We study associative multiplications in semi-simple associative algebras over C compatible with the usual one or, in other words, linear deformations of semi-simple associative algebras over C. It turns out that these deformations are in one-to-one correspondence with representations of certain algebraic structures, which we call M-structures in the matrix case and PM-structures in the case of direct sums of several matrix algebras. We also investigate various properties of PM-structures, provide numerous examples and describe an important class of PM-structures. The classification of these PM-structures naturally leads to affine Dynkin diagrams of A, D, E-type.
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 investigate in details the multiplications of the A-type and integrable matrix ODEs and PDEs generated by them.
Given an associative multiplication in matrix algebra compatible with the usual one or, in other words, linear deformation of matrix algebra, we construct a solution to the classical Yang-Baxter equation. We also develop a theory of such deformations and construct numerous examples. It turns out that these deformations are in one-to-one correspondence with representations of certain algebraic structures, which we call M-structures. We also describe an important class of M-structures related to the affine Dynkin diagrams of A, D, E-type. These M-structures and their representations are describe…
This paper studies the concepts of a totally compatible dialgebra and a totally compatible Lie dialgebra, defined to be a vector space with two binary operations that satisfy individual and mixed associativity conditions and Lie algebra conditions respectively. We show that totally compatible dialgebras are closely related to bimodule algebras and semi-homomorphisms. More significantly, Rota-Baxter operators on totally compatible dialgebras provide a uniform framework to generalize known results that Rota-Baxter related operators give tridendriform algebras. Free totally compatible dialgebras …
Pseudoalgebras, introduced in [BDK], are multi-dimensional analogues of conformal algebras, which provide an axiomatic description of the singular part of the operator product expansion. Our main interest in this paper is the pseudoalgebra Cend_n, which is the analogue of an algebra of endomorphisms of a finite module. We study its algebraic properties. In particular, we introduce the class of unital pseudoalgebras and describe their structure and representations. Also, we classify pseudoalgebras algebraically similar to Cend_n.
In this paper, we consider graded associative conformal algebras. The class of these objects includes pseudo-algebras over non-cocommutative Hopf algebras of regular functions on some linear algebraic groups. In particular, an associative conformal algebra which is graded by a finite group $Γ$ is a pseudo-algebra over the coordinate Hopf algebra of a linear algebraic group $G$ such that the identity component $G^0$ is the affine line and $G/G^0\simeq Γ$. A classification of simple and semisimple graded associative conformal algebras of finite type is obtained.
The paper is devoted to the Poisson brackets compatible with multiplication in associative algebras. These brackets are shown to be quadratic and their relations with the classical Yang--Baxter equation are revealed. The paper also contains a description of Poisson Lie structures on Lie groups whose Lie algebras are adjacent to an associative structure.
Quadratic Poisson brackets on associative algebras are studied. Such a bracket compatible with the multiplication is related to a differentiation in tensor square of the underlying algebra. Jacobi identity means that this differentiation satisfies a classical Yang--Baxter equation. Corresponding Lie groups are canonically equipped with a Poisson Lie structure. A way to quantize such structures is suggested.
We set up a strategy for studying large families of logarithmic conformal field theories by using the enlarged symmetries and non--semi-simple associative algebras appearing in their lattice regularizations (as discussed in a companion paper). Here we work out in detail two examples of theories derived as the continuum limit of XXZ spin-1/2 chains, which are related to spin chains with supersymmetry algebras gl($n|n$) and gl($n+1|n$), respectively, with open (or free) boundary conditions in all cases. These theories can also be viewed as vertex models, or as loop models. Their continuum limits…
Let X be a smooth algebraic variety over a field of characteristic 0. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf O_X. These are stack-like versions of usual deformations. We prove that there is a twisted quantization map from twisted Poisson deformations to twisted associative deformations, which is canonical and bijective on equivalence classes.
Let X be a smooth algebraic variety over a field K containing the real numbers. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf of X. These are stack-like versions of usual deformations. We prove that there is a twisted quantization operation from twisted Poisson deformations to twisted associative deformations, which is canonical and bijective on gauge equivalence classes. This result extends work of Kontsevich, and our own earlier work, on deformation quantization of algebraic varieties.
