Related papers: Manin Matrices for Quadratic Algebras

$W$-algebras are certain algebraic structures associated to a finite dimensional Lie algebra $\mathfrak g$ and a nilpotent element $f$ via Hamiltonian reduction. In this note we give a review of a recent approach to the study of (classical…

We consider linear bounded operators acting in Banach spaces with a basis, such operators can be represented by an infinite matrix. We prove that for an invertible operator there exists a sequence of invertible finite-dimensional operators…

$A_3$-associative algebra is a generalization of associative algebra and is one of the four remarkable types of Lie-admissible algebras, along with associative algebra, left-symmetric algebra and right-symmetric algebra. This paper develops…

We study associative multiplications in semi-simple associative algebras over C compatible with the usual one. An interesting class of such multiplications is related to the affine Dynkin diagrams of A, D, E-type. In this paper we…

The article is devoted to the problem of classification of Manin triples up to weak and gauge equivalence. The case of complex simple Lie algebras can be obtained by the papers of A.Belavin, V.Drinfel'd, M.Semenov-Tian-Shanskii. Studing the…

We introduce the notion of Hamiltonian spaces for Manin pairs over manifolds, using the so-called generalized Dirac structures. As an example, we describe Hamiltonian spaces of a quasi-Lie bialgebroid using this general framework. We also…

A Lax operator algebra is constructed for an arbitrary semi-simple Lie algebra over $\mathbb C$ equipped with a $\mathbb Z$-grading, and arbitrary compact Riemann surface with marked points. In this set-up, a treatment of almost graded…

Matrix properties are a type of property of categories which includes the ones of being Mal'tsev, arithmetical, majority, unital, strongly unital and subtractive. Recently, an algorithm has been developed to determine implications…

Lie admissible algebra structures, called center-symmetric algebras, are defined. Main properties and algebraic consequences are derived and discussed. Bimodules are given and used to build a center-symmetric algebra on the direct sum of…

We give canonical matrices of a pair (A,B) consisting of a nondegenerate form B and a linear operator A satisfying B(Ax,Ay)=B(x,y) on a vector space over F in the following cases: (i) F is an algebraically closed field of characteristic…

We study real and complex Manin triples for a complex reductive Lie algebra, $\g$. The first part includes, and extends to complex Manin triples, our earlier work [De]. First, we generalize results of E. Karolinsky, on the classification of…

Matrix conditions extend linear Mal'tsev conditions from Universal Algebra to exactness properties in Category Theory. Some can be stated in the finitely complete context while, in general, they can only be stated for regular categories. We…

Finite dimensional modular Lie superalgebras over algebraically closed fields with indecomposable Cartan matrices are classified under some technical, most probably inessential, hypotheses. If the Cartan matrix is invertible, the…

In this paper, first we introduce the notion of a Leibniz bialgebra and show that matched pairs of Leibniz algebras, Manin triples of Leibniz algebras and Leibniz bialgebras are equivalent. Then we introduce the notion of a (relative)…

We introduce quantum versions of Manin pairs and Manin triples and define quantum moment maps in this context. This provides a framework that incorporates quantum moment maps for actions of Lie algebras and quantum groups for any quantum…

A quadratic algebra is a homogeneous algebra generated by its elements of degree 1. Manin has endowed the category of quadratic algebras with two tensor products. These structures have been adapted to operads by Ginsburg and Kapranov.…

Generalizing the construction of the cyclotomic Gaudin algebra from arXiv:1409.6937, we define the universal cyclotomic Gaudin algebra. It is a cyclotomic generalization of the Gaudin models with irregular singularities defined in…

An algebraic investigation on bicomplex numbers is carried out here. Particularly matrices and linear maps defined on them are discussed. A new kind of cartesian product, referred to as an idempotent product, is introduced and studied. The…

In this paper, we give matrix formulae for non-orientable surfaces that provide the Laurent expansion for quasi-cluster variables, generalizing the orientable surface matrix formulae by Musiker-Williams. We additionally use our matrix…

We describe all linear operators which maps $n-1$-dimensional simplex of idempotent measures to itself. Such operators divided to two classes: the first class contains all $n\times n$-matrices with non-negative entries which has at least…