Graded Simple Algebras Research Articles

Mackey's obstruction map for discrete graded algebras

G.W. Mackey's celebrated obstruction theory for projective representations of locally compact groups was remarkably generalized by J. M. G. Fell and R. S. Doran to the wide area of saturated Banach *-algebraic bundles.Analogous obstruction is suggested here for discrete group graded algebras which are not necessarily saturated, i.e. strongly graded in the discrete context. The discrete obstruction is a map assigning a certain second cohomology class to every equivariance class of absolutely graded-simple modules.The set of equivariance classes of such modules is equipped with an appropriate multiplication, namely a graded product, such that the obstruction map is a homomorphism of abelian monoids. Graded products, essentially arising as pull-backs of bundles, admit many nice properties, including a way to twist graded algebras and their graded modules.The obstruction class turns out to determine the fine part that appears in the Bahturin-Zaicev-Sehgal decomposition, i.e. the graded Artin-Wedderburn theorem for graded-simple algebras which are graded Artinian, in case where the base algebras (i.e. the unit fiber algebras) are finite-dimensional over algebraically closed fields.

Codimension growth of Lie algebras with a generalized action

Let F F be a field of characteristic 0 0 and L L a finite dimensional Lie F F -algebra endowed with a generalized action by an associative algebra H H . We investigate the exponential growth rate of the sequence of H H -graded codimensions c n H ( L ) c_n^H(L) of L L which is a measure for the number of non-polynomial H H -identities of L L . More precisely, we construct an S S -graded Lie algebra (with S S a semigroup) which has an irrational exponential growth rate (the exact value is obtained). This is the first example of a graded Lie algebra with non-integer exponential growth rate. In addition, we prove an analogue of Amitsur’s conjecture (i.e. lim n → ∞ c n H ( L ) n ∈ Z \lim _{n\rightarrow \infty } \sqrt [n]{c_n^{H}(L)} \in \mathbb {Z} ) for general H H under the assumption that L L is both semisimple as Lie algebra and for the H H -action. Moreover if H = F S H=FS is a semigroup algebra the condition that L L is semisimple for the H H -action can be dropped. This is in strong contrast to the associative setting where an infinite family of graded-simple algebras with irrational graded PI-exponent was constructed.

On H-simple not necessarily associative algebras

An algebra [Formula: see text] with a generalized [Formula: see text]-action is a generalization of an [Formula: see text]-module algebra where [Formula: see text] is just an associative algebra with [Formula: see text] and a relaxed compatibility condition between the multiplication in [Formula: see text] and the [Formula: see text]-action on [Formula: see text] holds. At first glance, this notion may appear too general, however, it enables to work with algebras endowed with various kinds of additional structures (e.g. comodule algebras over Hopf algebras, graded algebras, algebras with an action of a semigroup by anti-endomorphisms). This approach proves to be especially fruitful in the theory of polynomial identities. We show that if [Formula: see text] is a finite dimensional (not necessarily associative) algebra over a field of characteristic [Formula: see text] and [Formula: see text] is simple with respect to a generalized [Formula: see text]-action, then there exists [Formula: see text] where [Formula: see text] is the sequence of codimensions of polynomial [Formula: see text]-identities of [Formula: see text]. In particular, if [Formula: see text] is a finite dimensional (not necessarily group graded) graded-simple algebra, then there exists [Formula: see text] where [Formula: see text] is the sequence of codimensions of graded polynomial identities of [Formula: see text]. In addition, we study the free-forgetful adjunctions corresponding to (not necessarily group) gradings and generalized [Formula: see text]-actions.

Graded-simple algebras and cocycle twisted loop algebras

The loop algebra construction by Allison, Berman, Faulkner, and Pianzola, describes graded-central-simple algebras with split centroid in terms of central simple algebras graded by a quotient of the original grading group. Here the restriction on the centroid is removed, at the expense of allowing some deformations (cocycle twists) of the loop algebras.

Identities and isomorphisms of finite-dimensional graded simple algebras

Let F be an algebraically closed field, G be an abelian group, and let U and V be arbitrary finite-dimensional G-graded simple algebras over F. We prove that U and V are isomorphic as graded algebras if, and only if, they satisfy the same graded polynomial identities.

Identities of graded simple algebras

We study identities of finite dimensional algebras over a field of characteristic zero, graded by an arbitrary groupoid . First, we prove that its graded colength has a polynomially bounded growth. For any graded simple algebra A, we prove the existence of the graded PI-exponent, provided that is a commutative semigroup. If A is simple in a non-graded sense, the existence of the graded PI-exponent is proved without any restrictions on .

Realization of graded-simple algebras as loop algebras

Multiloop algebras determined by n commuting algebra automorphisms of finite order are natural generalizations of the classical loop algebras that are used to realize affine Kac-Moody Lie algebras. In this paper, we obtain necessary and sufficient conditions for a Zn-graded algebra to be realized as a multiloop algebra based on a finite dimensional simple algebra over an algebraically closed field of characteristic 0. We also obtain necessary and sufficient conditions for two such multiloop algebras to be graded-isomorphic, up to automorphism of the grading group. We prove these facts as consequences of corresponding results for a generalization of the multiloop construction. This more general setting allows us to work naturally and conveniently with arbitrary grading groups and arbitrary base fields.

Semigroup gradings on associative rings

The main aim of this paper is the adaptation of the results of the papers [Y.A. Bahturin, S.K. Sehgal, M.V. Zaicev, Group gradings on associative algebras, J. Algebra 241 (2) (2001) 677–698; Y.A. Bahturin, S.K. Sehgal, M.V. Zaicev, Finite-dimensional graded simple algebras, preprint, and M.V. Zaĭtsev, S.K. Segal, Finite gradings of simple Artinian rings, Vestnik Moskov. Univ. Ser. I Mat. Mekh., vol. 3, 2001, pp. 21–24, 77 (in Russian); translation in Moscow Univ. Math. Bull. 56 (3) (2001) 21–24] about the structure of graded simple rings from the gradings by groups to the gradings by cancellative semigroups.

Simple Jordan Color Algebras Arising from Associative Graded Algebras

If R is a G -graded associative algebra, where G is an abelian group and ϵ is a bicharacter for G , then R also has the structure of a Jordan color algebra. In addition, if R is endowed with a color involution ∗ , then the symmetric elements S under ∗ are also a Jordan color algebra. Generalizing results of I. N. Herstein, we examine the Jordan color structure of R and S . In particular, we show that if R is a graded-simple algebra, then both R and S are simple Jordan color algebras, except for some special cases which cannot arise in the ordinary case.

Group Gradings on Associative Algebras

Let R=⊕g∈GRg be a G-graded ring. We describe all types of gradings on R if G is torsion free and R is Artinian semisimple. If R is a matrix algebra over an algebraically closed field F, then we give a description of all G-gradings on R provided that G is an abelian group. In the case of an abelian group G we also classify all finite-dimensional graded simple algebras and finite-dimensional graded division algebras over an algebraically closed field of characteristic zero.