Abstract:
Models of the exceptional simple modular Lie superalgebras in characteristic $p\geq 3$, that have appeared in the classification due to Bouarroudj, Grozman and Leites of the Lie superalgebras with indecomposable symmetrizable Cartan matrices, are provided. The models relate these exceptional Lie superalgebras to some low dimensional nonassociative algebraic systems.

Abstract:
Let $X$ be one of the finite-dimensional simple graded Lie superalgebras of Cartan type $W, S, H, K, HO, KO, SHO$ or $SKO$ over an algebraically closed field of characteristic $p>3$. In this paper we prove that $X$ can be generated by one element except the ones of type $W,$ $HO$, $KO$ or $SKO$ in certain exceptional cases, in which $X$ can be generated by two elements. As a subsidiary result, we also prove that certain classical Lie superalgebras or their relatives can be generated by one or two elements.

Abstract:
Two new simple modular Lie superalgebras are obtained in characteristics 3 and 5, which share the property that their even parts are orthogonal Lie algebras and the odd parts their spin modules. The characteristic 5 case is shown to be related, by means of a construction of Tits, to the exceptional ten dimensional Jordan superalgebra of Kac.

Abstract:
Over algebraically closed fields of characteristic p>2, prolongations of the simple finite dimensional Lie algebras and Lie superalgebras with Cartan matrix are studied for certain simplest gradings of these algebras. Several new simple Lie superalgebras are discovered, serial and exceptional, including superBrown and superMelikyan superalgebras. Simple Lie superalgebras with Cartan matrix of rank 2 are classified.

Abstract:
Over algebraically closed fields of positive characteristic, infinitesimal deformations of simple finite dimensional symmetric (the ones that with every root have its opposite of the same multiplicity) Lie algebras and Lie superalgebras are described for small ranks. The results are obtained by means of the Mathematica based code SuperLie. The infinitesimal deformation given by any odd cocycle is integrable. The moduli of the deformations form, in general, a supervariety. Not each even cocycle is integrable; but for those that are integrable, the global deforms (the results of deformations) are linear with respect to the parameter. In characteristic 2, the simple 3-dimensional Lie algebra admits a parametric family of non-isomorphic simple deforms. Some of Shen's "variations of G(2) theme" are interpreted as two global deforms corresponding to the several of the 20 infinitesimal deforms first found by Chebochko; we give their explicit form.

Abstract:
Over algebraically closed fields of positive characteristic, we list the outer derivations and non-trivial central extensions of simple finite dimensional Lie algebras and Lie superalgebras of the rank sufficiently small to allow computer-aided study using Mathematica-based code SuperLie. We consider algebras possessing a grading by integers such that the dimensions of the superspaces whose degrees are equal in absolute value coincide (symmetric algebras), and also non-trivial deforms of these algebras. When the answer is clear for the infinite series, it is given for any rank. We also consider the non-symmetric case of periplectic Lie superalgebras and the Lie algebras obtained from them by desuperization.

Abstract:
For modular Lie superalgebras, new notions are introduced: Divided power homology and divided power cohomology. For illustration, we give presentations (in terms of analogs of Chevalley generators) of finite dimensional Lie (super)algebras with indecomposable Cartan matrix in characteristic 2 (and in other characteristics for completeness of the picture). We correct the currently available in the literature notions of Chevalley generators and Cartan matrix in the modular and super cases, and an auxiliary notion of the Dynkin diagram. In characteristic 2, the defining relations of simple classical Lie algebras of the A, D, E types are not only Serre ones; these non-Serre relations are same for Lie superalgebras with the same Cartan matrix and any distribution of parities of the generators. Presentations of simple orthogonal Lie algebras having no Cartan matrix are also given..

Abstract:
In this paper we consider Lie superalgebras decomposable as the sum of two proper subalgebras. Any of these algebras has the form of the vector space sum $L=A+B$ where $A$ and $B$ are proper simple subalgebras which need not be ideals of $L$, and the sum need not be direct. The main result of this paper is the following: Let $S = {osp}(m,2n)$ be a Lie superalgebra such that $S=K+L$ where $K$, $L$ are two proper basic simple subalgebras. Then $m$ is even, $m=2k$ and $K \cong osp(2k-1,2n)$, $L \cong sl(k,n)$.

Abstract:
The notion of a Lie conformal superalgebra encodes an axiomatic descrption of singular parts of the operator product expansions of chiral fields in conformal field theory. In the paper we give a detailed proof of the classification of all finite simple Lie conformal superalgebras. We also classify all their central extensions.