On the module structure of free L ie algebras

We study the free Lie algebra over a field of non-zero characteristic as a module for the cyclic group of order acting on by cyclically permuting the elements of a free generating set. Our main result is a complete decomposition of as a direct sum of indecomposable modules.

     

The rings of quotients of free algebras

It is proved that a symmetric Utumi ring of quotients, U, of a free associative (noncommutative) algebra F(X) with unity coincides with the algebra itself, U=F(X). From this, we obtain a similar statement concerning a symmetric Martindale ring of quotients, Q(F(X))=F(X), which is well known. In addition, it is shown that a left Martindale ring of quotients, F(X)_F, of a free algebra is a prime algebra and, moreover, every homogeneous element in a free algebra has the right inverse in F(X)_F but does not have the left one (unless, of course, r belongs to an underlying field). Since a left Utumi ring of quotients and a left Martindale ring of quotients for a free algebra both appear prime, an interesting question arises as to whether or not they coincide. Supported by RFFR grant No. 95-01-01356 and by ISF grant RPS000-RPS300. Translated fromAlgebra i Logika, Vol. 35, No. 6, pp. 655–662, November–December, 1996.     

Left-symmetric algebras of derivations of free algebras

A structure of a left-symmetric algebra on the set of all derivations of a free algebra is introduced such that its commutator algebra becomes the usual Lie algebra of derivations. Left and right nilpotent elements of left-symmetric algebras of derivations are studied. Simple left-symmetric algebras of derivations and Novikov algebras of derivations are described. It is also proved that the positive part of the left-symmetric algebra of derivations of a free nonassociative symmetric m-ary algebra in one free variable is generated by one derivation and some right nilpotent derivations are described.     

Almost free algebras

We deal with the class of cardinalsk for which a given variety has an almost free, not free, even not essentially free algebra. The main result is that when enough reflection holds, this behaviour is determined by min{n:n=ω or the variety fail a condition CP _n }. Research supported by NSERC grant #9848. Prof. Mekler died on June 10, 1992. Publication #366. Research supported by the BSF.     

The structure of Hopf algebras

This survey considers work published between 1970 and 1990 on the algebraic aspects of Hopf theory. There are detailed discussions of the properties of antipodes, group and primitive elements, integrals, crossed products, Galois theory, Lie coalgebras, the category of Hopf algebras, quantum groups, etc.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 29, pp. 3–63, 1991.     

The structure of AS-Gorenstein algebras

In this paper, we define a notion of AS-Gorenstein algebra for N-graded algebras, and show that symmetric AS-regular algebras of Gorenstein parameter 1 are exactly preprojective algebras of quasi-Fano algebras. This result can be compared with the fact that symmetric graded Frobenius algebras of Gorenstein parameter −1 are exactly trivial extensions of finite-dimensional algebras. The results of this paper suggest that there is a strong interaction between classification problems in noncommutative algebraic geometry and those in representation theory of finite-dimensional algebras.     

Subalgebras of free algebras

We use non-commutative Jacobian matrix to get information on finitely generated subalgebras of a free Lie algebra. In particular, we show that the rank of such a subalgebra is equal to the left rank (i.e., to the maximal number of left independent rows) of the corresponding Jacobian matrix; this also yields an effective procedure for finding the rank of a finitely generated subalgebra.

     

A note on free algebras of discriminator algebras

Algebra universalis -     

The structure of saturated critical blocks

A 2‐connected graph G is a critical block if G ? v is not 2‐connected for every vertex v ∈ V(G). A critical block G is a saturated critical block if G + e is not a critical block for any new edge e. The structure of all saturated critical blocks and a procedure for constructing every saturated critical block are determined. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 223–237, 2003     

The cohomology structure of string algebras

We show that the graded commutative ring structure of the Hochschild cohomology HH^*(A) is trivial in case A is a triangular quadratic string algebra. Moreover, in case A is gentle, the Lie algebra structure on HH^*(A) is also trivial.     

The structure of solid binary algebras

A solid binary algebra in an abstract characterisation of the idempotents of a completely regular semigroup. We present here a structure theorem for solid binary algebras in terms of semilattices and rectangular bands. We also show that a free solid binary algebra can be embedded in a free completely regular semigroup; thus the word problem for the free solid binary algebra can be solved by using a solution of the word problem for the free completely regular semigroup.Presented by B. M. Schein.The author gratefully acknowledges the financial support of an Australian Postgraduate Research Award.