We develop the notion of deformations using a valuation ring as ring of coefficients. This permits to consider in particular the classical Gerstenhaber deformations of associative or Lie algebras as infinitesimal deformations and to solve the equation of deformations in a polynomial frame. We consider also the deformations of the enveloping algebra of a rigid Lie algebra and we define valued deformations for some classes of non associative algebras.
We give sufficient and necessary conditions to guarantee that a pseudo-effect algebra admits an $(n+1)$-valued discrete state. We introduce $n$-perfect pseudo-effect algebras as algebras which can be split into $n+1$ comparable slices. We prove that the category of strong $n$-perfect pseudo-effect algebras is categorically equivalent to the category of torsion-free directed partially ordered groups of a special type.
We show that every absolute-valued algebra with left-unit satisfying (x2; x2; x2) = 0 is finite-dimensional of degree at most 4: Next, we determine such an algebras. In addition to the already known algebras R; C; \astC; H; \astH; \astH(i; 1); O; \astO; \astO(i; 1) the list is completed by two new algebras not yet specified in the literature.
We show that three deformation functors (deformations of the product, flat deformations and deformations of the relations) assigned to an associative algebra are naturally isomorphic.
The aim of this paper is to extend Gerstenhaber formal deformations of algebras to the case of Hom-Alternative and Hom-Malcev algebras. We construct deformation cohomology groups in low dimensions. Using a composition construction, we give a procedure to provide deformations of alternative algebras (resp. Malcev algebras) into Hom-alternative algebras (resp. Hom-Malcev algebras). Then it is used to supply examples for which we compute some cohomology invariants.
We introduce an equivariant version of Hochschild cohomology as the deformation cohomology to study equivariant deformations of associative algebras equipped with finite group actions.
We study deformation quantization of nonassociative algebras whose associator satisfies some symmetric relations. This study is expanded to a larger class of nonassociative algebras includind Leibniz algebras. We apply also to this class the rule of polarization-depolarization.
By means of graphical calculus we prove that, over fields of characteristic zero, any bialgebra deformation of the universal enveloping algebra of the algebra of traceless octonions satisfying the dual of the left and right Moufang identities must be coassociative and cocommutative.
In this paper, we construct a differential graded Lie algebra whose Maurer-Cartan elements are given by crossed homomorphisms on Leibniz algebras. This allows us to define cohomology for a crossed homomorphism. Finally, we study linear deformations, formal deformations and extendibility of finite order deformations of a crossed homomorphism in terms of the cohomology theory.
In this note we consider low dimensional metric Leibniz algebras with an invariant inner product over the complex numbers up to five dimension. We study their deformations, and give explicit formulas for the cocycles and deformations. We identify among those the metric deformations.
In the algebraic theory of algebra-valued noncommutative probability spaces, for a unital algebra B, a mild reformulation of Speicher's noncrossing B-valued cumulants for random variables in these spaces is used to construct canonical random variables, acting on a Fock space over B, for arbitrary families of B-valued random variables. Also, a condition for traciality of a trace on B composed with the B-valued conditional expectation is given in terms of B-valued cumulants.
The aim of this paper is to give an algebraic characterization of the rings $C(X,\mathbb{Q}_p)$ of all continuous $\mathbb{Q}_p$-valued functions on a compact space $X$. The characterization is similar to that of M. Stone from 1940 for the case of $\mathbb{R}$-valued functions.
We prove formulas for the unitary SK_1 of a semiramified graded division algebra (or valued division algebra over a Henselian field) with a unitary involution. These formulas generalize earlier formulas of Yanchevskii, (and Platonov and Ershov for the nonunitary SK_1).
