Novikov algebras and Novikov structures on Lie algebras

We study ideals of Novikov algebras and Novikov structures on finite-dimensional Lie algebras. We present the first example of a three-step nilpotent Lie algebra which does not admit a Novikov structure. On the other hand we show that any free three-step nilpotent Lie algebra admits a Novikov structure. We study the existence question also for Lie algebras of triangular matrices. Finally we show that there are families of Lie algebras of arbitrary high solvability class which admit Novikov structures.     

Exponentiation of infinite dimensional Z-graded lie algebras

In this article we give a new technique for exponentiating infinite dimensional graded representations of graded Lie algebras that allows for the exponentiation of some non-locally nilpotent elements. Our technique is to naturally extend the representation of the Lie algebra g on the space V naturally to a representation on a subspace £ of the dual space V ^*. After introducing the technique, we prove that it enables the exponentiation of all elements of free Lie Algebras and afhne Kac-Moody Lie algebras.     

Braided symmetric and exterior algebras and quantizations of braided lie algebras

We study quantizations of braided symmetric and exterior algebras of graded vector spaces and of braided derivations on these algebras. We find quantizations of braided Lie algebras by considering quantizations of derivations on their braided exterior algebra. The text was submitted by the author in English.     

On the dimension of the Schur multiplier of nilpotent lie algebras

We give a bound on the dimension of the Schur multiplier of a finite dimensional nilpotent Lie algebra which sharpens the earlier known bounds.     

The algebraic classification of nilpotent Tortkara algebras

Abstract

We classify all complex 6-dimensional nilpotent Tortkara algebras.

Communicated by Alberto Facchini     

Coding the principal character formula for affine Kac-Moody lie algebras

In this paper, an algorithm for computing the principal character for affine Lie algebras is discussed and presented. The principal characters discovered using this program are given and/or proven. Results include level 2 and 3 character formulas in and the sole existence of the Rogers-Ramanujan products in , , , , , , .     

Enumeration of maximal subalgebras in free restricted lie algebras

Given a finitely generated restricted Lie algebra L over the finite field \(\mathbb{F}_q \), and n ≥ 0, denote by a _n (L) the number of restricted subalgebras H ? L with \(\dim _{\mathbb{F} _q} \) L/H = n. Denote by ã _n (L) the number of the subalgebras satisfying the maximality condition as well. Considering the free restricted Lie algebra L = F _d of rank d ≥ 2, we find the asymptotics of ã _n (F _d ) and show that it coincides with the asymptotics of a _n (F _d ) which was found previously by the first author. Our approach is based on studying the actions of restricted algebras by derivations on the truncated polynomial rings. We establish that the maximal subalgebras correspond to the so-called primitive actions. This means that “almost all” restricted subalgebras H ? F _d of finite codimension are maximal, which is analogous to the corresponding results for free groups and free associative algebras.     

Quandle cocycles from invariant theory

Let G$G$ be a group. Any G$G$-module M$M$ has an algebraic structure called a G$G$-family of Alexander quandles. Given a 2-cocycle of a cohomology associated with this G$G$-family, topological invariants of (handlebody) knots in the 3-sphere are defined. We develop a simple algorithm to algebraically construct n$n$-cocycles of this G$G$-family from G$G$-invariant group n$n$-cocycles of the abelian group M$M$. We present many examples of 2-cocycles of these G$G$-families using facts from (modular) invariant theory.     

Complex Lie algebras corresponding to weighted projective lines

The aim of this paper is to give an alternative proof of Kac’s theorem for weighted projective lines over the complex field. The geometric realization of complex Lie algebras arising from derived categories is essentially used.     

Twisted Steinberg algebras

We introduce twisted Steinberg algebras over a commutative unital ring R. These generalise Steinberg algebras and are a purely algebraic analogue of Renault's twisted groupoid C*-algebras. In particular, for each ample Hausdorff groupoid G and each locally constant 2-cocycle σ on G taking values in the units $R^{\times }$, we study the algebra $A_R(G,\sigma )$ consisting of locally constant compactly supported R-valued functions on G, with convolution and involution “twisted” by σ. We also introduce a “discretised” analogue of a twist Σ over a Hausdorff étale groupoid G, and we show that there is a one-to-one correspondence between locally constant 2-cocycles on G and discrete twists over G admitting a continuous global section. Given a discrete twist Σ arising from a locally constant 2-cocycle σ on an ample Hausdorff groupoid G, we construct an associated twisted Steinberg algebra $A_R(G;\mathrm{\Sigma })$, and we show that it coincides with $A_R(G,{\sigma }^{?1})$. Given any discrete field ${\mathbb{F}}_d$, we prove a graded uniqueness theorem for $A_{{\mathbb{F}}_d}(G,\sigma )$, and under the additional hypothesis that G is effective, we prove a Cuntz–Krieger uniqueness theorem and show that simplicity of $A_{{\mathbb{F}}_d}(G,\sigma )$ is equivalent to minimality of G.     

