A comparison of deformations and geometric study of associative algebras varieties (original) (raw)
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The aim of this paper is to give an overview and to compare the different deformation theories of algebraic structures. In each case we describe the corresponding notions of degeneration and rigidity. We illustrate these notions by examples and give some general properties. The last part of this work shows how these notions help in the study of varieties of associative algebras. The first and popular deformation approach was introduced by M. Gerstenhaber for rings and algebras using formal power series. A noncommutative version was given by Pinczon and generalized by F. Nadaud. A more general approach called global deformation follows from a general theory by Schlessinger and was developed by A. Fialowski in order to deform infinite-dimensional nilpotent Lie algebras. In a nonstandard framework, M. Goze introduced the notion of perturbation for studying the rigidity of finite-dimensional complex Lie algebras. All these approaches share the common fact that we make an "extension" of the field. These theories may be applied to any multilinear structure. In this paper, we will be dealing with the category of associative algebras.
A Comparison of Deformations and Geometric Study of Varieties of Associative Algebras
International Journal of Mathematics and Mathematical Sciences, 2007
The aim of this paper is to give an overview and to compare the different deformation theories of algebraic structures. We describe in each case the corresponding notions of degeneration and rigidity. We illustrate these notions with examples and give some general properties. The last part of this work shows how these notions help in the study of associative algebras varieties. The first and popular deformation approach was introduced by M. Gerstenhaber for rings and algebras using formal power series. A noncommutative version was given by Pinczon and generalized by F. Nadaud. A more general approach called global deformation follows from a general theory of Schlessinger and was developed by A. Fialowski in order to deform infinite dimensional nilpotent Lie algebras. In a Nonstandard framework, M.Goze introduced the notion of perturbation for studying the rigidity of finite dimensional complex Lie algebras. All these approaches has in common the fact that we make an "extension" of the field. These theories may be applied to any multilinear structure. We shall be concerned in this paper with the category of associative algebras.
Deformations of Complex 3-Dimensional Associative Algebras
Journal of Generalized Lie Theory and Applications, 2011
In this paper, we study deformations and the moduli space of 3-dimensional complex associative algebras. We use extensions to compute the moduli space, and then give a decomposition of this moduli space into strata consisting of complex projective orbifolds, glued together through jump deformations. The main purpose of this paper is to give a logically organized description of the moduli space, and to give an explicit description of how the moduli space is constructed by extensions.
Deformations and Contractions of Algebraic Structures
Труды Математического института им. Стеклова, 2014
We describe the basic notions of versal deformation theory of algebraic structures and compare it with the analytic theory. As a special case, we consider the notion of versal deformation used by Arnold. With the help of versal deformation we get a stratification of the moduli space into projective orbifolds. We compare this with Arnold's stratification in the case of similarity of matrices. The other notion we discuss is the opposite notion of contraction.
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 (K) and describe as formal deformations the q-deformed Witt algebra and Jackson sl 2 (K).
Speciality and deformations of algebras
2000
The notion of speciality has come from the theory of Jordan algebras. A Jordan algebra is called special if it admits an isomorphic embedding into an associative algebra with respect to a symmetrized multiplication a • b = 1/2(ab + ba). Not any Jordan algebra is special; moreover, the variety generated by all special algebras neither coincides with the class of all Jordan algebras, nor with the class of all special Jordan algebras. The algebras from this variety are called i-special. Both speciality and i-speciality can also be naturally defined for superalgebras.
The CROCs, non-commutative deformations, and (co)associative bialgebras
arXiv: Quantum Algebra, 2003
We compactify the spaces K(m,n)K(m,n)K(m,n) introduced by Maxim Kontsevich. The initial idea was to construct an LinftyL_\inftyLinfty 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 construct a dg Lie algebra conjecturally governing the commutative (usual) deformations of a (co)associative bialgebra. Philosophically, the main point is that for the associative bialgebras the non-commutative deformations is maybe a more fundamental object than the usual commutative ones.
Notes on formal deformations of Hom-associative and Hom-Lie algebras, to appear in Forum Math
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 (K) and describe as formal deformations the q-deformed Witt algebra and Jackson sl 2 (K).
Notes on 1-parameter formal deformations of Hom-associative and Hom-Lie algebras
2010
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 (K) and describe as formal deformations the q-deformed Witt algebra and Jackson sl 2 (K).