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Cartan described some of the finite dimensional simple Lie algebras and three of the four series of simple infinite dimensional vectorial Lie algebras with polynomial coefficients as prolongs, which now bear his name. The rest of the simple Lie algebras of these two types (finite dimensional and vectorial) are, if the depth of their grading is greater than 1, results of generalized Cartan–Tanaka–Shchepochkina (CTS) prolongs. Here we are looking for new examples of simple finite dimensional modular Lie (super)algebras in characteristic 2 obtained as Cartan prolongs. We consider pairs (an (ortho-)orthogonal Lie (super)algebra or its derived algebra, its irreducible module) and compute the Cartan prolongs of such pairs. The derived algebras of these prolongs are simple Lie (super)algebras. We point out several amazing phenomena in characteristic 2: a supersymmetry of representations of certain Lie algebras, latent or hidden over complex numbers, becomes manifest; the adjoint representation of some simple Lie...

The classification of simple finite dimensional modular Lie algebras over algebraically closed fields of characteristic p > 3 (described by the generalized Kostrikin–Shafarevich conjecture) being completed due to Block, Wilson, Premet and Strade (with contributions from other researchers) the next major classification problems are those of simple finite dimensional modular Lie algebras over fields of characteristic 3 and 2. For the latter, the Kochetkov–Leites conjecture involved classification of Lie superalgebras and their inhomogeneous with respect to parity subalgebras, called Volichenko algebras. In characteristic 2, we consider the result of application of the functor forgetting the superstructure to the simple serial vectorial Lie algebras known to us and their Volichenko subalgebras.

Felix Alexandrovich Berezin was born on April 25, 1931 and perished on July 14, 1980 (or several days before) under obscure circumstances while serving — during his summer holidays — as an auxiliary worker for a geological party near the Kolyma river. Wikipedia currently lists several researchers as discoverers of what we now call supersymmetry. Interestingly, Berezin is not even mentioned in this Wikipedia article. However, none of the authors mentioned so far as “pioneers of supersymmetry” qualify as such: Before Wess and Zumino have explained the meaning and importance of supersymmetry none of the earlier discoverers did fully appreciate the meaning and value of their own works, and some of them do not realize its full meaning even now judging by their later works. Contrariwise, Berezin definitely was the first to realize the real value of his discoveries, the fact that he stands on the threshold of the new branch of mathematics, currently referred to as “supermathematics” and guess its possible worthiness for physics, first consciously demonstrated by Wess and Zumino. In early 1960s, Berezin obtained a parallel description of the second quantization for Boze and Fermi particles summarized in his book [1] (where, among other things, Berezin introduces the Berezin integral, and gives a parallel description of the spinor and oscillator representations of, respectively, infinite dimensional orthogonal o(2∞) and symplectic Lie algebras sp(2∞), later rediscovered many times; for an overview of Berezin’s works, see the “scientific part” of [5]). In May 2007, World Scientific published the book “FELIX BEREZIN: Life and Death of the Mastermind of Supermathematics” edited by M. Shifman, [8]. The book consists of contributions of Berezin’s friends, colleagues and students; recollections of Berezin as a person, and some scientific reviews of his results and ideas, their role in mathematical physics and their developments. The core of the book is the skillfully written heart-breaking story of Berezin’s widow, Elena Karpel; and the two Berezin’s letters — to the Rector of Moscow State University and the Moscow Mathematical Society — on the situation in these institutions and in mathematical community in the Soviet Union in general (the problems pointed at in these letters did not lose their timeliness today).

Berger and Stassen reviewed skein relations for link invariants coming from the simple Lie algebras g. They related the invariants with decomposition of the tensor square of the g-module V of minimal dimension into irreducible components. (If V 6≃ V � , one should also consider the decompositions of V ⊗ Vand V � ⊗ V � .) Here we consider decompositions into irreducible components for g-modules V of minimal dimension over some simple and close to simple Lie superalgebras g. For the classical series (gl, sl, osp), as well as for the Poisson and Hamiltonian algebras — "quasi-classical" analogs of gl and sl — the answer is rather complicated due to the lack of complete reduciblity. Contrariwise, the case of exceptional Lie superalgebras g = ag2 and ab3 turned out to be similar to that of Lie algebras: The g-module g ⊗ g (here the representation of minimal dimension is the adjoint one) is completely reducible and, remarkably, the spectra of highest weights for ag2 are almost identical (in certain coordinates) to that for ab3! We also consider g = osp � (4|2) for � 6 0,1.

So far, there are described in the literature two ways to superize the Liouville equation: for a scalar field (for $N\leq 4$) and for a vector-valued field (analogs of the Leznov--Saveliev equations) for N=1. Both superizations are performed with the help of Neveu--Schwarz superalgebra. We consider another version of these superLiouville equations based on the Ramond superalgebra, their explicit solutions are given by Ivanov--Krivonos' scheme. Open problems are offered.