Papers by Alice Fialowski
Proceedings of the Steklov Institute of Mathematics, Oct 1, 2014
I describe the basic notions of versal deformation theory of algebraic structures and compare it ... more I describe the basic notions of versal deformation theory of algebraic structures and compare it with the analytic theory. As a special case, I 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. I compare this with Arnold's stratification in the case of similarity of matrices. The other notion I discuss is the opposite notion of contraction.
The formal rigidity of the Witt and Virasoro algebras was first established by the author in [4].... more The formal rigidity of the Witt and Virasoro algebras was first established by the author in [4]. The proof was based on some earlier results of the author and Goncharowa, and was not presented there. In this paper we give an elementary proof of these facts.
We develop deformation theory of algebras over quadratic operads where the parameter space is a c... more 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.
We study finite-dimensional commutative algebras, which satisfy the Jacobi identity. Such algebra... more 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.
In this paper, we study 4-dimensional nilpotent complex associative algebras. This is a continuat... more 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 algebras were analyzed in an earlier paper. Even though there are only 15 families of nilpotent 4-dimensional algebras, the complexity of their behavior warranted a separate study, which we give here.
Cornell University - arXiv, Nov 14, 2017
In this note we compute the homology of the Lie algebra gl(∞, R) where R is an associative unital... more In this note we compute the homology of the Lie algebra gl(∞, R) where R is an associative unital k-algebra which is used in higher dimensional soliton theory [Ta]. When k is a field of characteristic 0, our result justifies an old result of Feigin and Tsygan [FT]. The special case when R = k = C appeared first in soliton theory (cf. [JM]). Dedicated to the memory of our friend Jean-Louis Loday 0. Introduction Among several versions of the Lie algebra gl of infinite rank, the Lie algebra gl(∞), that has been extensively used to describe the soliton solutions of the Kadomtsev-Petviashvili (KP in short) hierarchy (see, e.g., [DJM] for detail) in the first half of 1980's, has a special feature. For example, the Lie algebra gl(∞) is neither ind-finite nor pro-finite. For this reason, it had been a difficult task to analyze its algebraic properties. In 1983, B. Feigin and B. Tsygan published a short note [FT] (only 2 pages long !) where they determined the homology of the Lie algebra gl(∞, k) where k is a field of characteristic 0. They denoted this Lie algebra by gJ(k) that is recalled in §1.1. Unfortunately, it seems that their paper is too dense to decompress, so that this article had not been studied carefully in the mathematical community. At the same time, their note generated much interest, and-even 35 years later-the statements are important. In this paper, we managed to compute the homology of the Lie algebra gl(∞, R), where R is an associative unital k-algebra and k is a field of characteristic 0. The case R = k was treated by B. Feigin and B. Tsygan briefly in their note [FT]. We hope that our paperbeside generalizing the case which shows up in soliton theory-also makes the article [FT] accessible to the mathematical community. By an argument similar to B. Feigin and B. Tsygan [FT], we have seen that the primitive part of the homology H • (gJ(R), k) is isomorphic to the cyclic homology HC •−1 (J(R)), where J(R) = gJ(R) (as k-vector space) viewed as an associative k-algebra. Hence, the real problem is to express this cyclic homology in terms of the homology theory of R, namely, without intervention of J. With the aid of an analogue of the Hochschild-Serre type spectral sequence due to D. Stefan [St], we see that this spectral sequence degenerates at E 2-term. This allows us to show in Theorem 6.1 that the homology HH • (J(R)) is isomorphic to HH •−1 (R). Furthermore, a detailed analysis of the above spectral sequence shows that a similar phenomena is valid, that is, the cyclic homology HC • (J(R)) is isomorphic to the cyclic homology HC •−1 (R).
Nonassociative algebra and its applications, 2019
This article explores \Z_2-graded L_\infinity algebra structures on a 2|1-dimensional vector spac... more This article explores \Z_2-graded L_\infinity algebra structures on a 2|1-dimensional vector space. The reader should note that our convention on the parities is the opposite of the usual one, because we define our structures on the symmetric coalgebra of the parity reversion of a space, so our 2|1-dimensional L_\infinity algebras correspond to the usual 1|2-dimensional algebras. We give a complete classification of all structures with a nonzero degree 1 term. We also classify all degree 2 codifferentials, which is the same as a classification of all 1|2-dimensional Z_2-graded Lie algebras. For each of these algebra structures, we calculate the cohomology and a miniversal deformation.
In this work we consider deformations of Leibniz algebras over a field of characteristic zero. Th... more In this work we consider deformations of Leibniz algebras over a field of characteristic zero. The main problem in deformation theory is to describe all non-equivalent deformations of a given object. We give a method to solve this problem completely, namely work out a construction of a versal deformation for a given Leibniz algebra, which induces all non-equivalent deformations and is unique on the infinitesimal level.
Rendiconti Lincei - Matematica e Applicazioni, 2020
In this note, we compute the homology with trivial coefficients of Lie algebras of generalized Ja... more 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.
EMS Newsletter, 2017
We are very glad to see you here in Reims at this very special conference dedicated to your 3 4 t... more We are very glad to see you here in Reims at this very special conference dedicated to your 3 4 th birthday. We would like to ask you a few questions. To start with, how did you choose to do mathematics? You mean, how did I choose to be a mathematician? Well, I think till the fifth grade at school, 1 my dream was to be a pilot. But then my eyesight was not very good; at the end of school I had-4 and at university-5, so the career of a pilot was closed for me. Also, I like very much to dive but I've never thought of being a professional diver. I was a member of the university diving team and even won the "All-Union Student's Game" ["Универсиада"-Universiade], which involved university teams from all over the Soviet Union competing in different forms of sports. That was a very interesting story but not for the official record. Starting from the 6th grade, I participated in the mathematical Olympiads: my teacher at school said that there was such a thing in the 6th grade, I think. I did not go to the mathematical circles 2 because I was rather shy and in mathematical circles they make you answer ques
Topics in Singularity Theory, 1997
We develop deformation theory of algebras over quadratic operads where the parameter space is a c... more 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 memory of our good friend Jean-Louis Loday
International Journal of Algebra and Computation, 2022
In this note, we consider low-dimensional metric Leibniz algebras with an invariant inner product... more In this note, we consider low-dimensional metric Leibniz algebras with an invariant inner product over the complex numbers up to dimension [Formula: see text]. We study their deformations, and give explicit formulas for the cocycles and deformations. We identify among those the metric deformations.
Applications of Mathematics and Informatics in Natural Sciences and Engineering, 2020
In this paper I consider deformations with the base being a general commutative algebra with iden... more In this paper I consider deformations with the base being a general commutative algebra with identity. It turns out that in infinite dimension such global deformations give a much richer picture, depending on the augmentation of the base algebra. It is of course not the case if one considers deformations with complete local algebra base. In infinite dimension, even rigidity is not kept by considering global deformations, as the Witt/Virasoro algebra shows. It is formally rigid, but it has plenty nontrivial nice global deformations, like the Krichever-Novikov type algebras.
This article explores L∞ algebra structures on a 2|1dimensional vector space. We do not give a co... more This article explores L∞ algebra structures on a 2|1dimensional vector space. We do not give a complete classification in all cases, but do determine all structures which begin with either a nontrivial first or second order term. In particular, we determine all extensions of a super Lie algebra as an L∞ algebra. The reader should note that our convention on the parities is the opposite of the usual one, because we define our structures on the symmetric coalgebra of the parity reversion of a space, so our 2|1-dimensional L∞ algebras correspond to the usual 1|2-dimensional algebras.
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Papers by Alice Fialowski