The concept of modulation is generalized to pseudo-modulation and its subclasses including pre-modulation, generalized modulation and regular modulation. The motivation is to define the valued analogue of natural quiver, called {\em natural valued quiver}, of an artinian algebra so as to correspond to its valued Ext-quiver when this algebra is not $k$-splitting over the field $k$. Moreover, we illustrate the relation between the valued Ext-quiver and the natural valued quiver. The interesting fact we find is that the representation categories of a pseudo-modulation and of a pre-modulation are …
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 $sl_2$.
Algebraic scheme for constructing deformations of structure constants for associative algebras generated by a deformation driving algebras (DDAs) is discussed. An ideal of left divisors of zero plays a central role in this construction. Deformations of associative three-dimensional algebras with the DDA being a three-dimensional Lie algebra and their connection with integrable systems are studied.
The purpose of this paper is to study deformation theory of Hom-associative algebra morphisms and Hom-Lie algebra morphisms. We introduce a suitable cohomology and discuss Infinitesimal deformations, equivalent deformations and obstructions. Moreover, we provide some examples.
We describe infinitesimal deformations of complex naturally graded filiform Leibniz algebras. It is known that any $n$-dimensional filiform Lie algebra can be obtained by a linear integrable deformation of the naturally graded algebra $F_n^3(0)$. We establish that in the same way any $n$-dimensional filiform Leibniz algebra can be obtained by an infinitesimal deformation of the filiform Leibniz algebras $F_{n}^1,$ $F_{n}^2$ and $F_{n}^3(α)$. Moreover, we describe the linear integrable deformations of above-mentioned algebras with a fixed basis of $HL^2$ in the set of all $n$-dimensional Leibni…
The deformation theory of Lie-Yamaguti algebras is developed by choosing a suitable cohomology. The relationship between the deformation and the obstruction of Lie-Yamaguti algebras is obtained.
Deformations of the structure constants for a class of associative noncommutative algebras generated by Deformation Driving Algebras (DDA's) are defined and studied. These deformations are governed by the Central System (CS). Such a CS is studied for the case of DDA being the algebra of shifts. Concrete examples of deformations for the three-dimensional algebra governed by discrete and mixed continuous-discrete Boussinesq (BSQ) and WDVV equations are presented. It is shown that the theory of the Darboux transformations, at least for the BSQ case, is completely incorporated into the proposed sc…
This paper discusses the notion of a deformation quantization for an arbitrary polynomial Poisson algebra A. We examine the Hochschild cohomology group H^3(A) and find that if a deformation of A exists it can be given by bidifferential operators. We then compute an explicit third order deformation quantization of A and show that it comes from a quantized enveloping algebra. We show that the deformation extends to a fourth order deformation if and only if the quantized enveloping algebra gives a fourth order deformation; moreover we give an example where the deformation does not extend. A corre…
In this paper we produce noncommutative algebras derived equivalent to deformations of schemes with tilting bundles. We do this in two settings, first proving that a tilting bundle on a scheme lifts to a tilting bundle on an infinitesimal deformations of that scheme and then expanding this result to $\mathbb{C}^*$-equivariant deformations over schemes with a good $\mathbb{C}^*$-action. In both these situations the endomorphism algebra of the lifted tilting bundle produces a deformation of the original endomorphism algebra, and this is a graded deformation in the $\mathbb{C}^*$-equivariant case…
The aim of this paper is to review the deformation theory of $n$-Lie algebras. We summarize the 1-parameter formal deformation theory and provide a generalized approach using any unital commutative associative algebra as a deformation base. Moreover, we discuss degenerations and quantization of $n$-Lie algebras.
The purpose of this paper is to introduce an algebraic cohomology and formal deformation theory of left alternative algebras. Connections to some other algebraic structures are given also.
In this article we establish an explicit link between the classical theory of deformations à la Gerstenhaber -- and a fortiori with the Hochschild cohomology-- and (weak) PBW-deformations of homogeneous algebras. Our point of view is of cohomological nature. As a consequence, we recover a theorem by R. Berger and V. Ginzburg, which gives a precise condition for a filtered algebra to satisfy the so-called PBW property, under certain assumptions.
