Isomorphism Classes Of Algebras Research Articles

The algebraic classification of nilpotent algebras

We give the complete algebraic classification of all complex 4-dimensional nilpotent algebras. The final list has 234 (parametric families of) isomorphism classes of algebras, 66 of which are new in the literature.

Isomorphism Classes of Algebras with Radical Cube Zero II

We study to classify, up to isomorphism, algebras Λ over a field k such that the radical cubed is zero and Λ modulo the radical is a product of copies of k. The number of local quasi-Frobenius k-algebras with the condition is shown to be not less than the cardinality of k. In particular, the canonical forms of those algebras of dimension 5 are presented and their isomorphism classes are completely determined under some conditions on k.

Isomorphism Classes of Rings Related to Heisenberg Algebras#

In this paper we examine isomorphism classes of algebras which are generalizations of the enveloping algebra of the Heisenberg Lie algebra.

Algebras with minimal spectrum

The spectrum of a class of algebras, ~ , is the set of cardinalities of the finite members of ~K; Spec (~K) denotes the spectrum of ~s If ~K is a variety (equational class), then Spec (~) is a multiplicative monoid of positive integers. Le t 92 be a finite non-trivial algebra and let ~s be the variety generated by 92; we say that 92 has minimal spectrum if Spec (~s = {1921 ~ [ n 0}; the aim of this paper is to characterize all algebras with minimal spectrum. Closely connected with this problem is the concept of the fine spectrum of ~ ; this was introduced and studied by Walter Taylor in [16]. The fine spectrum of ~K is the function fx such that fx(n) is the number of isomorphism classes of algebras in ~ of cardinality n. We will abbreviate fxr by f~; if 92 is finite, then fga(1) = 1 and f~(n) is finite. Note that if 92 has minimal spectrum, then fa(n)= 0 if n is not a power of [92 I. In [16] Taylor showed that there are exactly 9 2-e lement algebras with minimal spectrum (of course, up to polynomial equivalence). Moreover , he showed that for 7 of them, fga(2 "~) = 1 while for the other 2, fa(2 m) = 2 for m > 0. This property carries over to every algebra with minimal spectrum. Examples of the first kind are ~2, the 2-element boolean algebra, and @2, the 2-element group. An example of the second kind is ~2"=({0, 1};+, 1), the 2-element group in which the non-identity element is the value of a nuUary operation. The two isomorphism classes of cardinality 2 ~ are the one in which the added nullary operat ion is not the identity and the one in which it is the identity. For instance, if we factor ((~2") 2 by the congruence induced by the diagonal subgroup, then in the quotient the nullary is the identity. All 9 algebras are closely related to either ~2 or (~2. In particular, each example generates a congruence permutable variety. This property carries over to every algebra with minimal spectrum.