arXiv subjects
Publications and source records attributed to Salvatore Siciliano.
In this paper we investigate the problem of which Lie algebras appear as the derived algebra of a Lie algebra. We present new results that further develop this study and address two questions raised in a paper concerned with the corresponding problem for groups.
It is well known that a finite-dimensional Lie algebra over a field of characteristic zero is simple exactly when its derivation algebra is simple. In this paper we characterize those Lie algebras of arbitrary dimension over any field that have a simple derivation algebra. As an application we classify the Lie algebras that have a complete simple derivation algebra and are either finite-dimensional over an algebraically closed field of prime characteristic $p>3$ or $\mathbb{Z}$-graded of finite growth over an algebraically closed field of characteristic zero.
Let $FI(X,K)$ be the finitary incidence algebra of a poset $X$ over a field $K$. In this short note we establish when $FI(X,K)$ satisfies a polynomial identity and when its group of units $\mathcal{U}(FI(X,K))$ satisfies a group identity. The Lie derived length of $FI(X,K)$ and the derived length of $\mathcal{U}(FI(X,K))$ are also determined.
In this paper we continue the study of the subalgebra lattice of a Leibniz algebra. In particular, we find out that solvable Leibniz algebras with an upper semi-modular lattice are either almost-abelian or have an abelian ideal spanned by the elements with square zero. We also study Leibniz algebras in which every subalgebra is a weak quasi-ideal, as well as modular symmetric Leibniz algebras.
In this paper we study the lattice of restricted subalgebras of a restricted Lie algebra. In particular, we consider those algebras in which this lattice is dually atomistic, lower or upper semimodular, or in which every restricted subalgebra is a quasi-ideal. The fact that there are one-dimensional subalgebras which are not restricted results in some of these conditions being weaker than for the corresponding conditions in the non-restricted case.
In this paper we begin to study the subalgebra lattice of a Leibniz algebra. In particular, we deal with Leibniz algebras whose subalgebra lattice is modular, upper semi-modular, lower semi-modular, distributive, or dually atomistic. The fact that a non-Lie Leibniz algebra has fewer one-dimensional subalgebras in general results in a number of lattice conditions being weaker than in the Lie case.
Let $P$ be a Poisson algebra with a Lie bracket $\{, \}$ over a field $\F$ of characteristic $p\geq 0$. In this paper, the Lie structure of $P$ is investigated. In particular, if $P$ is solvable with respect to its Lie bracket, then we prove that the Poisson ideal $\mathcal{J}$ of $P$ generated by all elements $\{\{\{x_1, x_2\}, \{x_3, x_4\}\}, x_5\}$ with $x_1,\ldots ,x_5 \in P$ is associative nilpotent of index bounded by a function of the derived length of $P$. We use this result to further prove that if $P$ is solvable and $p\neq 2$, then the Poisson ideal $\{P,P\}P$ is nil.
Let $L$ be a restricted Lie algebra over a field of positive characteristic. We prove that the restricted enveloping algebra of $L$ is a principal ideal ring if and only if $L$ is an extension of a finite-dimensional torus by a cyclic restricted Lie algebra.
For a restricted Lie algebra $L$, the conditions under which its restricted enveloping algebra $u(L)$ is semiperfect are investigated. Moreover, it is proved that $u(L)$ is left (or right) perfect if and only if $L$ is finite-dimensional.
Let L be a Lie superalgebra over a fled of characteristic p\neq 2 with enveloping algebra U(L) or let L be a restricted Lie superalgebra over a field of characteristic p > 2 with restricted enveloping algebra u(L). In this note, we establish when u(L) or U(L) is bounded Lie Engel.
Let L be a restricted Lie algebra over a field of positive characteristic. We survey the known results about the Lie structure of the restricted enveloping algebra u(L) of L. Related results about the structure of the group of units and the symmetric and skew-symmetric elements of u(L) are also discussed. Moreover, a new theorem about an upper bound for the Lie nilpotency class of u(L) is proved.
Let $L$ be a restricted Lie algebra over a field of characteristic $p>2$ and denote by $u(L)$ its restricted enveloping algebra. We establish when the symmetric or skew elements of $u(L)$ under the principal involution are Lie metabelian.
In this paper we show that the projective cover of the trivial irreducible module of a finite-dimensional solvable restricted Lie algebra is induced from the one-dimensional trivial module of a maximal torus. As a consequence, we obtain that the number of the isomorphism classes of irreducible modules with a fixed p-character for a finite-dimensional solvable restricted Lie algebra L is bounded above by p^MT(L), where MT(L) denotes the largest dimension of a torus in L. Finally, we prove that in characteristic p>3 the projective cover of the trivial irreducible L-module is only induced from the one-dimensional trivial module of a torus of maximal dimension if L is solvable.
Lie solvable restricted enveloping algebras were characterized by Riley and Shalev except when the ground field is of characteristic 2. We resolve the characteristic 2 case here which completes the classification. As an application of our result, we obtain a characterization of ordinary Lie algebras over any field whose enveloping algebra is Lie solvable.
In this paper we investigate the relation between the multiplicities of split abelian chief factors of finite-dimensional Lie algebras and first degree cohomology. In particular, we obtain a characterization of modular solvable Lie algebras in terms of the vanishing of first degree cohomology or in terms of the multiplicities of split abelian chief factors. The analogues of these results are well known in the modular representation theory of finite groups. An important tool in the proof of these results is a refinement of a non-vanishing theorem of Seligman for the first degree cohomology of non-solvable finite-dimensional Lie algebras in prime characteristic. As applications we derive several results in the representation theory of restricted Lie algebras related to the principal block and the projective cover of the trivial irreducible module of a finite-dimensional restricted Lie algebra. In particular, we obtain a characterization of solvable restricted Lie algebras in terms of the second Loewy layer of the projective cover of the trivial irreducible module.
In this paper we investigate the relation between the multiplicities of split strongly abelian p-chief factors of finite-dimensional restricted Lie algebras and first degree restricted cohomology. As an application we obtain a characterization of solvable restricted Lie algebras in terms of the multiplicities of split strongly abelian p-chief factors. Moreover, we derive some results in the representation theory of restricted Lie algebras related to the principal block and the projective cover of the trivial irreducible module of a finite-dimensional restricted Lie algebra. In particular, we obtain a characterization of finite-dimensional solvable restricted Lie algebras in terms of the second Loewy layer of the projective cover of the trivial irreducible module. The analogues of these results are well known in the modular representation theory of finite groups.
In this paper we prove that every finite-dimensional nilpotent restricted Lie algebra over a field of prime characteristic has an outer restricted derivation whose square is zero unless the restricted Lie algebra is a torus or it is one-dimensional or it is isomorphic to the three-dimensional Heisenberg algebra in characteristic two as an ordinary Lie algebra. This result is the restricted analogue of a result of Tôgô on the existence of nilpotent outer derivations of ordinary nilpotent Lie algebras in arbitrary characteristic and the Lie-theoretic analogue of a classical group-theoretic result of Gaschütz on the existence of $p$-power automorphisms of $p$-groups. As a consequence we obtain that every finite-dimensional non-toral nilpotent restricted Lie algebra has an outer restricted derivation.
An algebra is called a GI-algebra if its group of units satisfies a group identity. We provide positive support for the following two open problems. 1. Does every algebraic GI-algebra satisfy a polynomial identity? 2. Is every algebraically generated GI-algebra locally finite?