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Register a new userExtremal projectors for contragredient Lie (super)symmetries (short review)
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V.N. Tolstoy
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A brief review of the extremal projectors for contragredient Lie (super)symmetries (finite-dimensional simple Lie algebras, basic classical Lie superalgebras, infinite-dimensional affine Kac-Moody algebras and superalgebras, as well as their quantum $q$-analogs) is given. Some bibliographic comments on the applications of extremal projectors are presented.
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We survey earlier results on factorizations of extremal projectors and relative extremal projectors and present preliminary results on non-commutative factorizations of relative extremal projectors: we deduce the existence of such factorizations for sl(4) and sl(5).
A brief review of the extremal projector method for Lie symmetries (Lie algebras and superalgebras as well as their quantum analogs) is given. A history of its discovery and some simplest applications are presented.
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.
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.
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.
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