We compactify the spaces $K(m,n)$ introduced by Maxim Kontsevich. The initial idea was to construct an $L_\infty$ algebra governing the deformations of a (co)associative bialgebra. However, this compactification leads not to a resolution of the PROP of (co)associative bialgebras, but to a new algebraic structure we call here a CROC. It turns out that these constructions are related to the non-commutative deformations of (co)associative bialgebras. We construct an associative dg algebra conjecturally governing the non-commutative deformations of a bialgebra. Then, using the Quillen duality, we …
Rota-Baxter operators and bialgebras go hand in hand in their applications, such as in the Connes-Kreimer approach to renormalization and the operator approach to the classical Yang-Baxter equation. We establish a bialgebra structure that is compatible with the Rota-Baxter operator, called the Rota-Baxter antisymmetric infinitesimal (ASI) bialgebra. This bialgebra is characterized by generalizations of matched pairs of algebras and double constructions of Frobenius algebras to the context of Rota-Baxter algebras. The study of the coboundary case leads to an enrichment of the associative Yang-B…
After a thorough treatment of all algebraic structures involved, we address two dimensional holonomy operators with values in crossed modules of Hopf algebras and in crossed modules of associative algebras (called here crossed modules of bare algebras.) In particular, we will consider two general formulations of the two-dimensional holonomy of a (fully primitive) Hopf 2-connection (exact and blur), the first being multiplicative the second being additive, proving that they coincide in a certain natural quotient (defining what we called the fuzzy holonomy of a fully primitive Hopf 2-connection)…
We apply discrete algebraic Morse theory to the computation of Hochschild cohomologies of associative conformal algebras. As an example, we evaluate the dimensions of the universal associative conformal envelope $U(3)$ of the Virasoro Lie conformal algebra relative to the associative locality $N=3$ on the generator with scalar coefficients.
The aim of the note is to provide an introduction to the algebraic, geometric and quantum field theoretic ideas that lie behind the Kontsevich-Cattaneo-Felder formula for the quantization of Poisson structures. We show how the quantization formula itself naturally arises when one imposes the following two requirements to a Feynman integral: on the one side it has to reproduce the given Poisson structure as the first order term of its perturbative expansion; on the other side its three-point functions should describe an associative algebra. It is further shown how the Magri-Koszul brackets on 1…
We address the general classification problem of all stable associative product structures in the complex cobordism theory. We show how to reduce this problem to the algebraic one in terms of the Hopf algebra $S$ (the Landweber-Novikov algebra) acting on its dual Hopf algebra $S^*$ with a distinguished ``topologically integral'' part $Λ$ that coincids with the coefficient ring of the complex cobordism. We describe the formal group and its logarithm in terms of representations of $S$. We introduce one-dimensional representations of a Hopf algebra. We give series of examples of such representati…
Let G be an affine algebraic group and let R be an associative algebra with a rational action of G by algebra automorphisms. We study the induced G-action on the spectrum Spec R of all prime ideals of R, viewed as a topological space with the Jacobson-Zariski topology, and on the subspace consisting of all rational ideals of R. Here, a prime ideal P of R is said to be rational if the extended centroid of R/P is equal to the base field. The main themes of the article are local closedness of G-orbits in Spec R and the so-called G-stratification of Spec R. This stratification plays a central role…
We develop in this paper the principles of an associative algebraic approach to bulk logarithmic conformal field theories (LCFTs). We concentrate on the closed $gl(1|1)$ spin-chain and its continuum limit - the $c=-2$ symplectic fermions theory - and rely on two technical companion papers, "Continuum limit and symmetries of the periodic gl(1|1) spin chain" [Nucl. Phys. B 871 (2013) 245-288] and "Bimodule structure in the periodic gl(1|1) spin chain" [Nucl. Phys. B 871 (2013) 289-329]. Our main result is that the algebra of local Hamiltonians, the Jones-Temperley-Lieb algebra JTL_N, goes over i…
The integrals of motion of the classical two dimensional superintegrable systems with quadratic integrals of motion close in a restrained quadratic Poisson algebra, whose the general form is investigated. Each classical superintegrable problem has a quantum counterpart, a quantum superintegrable system. The quadratic Poisson algebra is deformed to a quantum associative algebra, the finite dimensional representations of this algebra are calculated by using a deformed parafermion oscillator technique. It is shown that, the finite dimensional representations of the quadratic algebra are determine…
This paper is based on a talk given at the 14-th International Workshop on Differential Geometry and Its Applications, hosted by the Petroleum Gas University from Ploiesti, between July 9-th and July 11-th, 2019. After presenting some historical facts, we will consider some geometry problems related to unification approaches. Jordan algebras and Lie algebras are the main non-associative structures. Attempts to unify non-associative algebras and associative algebras led to UJLA structures. Another algebraic structure which unifies non-associative algebras and associative algebras is the Yang-Ba…
