records attributed to Vladimir Retakh.

At least 19 recordsLinked to original sources
arXiv2025

Noncommutative marked surfaces II: tagged triangulations, clusters, and their symmetries

The aim of the paper is to define noncommutative cluster structure on several algebras ${\mathcal A}$ related to marked surfaces possibly with orbifold points of various orders, which includes noncommutative clusters, i.e., embeddings of a given group $G$ into the multiplicative monoid ${\mathcal A}^\times$ and an action of a certain braid-like group $Br_{\mathcal A}$ by automorphisms of each cluster group in a compatible way. For punctured surfaces we construct new symmetries, noncommutative tagged clusters and establish a noncommutative Laurent Phenomenon.

math.RT↗
arXiv2024

Non-Abelian discrete Toda chains and related lattices

We have derived a non-abelian analog for the two-dimensional discrete Toda lattice which possesses solutions in terms of quasideterminants and admits Lax pairs of different forms. Its connection with non-abelian analogs for several well-known (1+1) and one-dimensional lattices is discussed. In particular, we consider a non-commutative analog of the scheme: discrete Toda equations $\rightarrow$ Somos-$N$ sequences $\rightarrow$ discrete Painlevé equations.

nlin.SI↗
arXiv2021

Symplectic groups over noncommutative algebras

We introduce the symplectic group $\mathrm{Sp}_2(A,σ)$ over a noncommutative algebra $A$ with an anti-involution $σ$. We realize several classical Lie groups as $\mathrm{Sp}_2$ over various noncommutative algebras, which provides new insights into their structure theory. We construct several geometric spaces, on which the groups $\mathrm{Sp}_2(A,σ)$ act. We introduce the space of isotropic $A$-lines, which generalizes the projective line. We describe the action of $\mathrm{Sp}_2(A,σ)$ on isotropic $A$-lines, generalize the Kashiwara-Maslov index of triples and the cross ratio of quadruples of isotropic $A$-lines as invariants of this action. When the algebra $A$ is Hermitian or the complexification of a Hermitian algebra, we introduce the symmetric space $X_{\mathrm{Sp}_2(A,σ)}$, and construct different models of this space. Applying this to classical Hermitian Lie groups of tube type (realized as $\mathrm{Sp}_2(A,σ)$) and their complexifications, we obtain different models of the symmetric space as noncommutative generalizations of models of the hyperbolic plane and of the three-dimensional hyperbolic space. We also provide a partial classification of Hermitian algebras in Appendix A.

math.DG↗
arXiv2020

On the rational relationships among pseudo-roots of a non-commutative polynomial

For a non-commutative ring R, we consider factorizations of polynomials in R[t] where t is a central variable. A pseudo-root of a polynomial p(t) is an element x in R, for which there exist polynomials q(t) and s(t) such that p(t)=q(t)(t-x)s(t). We investigate the rational relationships that hold among the pseudo-roots of p(t) by using the diamond operations for cover graphs of modular lattices.

math.RA↗
arXiv2019

Noncommutative Catalan numbers

The goal of this paper is to introduce and study noncommutative Catalan numbers $C_n$ which belong to the free Laurent polynomial algebra in $n$ generators. Our noncommutative numbers admit interesting (commutative and noncommutative) specializations, one of them related to Garsia-Haiman $(q,t)$-versions, another -- to solving noncommutative quadratic equations. We also establish total positivity of the corresponding (noncommutative) Hankel matrices $H_m$ and introduce accompanying noncommutative binomial coefficients.

math.QA↗
arXiv2019

Noncommutative cross-ratio and Schwarz derivative

We present here a theory of noncommutative cross-ratio, Schwarz derivative and their connections and relations to the operator cross-ratio. We apply the theory to "noncommutative elementary geometry" and relate it to noncommutative integrable systems. We also provide a noncommutative version of the celebrated "pentagramma mirificum".

