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Sergei Silvestrov

Publications and source records attributed to Sergei Silvestrov.

90 records · Page 5Linked to original sources
arXiv2011

Construction of n-Lie algebras and n-ary Hom-Nambu-Lie algebras

We present a procedure to construct (n+1)-Hom-Nambu-Lie algebras from n-Hom-Nambu-Lie algebras equipped with a generalized trace function. It turns out that the implications of the compatibility conditions, that are necessary for this construction, can be understood in terms of the kernel of the trace function and the range of the twisting maps. Furthermore, we investigate the possibility of defining (n+k)-Lie algebras from n-Lie algebras and a k-form satisfying certain conditions.

math.RA↗
arXiv2010

Irreducible wavelet representations and ergodic automorphisms on solenoids

We focus on the irreducibility of wavelet representations. We present some connections between the following notions: covariant wavelet representations, ergodic shifts on solenoids, fixed points of transfer (Ruelle) operators and solutions of refinement equations. We investigate the irreducibility of the wavelet representations, in particular the representation associated to the Cantor set, introduced in \cite{DuJo06b}, and we present several equivalent formulations of the problem.

math.FA↗
arXiv2009

Ternary Hom-Nambu-Lie algebras induced by Hom-Lie algebras

The purpose of this paper is to investigate ternary multiplications constructed from a binary multiplication, linear twisting maps and a trace function. We provide a construction of ternary Hom-Nambu and Hom-Nambu-Lie algebras starting from a binary multiplication of a Hom-Lie algebra and a trace function satisfying certain compatibility conditions involving twisting maps. We show that mutual position of kernels of twisting maps and the trace play important role in this context, and provide examples of Hom-Nambu-Lie algebras obtained using this construction.

math.RA↗
arXiv2009

Cohomology of 3-dimensional color Lie algebras

We develop the cohomology theory of color Lie superalgebras due to Scheunert--Zhang in a framework of nonhomogeneous quadratic Koszul algebras. In this approach, the Chevalley--Eilenberg complex of a color Lie algebra becomes a standard Koszul complex for its universal enveloping algebra. As an application, we calculate cohomologies with trivial coefficients of 3-dimensional color Lie superalgebras.

math.KT↗
arXiv2009

Affine transformation crossed product like algebras and noncommutative surfaces

Several classes of *-algebras associated to the action of an affine transformation are considered, and an investigation of the interplay between the different classes of algebras is initiated. Connections are established that relate representations of *-algebras, geometry of algebraic surfaces, dynamics of affine transformations, graphs and algebras coming from a quantization procedure of Poisson structures. In particular, algebras related to surfaces being inverse images of fourth order polynomials (in R^3) are studied in detail, and a close link between representation theory and geometric properties is established for compact as well as non-compact surfaces.

math-ph↗
arXiv2008

Hom-Algebras and Hom-Coalgebras

The aim of this paper is to develop the theory of Hom-coalgebras and related structures. After reviewing some key constructions and examples of quasi-deformations of Lie algebras involving twisted derivations and giving rise to the class of quasi-Lie algebras incorporating Hom-Lie algebras, we describe the notion and some properties of Hom-algebras and provide examples. We introduce Hom-coalgebra structures, leading to the notions of Hom-bialgebra and Hom-Hopf algebras and prove some fundamental properties and give examples. Finally, we define the concept of Hom-Lie admissible Hom-coalgebra and provide their classification based on subgroups of the symmetric group.

math.RA↗
arXiv2008

Algebraic curves for commuting elements in the q-deformed Heisenberg algebra

In this paper we extend the eliminant construction of Burchnall and Chaundy for commuting differential operators in the Heisenberg algebra to the q-deformed Heisenberg algebra and show that it again provides annihilating curves for commuting elements, provided q satisfies a natural condition. As a side result we obtain estimates on the dimensions of the eigenspaces of elements of this algebra in its faithful module of Laurent series.

math.RA↗
arXiv2008

On the Exel crossed product of topological covering maps

For dynamical systems defined by a covering map of a compact Hausdorff space and the corresponding transfer operator, the associated crossed product $C^*$-algebras $\cros$ introduced by Exel and Vershik are considered. An important property for homeomorphism dynamical systems is topological freeness. It can be extended in a natural way to in general non-invertible dynamical systems generated by covering maps. In this article, it is shown that the following four properties are equivalent: the dynamical system generated by a covering map is topologically free; the canonical imbedding of $C(X)$ into $\cros$ is a maximal abelian $C^*$-subalgebra of $\cros$; any nontrivial two sided ideal of $\cros$ has non-zero intersection with the imbedded copy of $C(X)$; a certain natural representation of $\cros$ is faithful. This result is a generalization to non-invertible dynamics of the corresponding results for crossed product $C^*$-algebras of homeomorphism dynamical systems.

math.OA↗
arXiv2007

Notes on Formal Deformations of Hom-associative and Hom-Lie Algebras

The aim of this paper is to extend to Hom-algebra structures the theory of formal deformations of algebras which was introduced by Gerstenhaber for associative algebras and extended to Lie algebras by Nijenhuis-Richardson. We deal with Hom-associative and Hom-Lie algebras. We construct the first groups of a deformation cohomology and give several examples of deformations. We provide families of Hom-Lie algebras deforming Lie algebra $sl_2$ and describe as formal deformations the q-deformed Witt algebra and Jackson $sl_2$.

math.RA↗
arXiv2007

Hom-Lie admissible Hom-coalgebras and Hom-Hopf algebras

The aim of this paper is to generalize the concept of Lie-admissible coalgebra introduced by Goze and Remm to Hom-coalgebras and to introduce Hom-Hopf algebras with some properties. These structures are based on the Hom-algebra structures introduced by the authors.

