Papers by Alice Fialowski, Fialovski
RESEARCH-EXPOSITORY PAPERS Chang, Sun-Yung A. and Fefferman, Robert, Some recent developments in ... more RESEARCH-EXPOSITORY PAPERS Chang, Sun-Yung A. and Fefferman, Robert, Some recent developments in Fourier analysis and Hp-theory on product domains, 12, 1 Fefferman, Robert. See Chang, Sun-Yung A. Fournier, John JF and Stewart, James, Amalgams of LP ...
arXiv (Cornell University), Jul 20, 2008
We develop the theory of versal deformations of dialgebras and describe a method for constructing... more We develop the theory of versal deformations of dialgebras and describe a method for constructing a miniversal deformation of a dialgebra.
arXiv (Cornell University), Nov 8, 2006
In this paper, we give a purely cohomological interpretation of the extension problem for (super)... more In this paper, we give a purely cohomological interpretation of the extension problem for (super) Lie algebras; that is the problem of extending a Lie algebra by another Lie algebra. We then give a similar interpretation of infinitesimal deformations of extensions. In particular, we consider infinitesimal deformations of representations of a Lie algebra.
EMS Newsletter, 2016
doctoral thesis was also related to Piatetski-Shapiro's question. It concerned the Selberg conjec... more doctoral thesis was also related to Piatetski-Shapiro's question. It concerned the Selberg conjecture about arithmeticity of discrete subgroups, so-called lattices, in semisimple Lie groups. Piatetski-Shapiro was interested in this problem but he did not manage to solve it, although he had some results in this direction. I developed a theory of hyperbolic reflection groups, which permitted me to construct many counterexamples at rank one. After that, Margulis proved his famous theorem for higher rank, answering affirmatively the Selberg conjecture. My work on hyperbolic reflection groups had two sources: one was Dynkin's theory of simple roots, which is tightly related to finite reflection groups; and the other was the theory of automorphic forms for lattices in semisimple Lie groups, which was a favourite subject of Piatetski-Shapiro. So in a sense I continued studying some questions they posed to me (both Dynkin and Piatetski-Shapiro). In 1966, there was the International Congress
Newsletter of the European Mathematical Society, 2016
doctoral thesis was also related to Piatetski-Shapiro's question. It concerned the Selberg conjec... more doctoral thesis was also related to Piatetski-Shapiro's question. It concerned the Selberg conjecture about arithmeticity of discrete subgroups, so-called lattices, in semisimple Lie groups. Piatetski-Shapiro was interested in this problem but he did not manage to solve it, although he had some results in this direction. I developed a theory of hyperbolic reflection groups, which permitted me to construct many counterexamples at rank one. After that, Margulis proved his famous theorem for higher rank, answering affirmatively the Selberg conjecture. My work on hyperbolic reflection groups had two sources: one was Dynkin's theory of simple roots, which is tightly related to finite reflection groups; and the other was the theory of automorphic forms for lattices in semisimple Lie groups, which was a favourite subject of Piatetski-Shapiro. So in a sense I continued studying some questions they posed to me (both Dynkin and Piatetski-Shapiro). In 1966, there was the International Congress
arXiv (Cornell University), Aug 7, 2022
Consider the maximal nilpotent subalgebra n + (A (1) 1) of the simplest affine algebra A (1) 1 wh... more Consider the maximal nilpotent subalgebra n + (A (1) 1) of the simplest affine algebra A (1) 1 which is one of the 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.
arXiv (Cornell University), Aug 7, 2022
Consider the maximal nilpotent subalgebra n + (A (1) 1) of the simplest affine algebra A (1) 1 wh... more Consider the maximal nilpotent subalgebra n + (A (1) 1) of the simplest affine algebra A (1) 1 which is one of the 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.
Communications in Contemporary Mathematics, Oct 1, 2009
In this paper, we give a purely cohomological interpretation of the extension problem for (super)... more In this paper, we give a purely cohomological interpretation of the extension problem for (super) Lie algebras; that is the problem of extending a Lie algebra by another Lie algebra. We then give a similar interpretation of infinitesimal deformations of extensions. In particular, we consider infinitesimal deformations of representations of a Lie algebra.
arXiv (Cornell University), Oct 16, 2005
A classical problem in algebraic deformation theory is whether an infinitesimal deformation can b... more 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.
arXiv (Cornell University), Sep 3, 2017
In this paper, we study moduli spaces of low dimensional complex Lie superalgebras. We discover a... more 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 unique manner of decomposing the moduli space which is consistent with deformation theory.
arXiv (Cornell University), Aug 3, 2017
In this paper, the authors apply a stratification of moduli spaces of complex Lie algebras to ana... more In this paper, the authors apply a stratification of moduli spaces of complex Lie algebras to analyzing the moduli spaces of n × n 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 Σ n+1 acting on P n by permuting the projective coordinates. For bilinear forms, we give a similar stratification up to dimension 4.
Journal of Algebra, Jun 1, 2008
Denote m0 the infinite dimensional N-graded Lie algebra defined by basis ei i ≥ 1 and relations [... more Denote m0 the infinite dimensional N-graded Lie algebra defined by basis ei i ≥ 1 and relations [e1, ei] = ei+1 for all i ≥ 2. We compute in this article the bracket structure on H 1 (m0, m0) , H 2 (m0, m0) and in relation to this, we establish that there are only finitely many true deformations of m0 in each nonpositive weight by constructing them explicitely. It turns out that in weight 0 one gets exactly the other two filiform Lie algebras.
Journal of Algebra, Apr 1, 2010
We compute the second Hochschild cohomology space HH 2 (H 1) of Connes-Moscovici's Hopf algebra H... more We compute the second Hochschild cohomology space HH 2 (H 1) of Connes-Moscovici's Hopf algebra H 1 , giving the infinitesimal deformations (up to equivalence) of the associative structure. The space HH 2 (H 1) is shown to be one-dimensional.
Journal of Algebra, Dec 1, 2007
Denote m 0 the infinite-dimensional N-graded Lie algebra defined by basis e i , i 1, and relation... more Denote m 0 the infinite-dimensional N-graded Lie algebra defined by basis e i , i 1, and relations [e 1 , e i ] = e i+1 for all i 2. We compute in this article the bracket structure on H 1 (m 0 , m 0), H 2 (m 0 , m 0) and in relation to this, we establish that there are only finitely many true deformations of m 0 in each nonpositive weight by constructing them explicitly. It turns out that in weight 0 one gets exactly the other two filiform Lie algebras.
HAL (Le Centre pour la Communication Scientifique Directe), 2008
Denote m0 the infinite dimensional N-graded Lie algebra defined by basis ei i ≥ 1 and relations [... more Denote m0 the infinite dimensional N-graded Lie algebra defined by basis ei i ≥ 1 and relations [e1, ei] = ei+1 for all i ≥ 2. We compute in this article the bracket structure on H 1 (m0, m0) , H 2 (m0, m0) and in relation to this, we establish that there are only finitely many true deformations of m0 in each nonpositive weight by constructing them explicitely. It turns out that in weight 0 one gets exactly the other two filiform Lie algebras.
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.
The aim of this paper is to compare stratifications of moduli spaces given by group actions in th... more The aim of this paper is to compare stratifications of moduli spaces given by group actions in the case of similarity of matrices introduced by Arnold and the author's stratification by projective orbifolds, and its relation to deformations of elements in the moduli space.
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Papers by Alice Fialowski, Fialovski