On free products in varieties of associative algebras

Varieties of associative algebras over a field of characteristic zero are considered. Belov recently proved that, in any variety of this kind, the Hilbert series of a relatively free algebra of finite rank is rational. At the same time, for three important varieties, namely, those of algebras with zero multiplication, of commutative algebras, and of all associative algebras, a stronger assertion holds: for these varieties, formulas that rationally express the Hilbert series of the free product algebra via the Hilbert series of the factors are well known. In the paper, a system of counterexamples is presented which shows that there is no formula of this kind in any other variety, even in the case of two factors one of which is a free algebra. However, if we restrict ourselves to the class of graded PI-algebras generated by their components of degree one, then there exist infinitely many varieties for each of which a similar formula is valid. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 693–702, May, 1999.     

On matrix types of prime varieties of associative algebras

Some answer is obtained to the problem by A. R. Kemer: We show that for a given number k and all sufficiently large p every prime variety of associative algebras of matrix type k over a field of characteristic p > 0 is regular.     

The unitary closure property of the prime varieties of associative algebras

We prove that every prime variety of associative algebras over an infinite field of characteristic p>0 is generated by either a unital algebra or a nilalgebra of bounded index. We show that the Engel verbally prime T-ideals remain verbally prime as we impose the identity $ x^{p^N } = 0 $ x^{p^N } = 0 for sufficiently large N. We then describe all prime varieties in an interesting class of varieties of associative algebras.     

Lie superalgebras whose enveloping algebras satisfy a non-matrix polynomial identity

Let L be a Lie superalgebra with its enveloping algebra U(L) over a field F. A polynomial identity is called non-matrix if it is not satisfied by the algebra of 2×2 matrices over F. We characterize L when U(L) satisfies a non-matrix polynomial identity. We also characterize L when U(L) is Lie solvable, Lie nilpotent, or Lie super-nilpotent.     

Lie algebras and algebras of associative type

In the paper, some properties of algebras of associative type are studied, and these properties are then used to describe the structure of finite-dimensional semisimple modular Lie algebras. It is proved that the homogeneous radical of any finite-dimensional algebra of associative type coincides with the kernel of some form induced by the trace function with values in a polynomial ring. This fact is used to show that every finite-dimensional semisimple algebra of associative type A = ⊕ _αεG A _α graded by some group G, over a field of characteristic zero, has a nonzero component A ₁ (where 1 stands for the identity element of G), and A ₁ is a semisimple associative algebra. Let B = ⊕ _αεG B _α be a finite-dimensional semisimple Lie algebra over a prime field F _p , and let B be graded by a commutative group G. If B = F _p ? _? A _L , where A _L is the commutator algebra of a ?-algebra A = ⊕ _αεG A _α ; if ? ? _? A is an algebra of associative type, then the 1-component of the algebra K ? _? B, where K stands for the algebraic closure of the field F _p , is the sum of some algebras of the form gl(n _i ,K).     

Homogeneously simple associative algebras

We prove that every finite-dimensional homogeneously simple associative algebra over an algebraically closed field is representable as the product of a full matrix algebra and a graded field.     

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.     

Power associative composition algebras

Composition algebras of arbitrary dimension over a field and satisfying the identities x ² x=x x ² and (x ²)²=(x ² x)x are shown to be precisely the well-known unital composition algebras, with the exception of three two dimensional algebras over the field of two elements. Received: 1 February 2000     

Octonion algebras obtained from associative algebras with involution

A natural octonion algebra structure on the symmetric elements of trace 0 of central simple associative algebras of degree 3 with involution of the second kind is obtained.

     

Permutation orbifolds and associative algebras

Let V be a vertex operator algebra and g =(1 2 ··· k) be a k-cycle which is viewed as an automorphism of the vertex operator algebra V?k. It is proved that Dong-Li-Mason’s associated associative algebra Ag(V?k) is isomorphic to Zhu’s algebra A(V) explicitly. This result recovers a previous result that there is a one-to-one correspondence between irreducible g-twisted V?k-modules and irreducible V-modules.     

Invariants of semisimple Lie algebras acting on associative algebras

If is a Lie algebra of derivations of an associative algebra , then the subalgebra of invariants is the set In this paper, we study the relationship between the structure of and the structure of , where is a finite dimensional semisimple Lie algebra over a field of characteristic zero acting finitely on , when is semiprime.

     

Construction of vertex operator algebras from commutative associative algebras

Given a commutative associative algebra A with an associative form (’), we construct a vertex operator algebra V with the weight two space V₂;? A If in addition the form (’) is nondegenerate, we show that there is a simple vertex operator algebra with V₂;? A We also show that if A is semisimple, then the vertex operator algebra constructed is the tensor products of a certain number of Virasoro vertex operator algebras.