arXiv subjects
Publications and source records attributed to Alice Fialowski.
In this note we explicitly give all the equivalent classes of deformations of the 5-dimensional Heisenberg Lie algebra $\mathfrak{h}_2$ over complex or real number fields. We show that there are altogether 20 infinitesimal deformations (families), 18 of them being extendable to real deformations and 2 of them are only infinitesimal.
We show that the Heisenberg Lie algebras over a field $\mathbb{F}$ of characteristic $p>0$ admit a family of restricted Lie algebras, and we classify all such non-isomorphic restricted Lie algebra structures. We use the ordinary 1- and 2-cohomology spaces with trivial coefficients to compute the restricted 1- and 2-cohomology spaces of these restricted Heisenberg Lie algebras. We describe the restricted 1-dimensional central extensions, including explicit formulas for the Lie brackets and $\cdot^{[p]}$-operators.
Consider the maximal nilpotent subalgebra $n_+(A_1^{(1)})$ of the simplest affine algebra $A_1^{(1)}$ which is one of the $\mathbb{N}$-graded Lie algebras with minimal number of generators. We show truncated versions of this algebra in positive characteristic admit the structure of a family of restricted Lie algebras. We compute the ordinary and restricted 1- and 2-cohomology spaces with trivial coefficients by giving bases. With these we explicitly describe the restricted 1-dimensional central extensions.
In this work we consider all metric Lie algebras, having a nondegenerate symmetric invariant bilinear form, over \C and \R up to dimension 5 and all metric Lie algebras over \C in dimension 6. We introduce cyclic and reduced cyclic cohomology to identify their metric deformations.
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.
This article is a collection of several memories for a special issue of SIGMA devoted to Dmitry Borisovich Fuchs.
In this lecutre note, we consider infinite dimensional Lie algebras of generalized Jacobi matrices $\mathfrak{g}J(k)$ and $\mathfrak{gl}_\infty(k)$, which are important in soliton theory, and their orthogonal and symplectic subalgebras. In particular, we construct the homology ring of the Lie algebra $\mathfrak{g}J(k)$ and of the orthogonal and symplectic subalgebras.
For the $p$-dimensional filiform Lie algebra ${\mathfrak m}_2(p)$ over a field ${\mathbb F}$ of prime characteristic $p\ge 5$ with nonzero Lie brackets $[e_1,e_i] = e_{i+1}$ for $1<i<p$ and $[e_2,e_i]=e_{i+2}$ for $2<i<p-1$, we show that there is a family ${\mathfrak m}_2^{\lambda}(p)$ of restricted Lie algebra structures parameterized by elements $\lambda \in {\mathbb F}^p$. We explicitly describe bases for the ordinary and restricted 1- and 2-cohomology spaces with trivial coefficients, and give formulas for the bracket and $[p]$-operations in the corresponding restricted one-dimensional central extensions.
We show, for a field ${\mathbb F}$ of prime characteristic $p>0$, that the truncated filiform Lie algebra ${\frak m}_0(p)$ admits a family ${\frak m}_0^\lambda(p)$ of restricted Lie algebra structures parameterized by elements $\lambda\in {\mathbb F}^p$. We compute the ordinary cohomology groups $H^q({\frak m}_0^\lambda(p))$ and restricted cohomology groups $H^q_*({\frak m}_0^\lambda(p))$ for $q=1, 2$, and we give explicit descriptions of bases for these cohomology spaces. We apply our results to restricted one-dimensional central Extensions of the algebras ${\frak m}_0^\lambda(p)$.
In this note, we compute the homology with trivial coefficients of Lie algebras of generalized Jacobi matrices of type $B, C$ and $D$ over an associative unital $k$-algebra with $k$ being a field of characteristic $0$.
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.
In this paper, we describe restricted one-dimensional central extensions of all finite dimensional simple restricted Lie algebras defined over fields of characteristic $p\ge 5$.
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.
We classify one-dimensional restricted central extensions of the modular Witt Lie algebra in characteristic $p>3$.
We study finite-dimensional commutative algebras, which satisfy the Jacobi identity. Such algebras are Jordan algebras. We describe some of their properties and give a classification in dimensions $n<7$ over algebraically closed fields of characteristic not $2$ or $3$.
We develop deformation theory of algebras over quadratic operads where the parameter space is a commutative local algebra. We also give a construction of a distinguised deformation of an algebra over a quadratic operad with a complete local algebra as its base--the so called `versal deformation'--which induces all other deformations of the given algebra.
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.
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.