A double algebra is a linear space $V$ equipped with linear map $V\otimes V\to V\otimes V$. Additional conditions on this map lead to the notions of Lie and associative double algebras. We prove that simple finite-dimensional Lie double algebras do not exist over an arbitrary field, and all simple finite-dimensional associative double algebras over an algebraically closed field are trivial. Over an arbitrary field, every simple finite-dimensional associative double algebra is commutative. A double algebra structure on a finite-dimensional space $V$ is naturally described by a linear operator $…
This paper studies operator forms of the nonhomogeneous associative classical Yang-Baxter equation (nhacYBe), extending and generalizing such studies for the classical Yang-Baxter equation and associative Yang-Baxter equation that can be tracked back to the works of Semonov-Tian-Shansky and Kupershmidt on Rota-Baxter Lie algebras and $\mathcal{O}$-operators. In general, solutions of the nhacYBe are characterized in terms of generalized $\mathcal{O}$-operators. The characterization can be given by the classical $\mathcal{O}$-operators precisely when the solutions satisfy an invariant condition.…
This paper establishes a uniform procedure to split the operations in any algebraic operad, generalizing previous known notions of splitting algebraic structures from the dendriform algebra of Loday that splits the associative operation to the successors that split any binary operad. Examples are provided for various $n$-associative algebras, $n$-Lie algebras, $A_\infty$ algebras and $L_\infty$ algebras. Further, the concept of a Rota-Baxter operator, first showing its importance in the associative and Lie algebra context and then generalized to any binary operads, is generalized to arbitrary …
Let X be a smooth affine algebraic variety over a field K of characteristic 0, and let R be a complete parameter K-algebra (e.g. R = K[[h]]). We consider associative (resp. Poisson) R-deformations of the structure sheaf O_X. The set of R-deformations has a crossed groupoid (i.e. strict 2-groupoid) structure. Our main result is that there is a canonical equivalence of crossed groupoids from the Deligne crossed groupoid of normalized polydifferential operators (resp. polyderivations) of X to the crossed groupoid of associative (resp. Poisson) R-deformations of O_X. The proof relies on a careful …
The associator of a non-associative algebra is the curvature of the Hochschild quasi-complex. The relationship ``curvature-associator'' is investigated. Based on this generic example, we extend the geometric language of vector fields to a purely algebraic setting, similar to the context of Gerstenhaber algebras. We interprete the elements of a non-associative algebra with a Lie bracket as ``vector fields'' and the multiplication as a connection. We investigate conditions for the existance of an ``algebra of functions'' having as algebra of derivations the original non-associative algebra.
We use monoidal category methods to study the noncommutative geometry of nonassociative algebras obtained by a Drinfeld-type cochain twist. These are the so-called quasialgebras and include the octonions as braided-commutative but nonassociative coordinate rings, as well as quasialgebra versions $\CC_{q}(G)$ of the standard q-deformation quantum groups. We introduce the notion of ribbon algebras in the category, which are algebras equipped with a suitable generalised automorphism $σ$, and obtain the required generalisation of cyclic cohomology. We show that this \emph{braided cyclic cocohomolo…
The associative operad is a certain algebraic structure on the sequence of group algebras of the symmetric groups. The weak order is a partial order on the symmetric group. There is a natural linear basis of each symmetric group algebra, related to the group basis by Mobius inversion for the weak order. We describe the operad structure on this second basis: the surprising result is that each operadic composition is a sum over an interval of the weak order. We deduce that the coradical filtration is an operad filtration. The Lie operad, a suboperad of the associative operad, sits in the first c…
Using methods of computer algebra, especially Gröbner bases for submodules of free modules over polynomial rings, we solve a classification problem in theory of algebraic operads: we show that the only nontrivial (possibly inhomogeneous) distributive law between the operad of Lie algebras and the operad of commutative associative algebras is given by the Livernet-Loday formula deforming the Poisson operad into the associative operad.
There is a notion of non-commutative Lie algebra called "Leibniz algebra", which is characterized by the condition: left bracketing is a derivation. The purpose of this article is to introduce and study a new notion of algebra, called "associative dialgebra", which is, to the Leibniz algebras, what associative algebras are to Lie algebras. An associative dialgebra is a vector space equipped with two associative operations satisfying three more conditions. For instance, any differential associative algebra gives rise to a dialgebra. In this article we construct and study a (co)homology theory f…
