Related papers: Central extensions of Steinberg Lie superalgebras …
In this paper we prove some general results on Leibniz 2-cocycles for simple Leibniz algebras. Applying these results we establish the triviality of the second Leibniz cohomology for a simple Leibniz algebra with coefficients in itself,…
This is an old paper put here for archeological purposes. We compute the second cohomology of current Lie algebras of the form $L\otimes A$, where $L$ belongs to some class of Lie algebras which includes classical simple and Zassenhaus…
Suppose the ground field to be algebraically closed and of characteristic different from $2$ and $3$. All Heisenberg Lie superalgebras consist of two super versions of the Heisenberg Lie algebras, $\frak{h}_{2m,n}$ and $\frak{ba}_n$ with…
In this paper, an explicit construction of a countable parafree Lie algebra over $\mathbb Z/2$ with nonzero second homology is given. It is also shown that the cohomological dimension of the pronilpotent completion of a free noncyclic…
Over algebraically closed fields of positive characteristic, for simple Lie (super)algebras, and certain Lie (super)algebras close to simple ones, with symmetric root systems (such that for each root, there is minus it of the same…
We exhibit an explicit construction for the second cohomology group $H^2(L, A)$ for a Lie ring $L$ and a trivial $L$-module $A$. We show how the elements of $H^2(L, A)$ correspond one-to-one to the equivalence classes of central extensions…
A Lie algebra $L$ of dimension $n \ge1 $ may be classified, looking for restrictions of the size on its second integral homology Lie algebra $H_2(L,\mathbb{Z})$, denoted by $M(L)$ and often called Schur multiplier of $L$. In case $L$ is…
In this expository article, we describe the classification of the subalgebras of the rank 2 semisimple Lie algebras. Their semisimple subalgebras are well-known, and in a recent series of papers, we completed the classification of the…
In this article, we explore the second integral homology, or Schur multiplier, of the special linear group ${\rm SL}_2(\mathbb{Z}[1/n])$ for a positive integer $n$. We definitively calculate the group structure of $H_2({\rm…
We compute the cohomology with trivial coefficients of Lie algebras $\mathfrak{m}_0$ and $\mathfrak{m}_2$ of maximal class over the field $\mathbb{Z}_2$. In the infinite-dimensional case, we show that the cohomology rings…
Over an algebraically closed ffeld F of characteristic p>0, the restricted twisted Heisenberg Lie algebras are studied. We use the Hochschild-Serre spectral sequence relative to its Heisenberg ideal to compute the trivial cohomology. The…
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…
In this thesis, we introduce a new cohomology theory associated to a Lie 2-algebras and a new cohomology theory associated to a Lie 2-group. These cohomology theories are shown to extend the classical cohomology theories of Lie algebras and…
In this article, we introduce a new cohomology theory associated to a Lie 2-algebras. This cohomology theory is shown to extend the classical cohomology theory of Lie algebras; in particular, we show that the second cohomology group…
Let $PU_n$ the projective unitary group of rank $n$ and $BPU_n$ its classifying space. For an odd prime $p$, we extend previous results to a compete description of $H^s(BPU_n;\mathbb{Z})_{(p)}$ for $s<2p+5$ by showing that the $p$-primary…
We study the homology and cohomology groups of super Lie algebra of supersymmetries and of super Poincare algebra. We discuss in detail the calculation in dimensions D=10 and D=6. Our methods can be applied to extended supersymmetry algebra…
We develop a theory of universal central extensions for Hom-Lie antialgebra. It is proved that a Hom-Lie antialgebra admits a universal central extension if and only if it is perfect. Moreover, we show that the kernel of the universal…
This paper mainly studies the ResLieDer pair in characteristic 2, that is, a restricted Lie algebra with a restricted derivation. We define the restricted representation of a ResLieDer pair and the corresponding cohomology complex. We show…
Let $R$ be a commutative ring that is free of rank $k$ as an abelian group, $p$ a prime, and $SL(n,R)$ the special linear group. We show that the Lie algebra associated to the filtration of $SL(n,R)$ by $p$-congruence subgroups is…
We study the homology and cohomology groups of super Lie algebra of supersymmetries and of super Poincare Lie algebra in various dimensions. We give complete answers for (non-extended) supersymmetry in all dimensions $\leq 11$. For…