We compute low-degree cohomology of current Lie algebras extended over the 3-dimensional simple algebra, compute deformations of related semisimple Lie algebras, and apply these results to classification of simple Lie algebras of absolute toral rank 2 and having a Cartan subalgebra of toral rank one. Everything is in characteristic 2.
Considering a Poisson algebra as a non associative algebra satisfying the Markl-Remm identity, we study deformations of Poisson algebras as deformations of this non associative algebra. This gives a natural interpretation of deformations which preserves the underlying associative structure and we study deformations which preserve the underlying Lie algebra.
We study formal deformations of hom-Lie-Rinehart algebras. The associated deformation cohomology that controls deformations is constructed using multiderivations of hom-Lie-Rinehart algebras.
The category of Frobenius Lie algebras is stable under deformation, and here we examine explicit infinitesimal deformations of four and six dimensional Frobenius Lie algebras with the goal of understanding if the spectrum of a Frobenius Lie algebra can evolve under deformation. It can.
The study of $n$-Lie algebras which are natural generalization of Lie algebras is motivated by Nambu Mechanics and recent developments in String Theory and M-branes. The purpose of this paper is to define cohomology complexes and study deformation theory of $n$-Lie algebra morphisms. We discuss infinitesimal deformations, equivalent deformations and obstructions. Moreover, we provide various examples.
The deformation bicomplex of a module-algebra over a bialgebra is constructed. It is then applied to study algebraic deformations in which both the module structure and the algebra structure are deformed. The cases of module-coalgebras, comodule-(co)algebras, and (co)module-bialgebras are also considered.
We study formal deformations of multiplication in an operad. This closely resembles Gerstenhaber's deformation theory for associative algebras. However, this applies to various algebras of Loday-type and their twisted analogs. We explicitly describe the cohomology of these algebras with coefficients in a representation. Finally, deformation of morphisms between algebras of the same Loday-type is also considered.
Let H be a Hopf algebra and let B be a Koszul H-module algebra. We provide necessary and sufficient conditions for a filtered algebra to be a Poincare-Birkhoff-Witt (PBW) deformation of the smash product algebra B#H. Many examples of these deformations are given.
In this review article, first we give the concrete formulas of representations and cohomologies of associative algebras, Lie algebras, pre-Lie algebras, Leibniz algebras and 3-Lie algebras and some of their strong homotopy analogues. Then we recall the graded Lie algebras and graded associative algebras that characterize these algebraic structures as Maurer-Cartan elements. The corresponding Maurer-Cartan element equips the graded Lie or associative algebra with a differential. Then the deformations of the given algebraic structures are characterized as the Maurer-Cartan elements of the result…
The notion of an F-manifold algebra is the underlying algebraic structure of an $F$-manifold. We introduce the notion of pre-Lie formal deformations of commutative associative algebras and show that F-manifold algebras are the corresponding semi-classical limits. We study pre-Lie infinitesimal deformations and extension of pre-Lie n-deformation to pre-Lie (n+1)-deformation of a commutative associative algebra through the cohomology groups of pre-Lie algebras. We introduce the notions of pre-F-manifold algebras and dual pre-F-manifold algebras, and show that a pre-F-manifold algebra gives rise …
An algebraic deformation theory of module-algebras over a bialgebra is constructed. The cases of module-coalgebras, comodule-algebras, and comodule-coalgebras are also considered.
We introduce an $A_\infty$-algebra structure on the Hochschild cohomology of the endomorphism bimodule of a finite-dimensional representation of an associative algebra. We prove that this structure determines a presentation for non-commutative deformations of the representation. From this, we deduce presentations of universal deformation rings of Galois representations. In turn, we apply these presentations in order to deduce universal deformation rings of Galois pseudorepresentations, supplying a a tangent and obstruction theory for pseudorepresentations. This generalizes the broadly used tan…
