Mathematicians going east
We survey emigration of mathematicians from Europe, before and during WWII, to Russia.
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arXiv subjects
Pasha Zusmanovich
Publications and source records attributed to Pasha Zusmanovich.
On the last question of Stefan Banach
We discuss the last question of Banach, posed by him in 1944, shortly before his death, about extension of a ternary map to superposition of a binary map. We try to put things into the context of Polish mathematics of that time, and touch upon subsequent developments in such diverse areas as multivalued logics, binary and ternary semigroups, theory of clones, and Hilbert's 13th problem. Making almost a full circle in time, we show how variants of Banach's question may be settled using a 1949 idea of Jacobson about envelopes of Lie triple systems.
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Nonexistence of invariant symmetric forms on generalized Jacobson-Witt algebras revisited
We provide a short alternative homological argument showing that any invariant symmetric bilinear form on simple modular generalized Jacobson-Witt algebras vanishes, and outline another, deformation-theoretic one, allowing to describe such forms on simple modular Lie algebras of contact type.
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Commutative 2-cocycles on Lie algebras
On Lie algebras, we study commutative 2-cocycles, i.e., symmetric bilinear forms satisfying the usual cocycle equation. We note their relationship with antiderivations and compute them for some classes of Lie algebras, including finite-dimensional semisimple, current and Kac-Moody algebras.
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Deformations of current Lie algebras. I. Small algebras in characteristic 2
We compute low-degree cohomology of current Lie algebras extended over the 3-dimensional simple algebra, compute deformations of related semisimple Lie algebras, and apply these results to classification of simple Lie algebras of absolute toral rank 2 and having a Cartan subalgebra of toral rank one. Everything is in characteristic 2.
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Low-dimensional cohomology of current Lie algebras and analogs of the Riemann tensor for loop manifolds
We obtain formulas for the first and second cohomology groups of a general current Lie algebra with coefficients in the "current" module, and apply them to compute structure functions for manifolds of loops with values in compact Hermitian symmetric spaces.
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Special and exceptional mock-Lie algebras
We observe several facts and make conjectures about commutative algebras satisfying the Jacobi identity. The central question is which of those algebras admit a faithful representation (i.e., in Lie parlance, satisfy the Ado theorem, or, in Jordan parlance, are special).
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Non-semigroup gradings of associative algebras
It is known that there are Lie algebras with non-semigroup gradings, i.e. such that the binary operation on the grading set is not associative. We provide a similar example in the class of associative algebras.
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Invariants of Lie algebras extended over commutative algebras without unit
We establish results about the second cohomology with coefficients in the trivial module, symmetric invariant bilinear forms and derivations of a Lie algebra extended over a commutative associative algebra without unit. These results provide a simple unified approach to a number of questions treated earlier in completely separated ways: periodization of semisimple Lie algebras (Anna Larsson), derivation algebras, with prescribed semisimple part, of nilpotent Lie algebras (Benoist), and presentations of affine Kac-Moody algebras.
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Lie algebras and around: selected questions
Several open questions are discussed. The topics include cohomology of current and related Lie algebras, algebras represented as the sum of subalgebras, structures and phenomena peculiar to characteristic $2$, and variations on themes of Ado, Whitehead, and Banach.
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On the utility of Robinson-Amitsur ultrafilters
An embedding theorem for algebraic systems is presented, basing on a certain old ultrafilter construction. As an application, we outline alternative proofs of some results from the theory of PI algebras, and establish some properties of Tarski's monsters.
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On the utility of Robinson-Amitsur ultrafilters. II
We extend the result of arXiv:0911.5414 about embedding of ideal-determined algebraic systems into ultraproducts, to arbitrary algebraic systems, and to ultraproducts over $κ$-complete ultrafilters. We also discuss the scope of applicability of this result, and correct a mistake from the previous paper concerning homomorphisms from ultraproducts.
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Melikyan algebra is a deformation of a Poisson algebra
We prove, using computer, that the restricted Melikyan algebra of dimension 125 is a deformation of a Poisson algebra.
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A Lie algebra that can be written as a sum of two nilpotent subalgebras, is solvable
This is an old paper put here for archeological purposes. It is proved that a finite-dimensional Lie algebra over a field of characteristic p>5, that can be written as a vector space (not necessarily direct) sum of two nilpotent subalgebras, is solvable. The same result (but covering also the cases of low characteristics) was established independently by V. Panyukov (Russ. Math. Surv. 45 (1990), N4, 181-182), and the homological methods utilized in the proof were developed later in arXiv:math/0204004. Many inaccuracies in the English translation are corrected, otherwise the text is identical to the published version.
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Deformations of $W_1(n) \otimes A$ and modular semisimple Lie algebras with a solvable maximal subalgebra
In one of his last papers, Boris Weisfeiler proved that if modular semisimple Lie algebra possesses a solvable maximal subalgebra which defines in it a long filtration, then associated graded algebra is isomorphic to one constructed from the Zassenhaus algebra tensored with the divided powers algebra. We completely determine such class of algebras, calculating in process low-dimensional cohomology groups of Zassenhaus algebra tensored with any associative commutative algebra.
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A converse to the Second Whitehead Lemma
We show that finite-dimensional Lie algebras over a field of characteristic zero such that the second cohomology group in every finite-dimensional module vanishes, are, essentially, semisimple.
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A compendium of Lie structures on tensor products
We demonstrate how a simple linear-algebraic technique used earlier to compute low-degree cohomology of current Lie algebras, can be utilized to compute other kinds of structures on such Lie algebras, and discuss further generalizations, applications, and related questions. While doing so, we touch upon such seemingly diverse topics as associative algebras of infinite representation type, Hom-Lie structures, Poisson brackets of hydrodynamic type, Novikov algebras, simple Lie algebras in small characteristics, and Koszul dual operads.
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Central extensions of current algebras
This is an old paper put here for archeological purposes. We compute the second cohomology of current Lie algebras of the form $L\otimes A$, where $L$ belongs to some class of Lie algebras which includes classical simple and Zassenhaus algebras, and of some modular semisimple Lie algebras. The results are largely superseded by subsequent papers, though, perhaps, some tricks and observations used here remain of minor interest.
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