Codimension growth of special simple Jordan algebras

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).

Journal of Algebra, 1996

Proceedings of the National Academy of …, 1966

Algebra and Logic, 1971

In 1948, A. A. Albert [6] defined a class of standard algebras. For finite-dimensional standard algebras of characteristic # 2 , fundamental structure theorems were proved, generalizing classical Wedderburn theorems for associative algebras.

Linear Algebra and its Applications, 2003

Let F be an algebraically closed field. This paper describes the possible numbers of nonconstant invariant polynomials of the Jordan product XA + AX, when A is fixed and X varies.

Proceedings of the American Mathematical Society, 2000

Let A be an associative algebras over a field of characteristic zero. We prove that the codimensions of A are polynomially bounded if and only if any finite dimensional algebra B with Id(A) = Id(B) has an explicit decomposition into suitable subalgebras; we also give a decomposition of the n-th cocharacter of A into suitable Sn-characters. We give similar characterizations of finite dimensional algebras with involution whose *-codimension sequence is polynomially bounded. In this case we exploit the representation theory of the hyperoctahedral group.

Journal of Pure and Applied Algebra, 2005

In this paper we prove the inheritance of polynomial identities by covers of nondegenerate Jordan algebras satisfying certain ideal absorption properties. As a consequence we obtain the inheritance of speciality by Martindale-like covers, proving, in particular, that a Jordan algebra having a nondegenerate essential ideal which is special must be special.

Transactions of the American Mathematical Society

In this paper we study Gelfand-Kirillov dimension in Jordan algebras. In particular we will relate Gelfand-Kirillov (GK for short) dimensions of a special Jordan algebra and its associative enveloping algebra and also the GK dimension of a Jordan algebra and the GK dimension of its universal multiplicative enveloping algebra.

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, 1999