math.RA↗
arXiv2019

Noncommutative Shifted Symmetric Functions

We introduce a ring of noncommutative shifted symmetric functions based on an integer-indexed sequence of shift parameters. Using generating series and quasideterminants, this multiparameter approach produces deformations of the ring of noncommutative symmetric functions. Shifted versions of ribbon Schur functions are defined and form a basis for the ring. Further, we produce analogues of Jacobi-Trudi and Nägelsbach-Kostka formulas, a duality anti-algebra isomorphism, shifted quasi-Schur functions, and Giambelli's formula in this setup. In addition, an analogue of power sums is provided, satisfying versions of Wronski and Newton formulas. Finally, a realization of these noncommutative shifted symmetric functions as rational functions in noncommuting variables is given. These realizations have a shifted symmetry under exchange of the variables and are well-behaved under extension of the list of variables.

math.RA↗
arXiv2018

Noncommutative marked surfaces

The aim of the paper is to attach a noncommutative cluster-like structure to each marked surface $Σ$. This is a noncommutative algebra ${\mathcal A}_Σ$ generated by "noncommutative geodesics" between marked points subject to certain triangle relations and noncommutative analogues of Ptolemy-Plücker relations. It turns out that the algebra ${\mathcal A}_Σ$ exhibits a noncommutative Laurent Phenomenon with respect to any triangulation of $Σ$, which confirms its "cluster nature". As a surprising byproduct, we obtain a new topological invariant of $Σ$, which is a free or a 1-relator group easily computable in terms of any triangulation of $Σ$. Another application is the proof of Laurentness and positivity of certain discrete noncommutative integrable systems.

math.QA↗
arXiv2017

Algebras of Quasi-Plücker Coordinates are Koszul

Motivated by the theory of quasi-determinants, we study non-commutative algebras of quasi-Plücker coordinates. We prove that these algebras provide new examples of non-homogeneous quadratic Koszul algebras by showing that their quadratic duals have quadratic Gröbner bases.

math.RA↗
arXiv2015

Generalized adjoint actions

The aim of this paper is to generalize the classical formula $e^xye^{-x}=\sum\limits_{k\ge 0} \frac{1}{k!} (ad~x)^k(y)$ by replacing $e^x$ with any formal power series $\displaystyle {f(x)=1+\sum_{k\ge 1} a_kx^k}$. We also obtain combinatorial applications to $q$-exponentials, $q$-binomials, and Hall-Littlewood polynomials.

math.QA↗
arXiv2014

Noncommutative Cross-Ratios

We present a definition of and discuss basic properties of cross-ratios over noncommutative skew-fields. A new theorem was added.

math.RA↗
arXiv2013

The Reciprocal of $\sum_{n\geq 0}a^nb^n$ for non-commuting $a$ and $b$, Catalan numbers and non-commutative quadratic equations

The aim of this paper is to describe the inversion of the sum $\sum_{n\geq 0}a^nb^n$ where $a$ and $b$ are non-commuting variables as a formal series in $a$ and $b$. We show that the inversion satisfies a non-commutative quadratic equation and that the number of certain monomials in its homogeneous components equals to a Catalan number. We also study general solutions of similar quadratic equations.

math.CO↗
arXiv2011

Hilbert series of algebras associated to direct graphs and order homology

We give a homological interpretation of the coefficients of the Hilbert series for an algebra associated with a directed graph and its dual algebra. This allows us to obtain necessary conditions for Koszulity of such algebras in terms of homological properties of the graphs. We use our results to construct algebras with a prescribed Hilbert series.

math.RA↗
arXiv2011

A Spectral Theory for Tensors

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.

math.SP↗
arXiv2010

A short proof of Kontsevich cluster conjecture

We give an elementary proof of the Kontsevich conjecture that asserts that the iterations of the noncommutative rational map K_r:(x,y)-->(xyx^{-1},(1+y^r)x^{-1}) are given by noncommutative Laurent polynomials.

math.QA↗