We consider versal deformations of 0|3-dimensional L-infinity algebras, which correspond precisely to ordinary (non-graded) three dimensional Lie algebras. The classification of such algebras over C is well known, although we shall give a derivation of this classification using an approach of treating them as L-infinity algebras. Because the symmetric algebra of a three dimensional odd vector space contains terms only of exterior degree less than or equal to three, the construction of versal deformations can be carried out completely.
We give a characterization of the moduli space of Lie algebras using deformation theory as a guide to understanding the picture.
This article explores some simple examples of L-infinity algebras and the construction of miniversal deformations of these structures. Among other things, it is shown that there are two families of nonequivalent L-infinity structures on a 1|1 dimensional vector space, two of which are Lie algebra structures. The main purpose of this work is to provide a simple effective procedure for constructing miniversal deformations, using the examples to illustrate the general technique. The same method can be applied directly to construct versal deformations of Lie algebras.
The purpose of this paper is to study global deformations of Hom-Leibniz algebras. We introduce a cohomology for Hom-Leibniz algebras with values in a Hom-module, characterize versal deformations and provide examples.
For a pair (algebra, module) with equidimensional and isolated singularity we establish the existence of a versal henselian deformation. Obstruction theory in terms of an André-Quillen cohomology for pairs is a central ingredient in the Artin theory used. In particular we give a long exact sequence relating the algebra cohomology and the module cohomology with the cohomology of the pair and define a Kodaira-Spencer class for pairs. Cohen-Macaulay approximation induces maps between versal base spaces for pairs and cohomology conditions imply properties like smoothness, isomorphism and linear se…
In this paper we study cohomology and deformations of Jacobi-Jordan algebras. We develop their formal deformation theory. In particular, we introduce a method to construct a versal deformation for a given Jacobi-Jordan algebra, which can induce all deformations and is unique on the infinitesimal level. We construct formal 1-parameter deformations of Jacobi-Jordan algebras up to dimension 5 and versal deformations for 3-dimensional Jacobi-Jordan algebras.
A rather complete phenomenology of the singularities is developed according to a new algebraic point of view in the frame of Langlands global correspondences. That is to say,a process of: -singularizations and versal deformations of these, -singularizations and monodromies of these, is envisaged on all the sections of sheaves of differentiable (bi)functions on (bi)linear algebraic (semi)groups constituting the n-dimensional representations of the global Weil groups. To get the searched holomorphic and cuspidal representations,it is necessary to consider: -the resolutions of the singularities a…
Let $\mathbf{k}$ be field of arbitrary characteristic and let $Λ$ be a finite dimensional $\mathbf{k}$-algebra. From results previously obtained by F.M Bleher and the author, it follows that if $V^\bullet$ is an object of the bounded derived category $\mathcal{D}^b(Λ\textup{-mod})$ of $Λ$, then $V^\bullet$ has a well-defined versal deformation ring $R(Λ, V^\bullet)$, which is complete local commutative Noetherian $\mathbf{k}$-algebra with residue field $\mathbf{k}$, and which is universal provided that $\textup{Hom}_{\mathcal{D}^b(Λ\textup{-mod})}(V^\bullet, V^\bullet)=\mathbf{k}$. Let $\mathc…
We study the moduli space of four dimensional ordinary Lie algebras, and their versal deformations. Their classification is well known; our focus in this paper is on the deformations, which yield a picture of how the moduli space is assembled. Surprisingly, we get a nice geometric description of this moduli space essentially as an orbifold, with just a few exceptional points.
Let $Λ$ be a finite-dimensional algebra over a fixed algebraically closed field $\mathbf{k}$ of arbitrary characteristic, and let $V$ be a finitely generated $Λ$-module. It follows from results previously obtained by F.M. Bleher and the third author that $V$ has a well-defined versal deformation ring $R(Λ, V)$, which is a complete local commutative Noetherian $\mathbf{k}$-algebra with residue field $\mathbf{k}$. The third author also proved that if $Λ$ is a Gorenstein $\mathbf{k}$-algebra and $V$ is a Cohen-Macaulay $Λ$-module whose stable endomorphism ring is isomorphic to $\mathbf{k}$, then …
Let $k$ be a field and let $Λ$ be a finite dimensional $k$-algebra. We prove that every bounded complex $V^\bullet$ of finitely generated $Λ$-modules has a well-defined versal deformation ring $R(Λ,V^\bullet)$ which is a complete local commutative Noetherian $k$-algebra with residue field $k$. We also prove that nice two-sided tilting complexes between $Λ$ and another finite dimensional $k$-algebra $Γ$ preserve these versal deformation rings. Additionally, we investigate stable equivalences of Morita type between self-injective algebras in this context. We apply these results to the derived eq…
Let $\mathbf{k}$ be a fixed field of arbitrary characteristic, and let $Λ$ be a finite dimensional $\mathbf{k}$-algebra. Assume that $V$ is a left $Λ$-module of finite dimension over $\mathbf{k}$. F. M. Bleher and the author previously proved that $V$ has a well-defined versal deformation ring $R(Λ,V)$ which is a local complete commutative Noetherian ring with residue field isomorphic to $\mathbf{k}$. Moreover, $R(Λ,V)$ is universal if the endomorphism ring of $V$ is isomorphic to $\mathbf{k}$. In this article we prove that if $Λ$ is a basic connected cycle Nakayama algebra without simple modu…
In this paper, we consider the versal deformations of three dimensional Lie algebras. We classify Lie algebras and study their deformations by using linear algebra techniques to study the cohomology. We will focus on how the deformations glue the space of all such structures together. This space is known as the moduli space. We will give a geometric description of this space, derived from deformation theory, in order to illustrate general features of moduli spaces of Lie algebras.
