The prime spectrum of algebras of quadratic growth

Advances in Applied Mathematics, 2006

We give a simple construction of a prime monomial algebra with quadratic growth, which is neither primitive nor PI.

The São Paulo Journal of Mathematical Sciences, 2007

The main goal of this paper is to prove that the five algebras which were used in [3] to classify (up to PI-equivalence) the algebras whose sequence of codimensions is bounded by a linear function generate the only five minimal varieties of quadratic growth.

Transactions of the American Mathematical Society, 1985

Let G G be an associative monomial k {\mathbf {k}} -algebra. If G G is assumed to be finitely presented, then either G G contains a free subalgebra on two monomials or else G G has polynomial growth. If instead G G is assumed to have finite global dimension, then either G G contains a free subalgebra or else G G has a finite presentation and polynomial growth. Also, a graded Hopf algebra with generators in degree one and relations in degree two contains a free Hopf subalgebra if the number of relations is small enough.

2011

Advances in Applied Mathematics, 2011

Let A be a (non-necessarily associative) finite-dimensional algebra over a field of characteristic zero. A quantitative estimate of the polynomial identities satisfied by A is achieved through the study of the asymptotics of the sequence of codimensions of A. It is well known that for such an algebra this sequence is exponentially bounded. Here we capture the exponential rate of growth of the sequence of codimensions for several classes of algebras including simple algebras with a special non-degenerate form, finite-dimensional Jordan or alternative algebras and many more. In all cases such rate of growth is integer and is explicitly related to the dimension of a subalgebra of A. One of the main tools of independent interest is the construction in the free non-associative algebra of multialternating polynomials satisfying special properties. addresses: [email protected] (A. Giambruno), [email protected] (I. Shestakov), [email protected] (M. Zaicev).

Advances in Mathematics

Ufnarovski remarked in 1990 that it is unknown whether any finitely presented associative algebra of linear growth is automaton, that is, whether the set of normal words in the algebra form a regular language. If the algebra is graded, then the rationality of the Hilbert series of the algebra follows from the affirmative answer to Ufnarovski's question. Assuming that the ground field has a positive characteristic, we show that the answer to Ufnarovskii's question is positive if and only if the basic field is an algebraic extension of its prime subfield. Moreover, in the "only if" part we show that there exists a finitely presented graded algebra of linear growth with irrational Hilbert series. In addition, over an arbitrary infinite basic field, the set of Hilbert series of the quadratic algebras of linear growth with 5 generators is infinite. Our approach is based on a connection with the dynamical Mordell-Lang conjecture. This conjecture describes the intersection of an orbit of an algebraic variety endomorphism with a subvariety. We show that the positive answer to Ufnarovski's question implies some known cases of the dynamical Mordell-Lang conjecture. In particular, the positive answer for a class of algebras is equivalent to the Skolem-Mahler-Lech theorem which says that the set of the zero elements of any linear recurrent sequence over a zero characteristic field is the finite union of several arithmetic progressions. In particular, the counterexamples to this theorem in the finite characteristic case give examples of algebras with irrational Hilbert series.

Advances in Mathematics, 2008

Let A be an algebra over a field F of characteristic zero and let c n (A), n = 1, 2,. .. , be its sequence of codimensions. We prove that if c n (A) is exponentially bounded, its exponential growth can be any real number > 1. This is achieved by constructing, for any real number α > 1, an F-algebra A α such that lim n→∞ n √ c n (A α) exists and equals α. The methods are based on the representation theory of the symmetric group and on properties of infinite Sturmian and periodic words.

Advances in Applied Mathematics, 2006

Let F be a field of characteristic zero and let A be an F-algebra. The polynomial identities satisfied by A can be measured through the asymptotic behavior of the sequence of codimensions and the sequence of colengths of A. For finite dimensional algebras we show that the colength sequence of A is polynomially bounded and the codimension sequence cannot have intermediate growth. We then prove that for general nonassociative algebras intermediate growth of the codimensions is allowed. In fact, for any real number 0 < β < 1, we construct an algebra A whose sequence of codimensions grows like n n β .

Journal of Algebra, 1988

Journal of the London Mathematical Society, 1991

Kac has introduced the notion of (polynomial) growth for a graded Lie algebra. Here we consider Lie algebras L that occur as ideals either in the rational homotopy Lie algebra of a simply connected CW complex of finite type and finite category or as ideals in the homotopy Lie algebra of a local noetherian ring. Theorem. If these ideals have {finite) polynomial growth, then they are finite dimensional.