math.RA↗
arXiv2007

Dynamical Systems and Commutants in Crossed Products

In this paper we describe the commutant of an arbitrary subalgebra $A$ of the algebra of functions on a set $X$ in a crossed product of $A$ with the integers, where the latter act on $A$ by a composition automorphism defined via a bijection of $X$. The resulting conditions which are necessary and sufficient for $A$ to be maximal abelian in the crossed product are subsequently applied to situations where these conditions can be shown to be equivalent to a condition in topological dynamics. As a further step, using the Gelfand transform we obtain for a commutative completely regular semi-simple Banach algebra a topological dynamical condition on its character space which is equivalent to the algebra being maximal abelian in a crossed product with the integers.

math.DS↗
arXiv2007

Algebraic dependence of commuting elements in algebras

The aim of this paper to draw attention to several aspects of the algebraic dependence in algebras. The article starts with discussions of the algebraic dependence problem in commutative algebras. Then the Burchnall-Chaundy construction for proving algebraic dependence and obtaining the corresponding algebraic curves for commuting differential operators in the Heisenberg algebra is reviewed. Next some old and new results on algebraic dependence of commuting q-difference operators and elements in q-deformed Heisenberg algebras are reviewed. The main ideas and essence of two proofs of this are reviewed and compared. One is the algorithmic dimension growth existence proof. The other is the recent proof extending the Burchnall-Chaundy approach from differential operators and the Heisenberg algebra to the q-deformed Heisenberg algebra, showing that the Burchnall-Chaundy eliminant construction indeed provides annihilating curves for commuting elements in the q-deformed Heisenberg algebras for q not a root of unity.

math.RA↗
arXiv2006

Quasi-Lie structure of twisted derivations of Laurent polynomials

Hartwig, Larsson and the second author in [J. Algebra, 2005] defined a bracket on sigma-derivations of a commutative algebra. We show that this bracket preserves inner derivations, and based on this obtain some structural results on sigma-derivations on Laurent polynomials in one variable.

math.QA↗
arXiv2006

Monotone operator functions, gaps and power moment problem

The article is devoted to investigation of the classes of functions belonging to the gaps between classes $P_{n+1}(I)$ and $P_{n}(I)$ of matrix monotone functions for full matrix algebras of successive dimensions. In this paper we address the problem of characterizing polynomials belonging to the gaps $P_{n}(I) \setminus P_{n+1}(I)$ for bounded intervals $I$. We show that solution of this problem is closely linked to solution of truncated moment problems, Hankel matrices and Hankel extensions. Namely, we show that using the solutions to truncated moment problems we can construct continuum many polynomials in the gaps. We also provide via several examples some first insights into the further problem of description of polynomials in the gaps that are not coming from the truncated moment problem. Also, in this article, we deepen further in another way into the structure of the classes of matrix monotone functions and of the gaps between them by considering the problem of position in the gaps of certain interesting subclasses of matrix monotone functions that appeared in connection to interpolation of spaces and in a prove of the L{ö}wner theorem on integral representation of operator monotone functions.

math.OA↗
arXiv2005

C*-crossed products and shift spaces

In this article, we use Exel's construction to associate a C*-algebra to every shift space. We show that it has the C*-algebra defined in [Carlsen and Matsumoto: Some remarks on the C*-algebras associated with subshifts] as a quotient, and possesses properties indicating that it can be thought of as the universal C*-algebra associated to a shift space. We also consider its representations, relationship to other C*-algebras associated to shift spaces, show that it can be viewed as a generalization of the universal Cuntz-Krieger algebra, discuss uniqueness and a faithful representation, provide conditions for it being nuclear, for satisfying the UCT, for being simple, and for being purely infinite, show that the constructed algebras and thus their K-theory, $K_0$ and $K_1$, are conjugacy invariants of one-sided shift spaces, present formulas for those invariants, and also present a description of the structure of gauge invariant ideals.

math.OA↗
arXiv2004

Quasi-hom-Lie Algebras, Central Extensions and 2-cocycle-like Identities

This paper begins by introducing the concept of a quasi-hom-Lie algebra which is a natural generalization of hom-Lie algebras introduced in a previous paper by the authors. Quasi-hom-Lie algebras include also as special cases (color) Lie algebras and superalgebras, and can be seen as deformations of these by homomorphisms, twisting the Jacobi identity and skew-symmetry. The natural realm for these quasi-hom-Lie algebras is as a generalization-deformation of the Witt algebra $\Witt$ of derivations on the Laurent polynomials $\C[t,t^{-1}]$. We also develop a theory of central extensions for qhl-algebras which can be used to deform and generalize the Virasoro algebra by centrally extending the deformed Witt type algebras constructed here. In addition, we give a number of other interesting examples of quasi-hom-Lie algebras, among them a deformation of the loop algebra.

math.RA↗
arXiv2004

Deformations of Lie Algebras using $σ$-derivations

In this article we develop an approach to deformations of the Witt and Virasoro algebras based on $σ$-derivations. We show that $σ$-twisted Jacobi type identity holds for generators of such deformations. For the $σ$-twisted generalization of Lie algebras modeled by this construction, we develop a theory of central extensions. We show that our approach can be used to construct new deformations of Lie algebras and their central extensions, which in particular include naturally the $q$-deformations of the Witt and Virasoro algebras associated to $q$-difference operators, providing also corresponding q-deformed Jacobi identities.

math.QA↗