Let $\mathbf{k}$ be a field of arbitrary characteristic, let $Λ$ be a finite dimensional $\mathbf{k}$-algebra, and let $V$ be a finitely generated $Λ$-module. F. M. Bleher and the third author previously proved that $V$ has a well-defined versal deformation ring $R(Λ,V)$. If the stable endomorphism ring of $V$ is isomorphic to $\mathbf{k}$, they also proved under the additional assumption that $Λ$ is self-injective that $R(Λ,V)$ is universal. In this paper, we prove instead that if $Λ$ is arbitrary but $V$ is Gorenstein-projective then $R(Λ,V)$ is also universal when the stable endomorphism ri…
In this paper, we give a classification of the 3-dimensional associative algebras over the complex numbers, including a construction of the moduli space, using versal deformations to determine how the space is glued together.
A classical problem in algebraic deformation theory is whether an infinitesimal deformation can be extended to a formal deformation. The answer to this question is usually given in terms of Massey powers. If all Massey powers of the cohomology class determined by the infinitesimal deformation vanish, then the deformation extends to a formal one. We consider another approach to this problem, by constructing a miniversal deformation of the algebra. One advantage of this approach is that it answers not only the question of existence, but gives a construction of an extension as well.
Let $\mathbf{k}$ be a field of arbitrary characteristic, let $Λ$ be a finite dimensional $\mathbf{k}$-algebra, and let $V$ be an indecomposable Gorenstein-projective $Λ$-module with finite dimension over $\mathbf{k}$. It follows that $V$ has a well-defined versal deformation ring $R(Λ, V)$, which is complete local commutative Noetherian $\mathbf{k}$-algebra with residue field $\mathbf{k}$, and which is universal provided that the stable endomorphism ring of $V$ is isomorphic to $\mathbf{k}$. We prove that if $Λ$ is a monomial algebra without overlaps, then $R(Λ,V)$ is universal and isomorphic …
Let $Λ$ be a basic finite dimensional algebra over an algebraically closed field $\mathbf{k}$, and let $\widehatΛ$ be the repetitive algebra of $Λ$. In this article, we prove that if $\widehat{V}$ is a left $\widehatΛ$-module with finite dimension over $\mathbf{k}$, then $\widehat{V}$ has a well-defined versal deformation ring $R(\widehatΛ,\widehat{V})$, which is a local complete Noetherian commutative $\mathbf{k}$-algebra whose residue field is also isomorphic to $\mathbf{k}$. We also prove that $R(\widehatΛ,\widehat{V})$ is universal provided that $\underline{\mathrm{End}}_{\widehatΛ}(\wideh…
This work explores the deformation theory of algebraic structures in a very general setting. These structures include commutative, associative algebras, Lie algebras, and the infinity versions of these structures, the strongly homotopy associative and Lie algebras. In all these cases the algebra structure is determined by an element of a certain graded Lie algebra which plays the role of a differential on this algebra. We work out the deformation theory in terms of the Lie algebra of coderivations of an appropriate coalgebra structure and construct a universal infinitesimal deformation as well…
We consider the algebra $\mathcal O(\mathsf M)$ of observables and the (formally) versal morphism $η: A \to \mathcal O(\mathsf M)$ defined by the noncommutative deformation functor $\mathsf{Def}_{\mathsf M}$ of a family $\mathsf M = \{ M_1, \dots, M_r \}$ of right modules over an associative $k$-algebra $A$. By the Generalized Burnside Theorem, due to Laudal, $η$ is an isomorphism when $A$ is finite dimensional, $\mathsf M$ is the family of simple $A$-modules, and $k$ is an algebraically closed field. The purpose of this paper is twofold: First, we prove a form of the Generalized Burnside Theo…
Let $\bold G$ be a reductive algebraic group defined over $\Q$, and let $Γ$ be an arithmetic subgroup of $\bold G(\Q)$. Let $X$ be the symmetric space for $\bold G(\R)$, and assume $X$ is contractible. Then the cohomology (mod torsion) of the space $X/Γ$ is the same as the cohomology of $Γ$. In turn, $X/Γ$ will have the same cohomology as $W/Γ$, if $W$ is a ``spine'' in $X$. This means that $W$ (if it exists) is a deformation retract of $X$ by a $Γ$-equivariant deformation retraction, that $W/Γ$ is compact, and that $\dim W$ equals the virtual cohomological dimension (vcd) of $Γ$. Then $W$ can…
