Minimal Superalgebra Research Articles

ℤ2-gradings and minimal varieties of ℤ2-graded PI algebras

Let [Formula: see text] be an algebraically closed field of characteristic zero and let [Formula: see text] be a minimal superalgebra with semisimple part [Formula: see text], where each [Formula: see text] is a simple superalgebra. We say that [Formula: see text] is [Formula: see text]-special if [Formula: see text] and there exists [Formula: see text] such that [Formula: see text], with [Formula: see text], whereas, for the remaining indices [Formula: see text], [Formula: see text]. In [1], Avelar et al. classified the [Formula: see text]-special minimal superalgebras and described some necessary and sufficient conditions for the varieties generated by such superalgebras to be minimal of [Formula: see text]-exponent [Formula: see text]. In this paper, we give continuity to a such study and we made explicit that the [Formula: see text]-gradings play an important role in the minimality of some varieties. We provide sufficient conditions, depending on the [Formula: see text]-grading of [Formula: see text], for the variety generated by [Formula: see text] to be minimal. Moreover, we characterize two families of [Formula: see text]-special minimal superalgebras and we give necessary and sufficient conditions (depending on their [Formula: see text]-gradings) for the varieties generated by them to be minimal.

On the classification of a family of minimal superalgebras

Let F be an algebraically closed field of characteristic zero and let be an n-tuple of simple superalgebras such that and there exists such that , with , whereas for all remaining indices. In this paper, we classify the minimal F-superalgebras such that its graded semisimple part is given by . Moreover, we obtain necessary and sufficient conditions to the supervariety generated by such minimal superalgebra be or not minimal of superexponent

Minimal superalgebras generating minimal supervarieties

It has been shown that in characteristic zero the generators of the minimal supervarieties of finite basic rank belong to the class of minimal superalgebras introduced by Giambruno and Zaicev (Trans Am Math Soc 355:5091–5117, 2003). In the present paper the complete list of minimal supervarieties generated by minimal superalgebras whose maximal semisimple homogeneous subalgebra is the sum of three graded simple algebras is provided. As a consequence, we negatively answer the question of whether any minimal superalgebra generates a minimal supervariety.

Comparing the ℤ 2-Graded Identities of Two Minimal Superalgebras with the Same Superexponent

Let F be a field of characteristic zero. We study two minimal superalgebras A and B having the same superexponent but such that T 2 (A) ⫋ T 2 (B), thus providing the first example of a minimal superalgebra generating a non minimal supervariety. We compare the structures and codimension sequences of A and B.

Minimal varieties of associative PI (super)-algebras with respect to their (graded) exponent

Giambruno and Zaicev (Trans Am Math Soc 355:5091–5117, 2003) characterized varieties of associative PI-algebras over a field of characteristic zero which are minimal of fixed exponent. The aim of the present survey paper is to discuss this result and the recent developments on the corresponding problem for PI-algebras endowed with a \(\mathbb {Z}_2\)-grading. In particular, we provide an example of minimal superalgebra not generating a minimal supervariety of fixed superexponent, thereby partially answering the question of which actually are the generators of minimal supervarieties of finite basic rank.

Minimal supervarieties with factorable ideal of graded polynomial identities

Given a minimal superalgebra A=Ass⊕J(A), any subsequence of the graded simple summands of Ass determines a homogeneous subalgebra of A which is still a minimal superalgebra. In the present paper we provide a sufficient condition so that the TZ2-ideal of graded polynomial identities satisfied by A factorizes as the product of the TZ2-ideals associated to its suitable homogeneous subalgebras of such a type. We use this fact to show that in this event A generates a minimal supervariety of fixed superexponent.

On some minimal supervarieties of exponential growth

In the present paper we investigate minimal supervarieties of given superexponent over fields of characteristic zero. We show that any minimal supervariety of finite basic rank is generated by one of the minimal superalgebras, introduced by Giambruno and Zaicev in 2003. Furthermore it is proved that any minimal superalgebra, whose graded simple components of the semisimple part are simple, generates a minimal supervariety. Finally we state that the same conclusion holds when the semisimple part of a minimal superalgebra has exactly two arbitrary graded simple components.

Codimension growth and minimal superalgebras

A celebrated theorem of Kemer (1978) states that any algebra satisfying a polynomial identity over a field of characteristic zero is PI-equivalent to the Grassmann envelope G(A) of a finite dimensional superalgebra A. In this paper, by exploiting the basic properties of the exponent of a PI-algebra proved by Giambruno and Zaicev (1999), we define and classify the minimal superalgebras of a given exponent over a field of characteristic zero. In particular we prove that these algebras can be realized as block-triangular matrix algebras over the base field. The importance of such algebras is readily proved: A is a minimal superalgebra if and only if the ideal of identities of G(A) is a product of verbally prime T-ideals. Also, such superalgebras allow us to classify all minimal varieties of a given exponent i.e., varieties V such that exp(V) = d > 2 and exp(U) < d for all proper subvarieties U of V. This proves in the positive a conjecture of Drensky (1988). As a corollary we obtain that there is only a finite number of minimal varieties for any given exponent. A classification of minimal varieties of finite basic rank was proved by the authors (2003). As an application we give an effective way for computing the exponent of a T-ideal given by generators and we discuss the problem of what functions can appear as growth functions of varieties of algebras.

Douglas algebras without maximal subalgebra and without minimal superalgebra

We give several examples of Douglas algebras that do not have any maximal subalgebra. We find a condition on these algebras that guarantees that some do not have any minimal superalgebra. We also show that if A is the only maximal subalgebra of a Douglas algebra B, then the algebra A does not have any maximal subalgebra.

Minimal Superalgebras of Weak- ∗ Dirichlet Algebras

Minimal Superalgebras of Weak- ∗ Dirichlet Algebras

Minimal superalgebras of weak-* Dirichlet algebras

Let A A be a weak- ∗ * Dirichlet algebra in L ∞ ( m ) {L^\infty }(m) and let H ∞ ( m ) {H^\infty }(m) be the weak- ∗ * closure of A A in L ∞ ( m ) {L^\infty }(m) . It may happen that there are minimal weak- ∗ * closed subalgebras of L ∞ ( m ) {L^\infty }(m) that contain H ∞ ( m ) {H^\infty }(m) properly. In this paper it is shown that if there is a minimal, proper, weak- ∗ * closed superalgebra of H ∞ ( m ) {H^\infty }(m) , then, in fact, that algebra is the unique least element in the lattice of all proper weak- ∗ * closed superalgebras of H ∞ ( m ) {H^\infty }(m) .