9,136 research outputs found
The moduli space of 3-dimensional associative algebras
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.Comment: 24 pages, 1 figur
Examples of infinity and Lie algebras and their versal deformations
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.Comment: 20 page
Graphs with girth at least 5 with orders between 20 and 32
We prove properties of extremal graphs of girth 5 and order 20 <=v <= 32. In
each case we identify the possible minimum and maximum degrees, and in some
cases prove the existence of (non-trivial) embedded stars. These proofs allow
for tractable search for and identification of all non isomorphic cases
The moduli space of complex 5-dimensional Lie algebras
In this paper, we study the moduli space of all complex 5-dimensional Lie
algebras, realizing it as a stratification by orbifolds, which are connected by
jump deformations. The orbifolds are given by the action of finite groups on
very simple complex manifolds. Our method of determining the stratification is
based on the construction of versal deformations of the Lie algebras, which
allow us to identify natural neighborhoods of the elements in the moduli space
Examples of Miniversal Deformations of Infinity Algebras
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.Comment: 25 page
The moduli space of 4-dimensional non-nilpotent complex associative algebras
In this paper we study the moduli space of 4-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. Because the space of
4-dimensional algebras is large, we only classify the non-nilpotent algebras in
this paper.Comment: 26 pages, 2 figure
Moduli spaces of low dimensional Lie superalgebras
In this paper, we study moduli spaces of low dimensional complex Lie
superalgebras. We discover a similar pattern for the structure of these moduli
spaces as we observed for ordinary Lie algebras, namely, that there is a
stratification of the moduli space by projective orbifolds. The moduli spaces
consist of some families as well as some singleton elements. The different
strata are linked by jump deformations, which gives a uniques manner of
decomposing the moduli space which is consistent with deformation theory.Comment: 28 page
Stratification of moduli spaces of Lie algebras, similar matrices and bilinear forms
In this paper, the authors apply a stratification of moduli spaces of complex
Lie algebras to analyzing the moduli spaces of nxn matrices under scalar
similarity and bilinear forms under the cogredient action. For similar
matrices, we give a complete description of a stratification of the space by
some very simple projective orbifolds of the form P^n/G, where G is a subgroup
of the symmetric group sigma_{n+1} acting on P^n by permuting the projective
coordinates. For bilinear forms, we give a similar stratification up to
dimension 3.Comment: 18 pages, 4 figure
Versal Deformations of Three Dimensional Lie Algebras as L-infinity Algebras
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.Comment: 21 page
The moduli space of 4-dimensional nilpotent complex associative algebras
In this paper, we study 4-dimensional nilpotent complex associative algebras.
This is a continuation of the study of the moduli space of 4-dimensional
algebras. The non-nilpotent algeras were analyzed in an earlier paper. Even
though there are only 15 families of nilpotent 4-dimensional algebras, the
complexity of their behaviour warrented a separate study, which we give here.Comment: 17 page
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