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A double extension ($mathscr{D}$ extension) of a Lie (super)algebra $mathfrak a$ with a non-degenerate invariant symmetric bilinear form $mathscr{B}$, briefly: a NIS-(super)algebra, is an enlargement of $mathfrak a$ by means of a central extension and a derivation; the affine Kac-Moody algebras are the best known examples of double extensions of loops algebras. Let $mathfrak a$ be a restricted Lie (super)algebra with a NIS $mathscr{B}$. Suppose $mathfrak a$ has a restricted derivation $mathscr{D}$ such that $mathscr{B}$ is $mathscr{D}$-invariant. We show that the double extension of $mathfrak a$ constructed by means of $mathscr{B}$ and $mathscr{D}$ is restricted. We show that, the other way round, any restricted NIS-(super)algebra with non-trivial center can be obtained as a $mathscr{D}$-extension of another restricted NIS-(super)algebra subject to an extra condition on the central element. We give new examples of $mathscr{D}$-extensions of restricted Lie (super)algebras, and pre-Lie superalgebras indigenous to characteristic 3.
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We give explicit formulas proving restrictedness of the following Lie (super)algebras: known exceptional simple vectorial Lie (super)algebras in characteristic 3, deformed Lie (super)algebras with indecomposable Cartan matrix, and (under certain conditions) their simple subquotients over an algebraically closed field of characteristic 3, as well as one type of the deformed divergence-free Lie superalgebras with any number of indeterminates in any characteristic.
We say that a~Lie (super)algebra is ``symmetric if with every root (with respect to the maximal torus) it has its opposite of the same multiplicity. Over algebraically closed fields of positive characteristics we describe the deforms (results of deformations) of all known simple finite-dimensional symmetric Lie (super)algebras of rank $<9$, except for superizations of the Lie algebras with ADE root systems. The moduli of deformations of any Lie superalgebra constitute a supervariety. Any infinitesimal deformation given by any odd cocycle is integrable with an odd parameter running over a~supervariety. All deforms corresponding to odd cocycles are new. Among new results are classification of the deforms of the 29-dimensional Brown algebra in characteristic 3, of Weisfeiler-Kac algebras and orthogonal Lie algebras without Cartan matrix in characteristic 2. For the Lie (super)algebras considered, all cocycles are integrable, the deforms corresponding to the weight cocycles are usually linear in the parameter. Problem: describe isomorphic deforms. Appendix: For several modular analogs of complex simple Lie algebras, and simple Lie algebras indigenous to characteristics 3 and 2, we describe the space of cohomology with trivial coefficients. We show that the natural multiplication in this space is very complicated.
A Lie (super)algebra with a non-degenerate invariant symmetric bilinear form $B$ is called a nis-(super)algebra. The double extension $mathfrak{g}$ of a nis-(super)algebra $mathfrak{a}$ is the result of simultaneous adding to $mathfrak{a}$ a central element and a derivation so that $mathfrak{g}$ is a nis-algebra. Loop algebras with values in simple complex Lie algebras are most known among the Lie (super)algebras suitable to be doubly extended. In characteristic 2 the notion of double extension acquires specific features. Restricted Lie (super)algebras are among the most interesting modular Lie superalgebras. In characteristic 2, using Grozmans Mathematica-based package SuperLie, we list double extensions of restricted Lie superalgebras preserving the non-degenerate closed 2-forms with constant coefficients. The results are proved for the number of indeterminates ranging from 4 to 7 - sufficient to conjecture the pattern for larger numbers. Considering multigradings allowed us to accelerate computations up to 100 times.
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 multiplicity) and of ranks most needed in an approach to the classification of simple vectorial Lie superalgebras, we list the outer derivations and nontrivial central extensions. When the answer is clear for the infinite series, it is given for any rank. We also list the outer derivations and nontrivial central extensions of one series of nonsymmetric, namely, periplectic Lie superalgebras (of any rank) preserving the nondegenerate supersymmetric odd bilinear forms, and of the Lie algebras obtained from periplectic Lie superalgebras by desuperization when the characteristic of the ground field is equal to 2. We also list the outer derivations and nontrivial central extensions of an analog of the rank 2 exceptional Lie algebra discovered by Shen Guangyu. Several results are counterintuitive.
We review the list of non-degenerate invariant (super)symmetric bilinear forms (briefly: NIS) on the following simple (relatives of) Lie (super)algebras: (a) with symmetrizable Cartan matrix of any growth, (b) with non-symmetrizable Cartan matrix of polynomial growth, (c) Lie (super)algebras of vector fields with polynomial coefficients, (d) stringy a.k.a. superconformal superalgebras, (e) queerifications of simple restricted Lie algebras. Over algebraically closed fields of positive characteristic, we establish when the deform (i.e., the result of deformation) of the known finite-dimensional simple Lie (super)algebra has a NIS. Amazingly, in most of the cases considered, if the Lie (super)algebra has a NIS, its deform has a NIS with the same Gram matrix after an identification of bases of the initial and deformed algebras. We do not consider odd parameters of deformations. Closely related with simple Lie (super)algebras with NIS is the notion of doubly extended Lie (super)algebras of which affine Kac--Moody (super)algebras are the most known examples.
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