Banach Lie algebras with Lie subalgebras of finite codimension: Their invariant subspaces and Lie ideals

The paper studies the existence of closed invariant subspaces for a Lie algebra L of bounded operators on an infinite-dimensional Banach space X. It is assumed that L contains a Lie subalgebra L₀ that has a non-trivial closed invariant subspace in X of finite codimension or dimension. It is proved that L itself has a non-trivial closed invariant subspace in the following two cases: (1) L₀ has finite codimension in L and there are Lie subalgebras L₀=L⁰⊂L¹⊂?⊂Lp=L such that Li+1=Li+[Li,Li+1] for all i; (2) L₀ is a Lie ideal of L and dim(L₀)=∞. These results are applied to the problem of the existence of non-trivial closed Lie ideals and closed characteristic Lie ideals in an infinite-dimensional Banach Lie algebra L that contains a non-trivial closed Lie subalgebra of finite codimension.     

Characterizing n-Isoclinism Classes of Lie Algebras

In this article, we introduce the notion of the equivalence relation, n-isoclinism, between Lie algebras, and obtain some criterions under which Lie algebras are n-isoclinic. In particular, we show that n-isoclinic Lie algebras can be isoclinically embedded into one Lie algebra. Also, we present the notion of an n-stem Lie algebra and prove its existence within an arbitrary n-isoclinism class. In addition, similar to a result of Hekster [6 Hekster , N. S. ( 1986 ). On the structure of n-isoclinam classes of groups . J. Pure Appl. Algebra 40 : 63 – 85 .[Crossref], [Web of Science ®] , [Google Scholar]] in the group case, we characterize the n-stem Lie algebras in the n-isoclinism classes which contains at least one finitely generated Lie algebra L with dim (L ⁿ⁺¹) finite.     

Twisted hopf comodule algebras

For k a commutative ring, H a k‐bialgebra and A a right H‐comodule k‐algebra, we define a new multiplication on the H‐comodule A to obtain a twisted algebra” A^T, T sumHom(H,End (A)). If ^T is convolution invertible, the categories of relative right Hopf modules over A and A^Tare isomorphic. Similarly a convolution invertible left twisting gives an isomorphism of the categories of relative left Hopf modules. We show that crossed products are invertible twistings of the tensor product, and obtain, as a corollary, a duality theorem for crossed products     

Nilpotent n-Lie Algebras

We find examples of nilpotent n-Lie algebras and prove n-Lie analogs of classical group theory and Lie algebra results. As an example we show that a nilpotent ideal I of class c in a n-Lie algebra A with A/I ² nilpotent of class d is nilpotent and find a bound on the class of A. We also find that some classical group theory and Lie algebra results do not hold in n-Lie algebras. In particular, non-nilpotent n-Lie algebras can admit a regular automorphism of order p, and the sum of nilpotent ideals need not be nilpotent.     

Novikov-Poisson algebras and associative commutative derivation algebras

We describe Novikov-Poisson algebras in which a Novikov algebra is not simple while its corresponding associative commutative derivation algebra is differentially simple. In particular, it is proved that a Novikov algebra is simple over a field of characteristic not 2 iff its associative commutative derivation algebra is differentially simple. The relationship is established between Novikov-Poisson algebras and Jordan superalgebras. Supported by RFBR (grant No. 05-01-00230), by SB RAS (Integration project No. 1.9), and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (project NSh-344.2008.1). __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 186–202, March–April, 2008.     

Simple sagle algebras

It is proved that ifS is a simple finite-dimensional anticommutative algebra over a field ϕ of characteristic zero satisfying the identityJ(x, y, z)t=J(t, z, xy)+J(t, y, zx)+J(t, x,yz), whereJ(x, y, z)=(xy)z+(zx)y+(yz)x, thenS is a Lie algebra. Translated fromMatematicheskie Zametki, Vol. 65, No. 4, pp. 607–611, April, 1999.     

COMPLETE LIE ALGEBRAS WITH l-STEP NILPOTENT RADICALS

The authors first give a necessary and sufficient condition for some solvable Lie algebras with l-step nilpotent radicals to be complete, and then construct a new class of infinite dimensional complete Lie algebras by using the modules of simple Lie algebras. The quotient algebras of this new constructed Lie algebras are non-solvable complete Lie algebras with l-step nilpotent radicals.