arXiv subjects
Publications and source records attributed to Toukaiddine Petit.
Let $\mathfrak{g}$ be a Color Lie Algebra and $\mathcal{U}(\mathfrak{g})$ its the universal Enveloping Algebra. We define the notion of graded deformations and we give explicit graded deformations of the universal Enveloping Algebra of $\mathfrak{g}$.
We compute the index of a Lie Borel Lie Algbra of a simple Lie algebra.
Let $Ł_m$ be the scheme of the laws defined by the Jacobi's identities on $\K^m$ with $\K$ a field. A deformation of $\g\inŁ_m$, parametrized by a local $\K$-algebra $\A$, is a local $\K$-algebra morphism from the local ring of $Ł_m$ at $ϕ_m$ to $\A$. The problem to classify all the deformation equivalence classes of a Lie algebra with given base is solved by "versal" deformations. First, we give an algorithm for computing versal deformations. Second, we prove there is a bijection between the deformation equivalence classes of an algebraic Lie algebra $ϕ_m=\mathrm{R}\ltimesϕ_n$ in $Ł_m$ and its nilpotent radical $ϕ_n$ in the $\mathrm{R}$-invariant scheme $Ł_n^{\mathrm{R}}$ with reductive part $\mathrm{R}$, under some conditions. So the versal deformations of $ϕ_m$ in $Ł_m$ is deduced to those of $ϕ_n$ in $Ł_n^{\mathrm{R}}$, which is a more simple problem. Third, we study versality in central extensions of Lie algebras. Finally, we calculate versal deformations of some Lie algebras.
Let $Ł_m$ be the scheme of the laws defined by the identities of Jacobi on $\K^m$. The local studies of an algebraic Lie algebra $\g=\mathrm{R}\ltimes\n$ in $Ł_m$ and its nilpotent part $\n$ in the scheme $Ł_n^{\mathrm{R}}$ of $\mathrm{R}$-invariant Lie algebras on $\K^n$ are linked. This comparison is made by means of slices, which are transversal subschemes to the orbits of $\g$ and $\n$ under the classical groups acting on $Ł_m$ and $Ł_n^{\mathrm{R}}$ respectively. We prove a reduction theorem saying that, under certain conditions on $\g$, the local rings of the slices at $\g$ and $\n$ are isomorphic. In particular, $\g$ is rigid if and only if is $\n$. In the formalism developed at beginning of this paper, a deformation of $\g$ with base a local ring $\A$ is a local morphism from the local ring of $Ł_m$ at $\g$ to $\A$. So the study of deformations for a large class of Lie algebras $\g$ in $Ł_m$ is equivalent to that of $\n$ in $Ł_n^{\mathrm{R}}$ "modulo" the actions of groups, which is a more simple problem. The laws of $Ł_n^{\mathrm{R}}$ are nilpotent with the choice of $\mathrm{R}$ and then we can construct these laws by central extensions. This corresponds to an induction on the schemes themselves $Ł_n^{\mathrm{R}}\toŁ_{n+1}^{\mathrm{R}}$. We restrict this study to a torus $\mathrm{R}=\mathrm{T}$ for certain slices. This leads to a concept of continuous families with the possibility to have nilpotent parameters $t$ (the schemes are generally not reduced). This gives an alternative formalism for the problem of obstructions classes in the theory of formal deformations of M.Gerstenhaber. Examples are given with $t^2=0$ ($t\neq 0$) and $t^5=0$ ($t^4\neq 0$).
Let $G$ be an abelien group, $ε$ an anti-bicharacter of $G$ and $L$ a $G$-graded $ε$ Lie algebra (color Lie algebra) over $\K$ a field of characteristic zero. We prove that all $G$-graded, positive filtered $A$ such that the associated graded algebra is isomorphic to the $G$-graded $ε$-symmetric algebra $S(L)$, there is a $G$- graded $ε$-Lie algebra $L$ and a $G$-graded scalar two cocycle $ω\in\mathrm{Z}_{gr}^2(L,\K)$, such that $A$ is isomorphic to $ U_ω(L)$ the generalized enveloping algebra of $L$ associated with $ω$. We also prove there is an isomorphism of graded spaces between the Hochschild cohomology of the generalized universal enveloping algebra $U(L)$ and the generalized cohomology of color Lie algebra $L$.
Let $A$ be a $(G, χ)$-Hopf algebra with bijection antipode and let $M$ be a $G$-graded $A$-bimodule. We prove that there exists an isomorphism \mathrm{HH}^*_{\rm gr}(A, M)\cong{\rm Ext}^*_{A{-}{\rm gr}} (\K, {^{ad}(M)}), where $\K$ is viewed as the trivial graded $A$-module via the counit of $A$, $^{ad} M$ is the adjoint $A$-module associated to the graded $A$-bimodule $M$ and $\mathrm{HH}_{\rm gr}$ denotes the $G$-graded Hochschild cohomology. As an application, we deduce that the cohomology of color Lie algebra $L$ is isomorphic to the graded Hochschild cohomology of the universal enveloping algebra $U(L)$, solving a question of M. Scheunert.
We provide an algorithm for decomposing a finite-dimensional Lie algebra over a field of characteristic 0 permitting to generalize the derivation tower theorem for Lie algebras, is proved by E. Schenkman \cite{Sc}.
We consider filtered or graded algebras $A$ over a field $K$. Assume that there is a discrete valuation $O_v$ of $K$ with $m_v$ its maximal ideal and $k_v:=O_v/m_v$ its residue field. Let $Λ$ be $O_v$-order such that $ΛK=A$ and $\barΛ:=k_v\otimes_{O_v}Λ$ the $Λ$-reduction of $A$ at the place $K\leadsto k_v$. Using the filtration of $A$ induced by $Λ$ we shall prove that for certain algebras $A$ their properties are related to $\barΛ$.