Capelli identity on multiparameter quantum linear groups

A quantum Capelli identity is given on the multiparameter quantum general linear group based on the (p _ij , u)-condition. The multiparameter quantum Pfaffan of the (p _ij , u)-quantum group is also introduced and the transformation under the congruent action is given. Generalization to the multiparameter hyper-Pfaffan and relationship with the quantum minors are also investigated.     

Algebraic derivations with constants satisfying a polynomial identity

We prove that if a semiprime ringR possesses a derivation which is integral over its extended centroidC and whose constants satisfy a polynomial identity, thenR itself is a PI-ring. This answers affirmatively a problem raised by M. Smith in 1975 and recently again by Bergen and Grzeszczuk [4].     

Group algebras with units satisfying a group identity

Let be the group algebra of a group over a field , and let be its group of units. A conjecture by Brian Hartley asserts that if is a torsion group and satisfies a group identity, then satisfies a polynomial identity. This was verified earlier in case is an infinite field. Here we modify the original proof so that it handles fields of all sizes.

     

Asymptotic behaviour of codimensions of p. i. algebras satisfying Capelli identities

Let be a p. i. algebra with 1 in characteristic zero, satisfying a Capelli identity. Then the cocharacter sequence is asymptotic to a function of the form , where and .

     

Group algebras with symmetric units satisfying a group identity

We study group algebras FG for which the symmetric units under the natural involution: g^*=g⁻¹ satisfy a group identity. For infinite fields F of characteristic ≠2, a classification of torsion groups G whose symmetric units U⁺(FG) satisfy a group identity was given in [3] by Giambruno-Sehgal-Valenti. We extend this work to non torsion groups. Research supported by NSERC of Canada and MIUR of Italy.     

Group algebras with units satisfying a group identity II

We classify group algebras of torsion groups over a field of characteristic with units satisfying a group identity.

     

The Capelli identity,the double commutant theorem,and multiplicity-free actions

Partially supported by NSF grant #DMS-8807336     

Rational orthogonal bases satisfying the Bedrosian identity

We develop a necessary and sufficient condition for the Bedrosian identity in terms of the boundary values of functions in the Hardy spaces. This condition allows us to construct a family of functions such that each of which has non-negative instantaneous frequency and is the product of two functions satisfying the Bedrosian identity. We then provide an efficient way to construct orthogonal bases of L ²(ℝ) directly from this family. Moreover, the linear span of the constructed basis is norm dense in L ^p (ℝ), 1 < p < ∞. Finally, a concrete example of the constructed basis is presented.     

Characterization of bands satisfying no non-trivial identity

The object of this paper is to describe the class of bands which satisfy no non-trivial identity, and to give some examples of bands in this class.     

Bounded Engel elements in groups satisfying an identity

We prove that a residually finite group G satisfying an identity \(w\equiv 1\) and generated by a commutator closed set X of bounded left Engel elements is locally nilpotent. We also extend such a result to locally graded groups, provided that X is a normal set. As an immediate consequence, we obtain that a locally graded group satisfying an identity, all of whose elements are bounded left Engel, is locally nilpotent.     

Group algebras with units satisfying an Engel identity

We classify group algebras of periodic groups over a field of positive characteristic with units satisfying an Engel identity.     

Algebras with skew-symmetric identity of degree 3

Algebras with one of the following identities are considered:

A Capelli Harish-Chandra homomorphism

For a real reductive dual pair the Capelli identities define a homomorphism from the center of the universal enveloping algebra of the larger group to the center of the universal enveloping algebra of the smaller group. In terms of the Harish-Chandra isomorphism, this map involves a -shift. We view a dual pair as a Lie supergroup and offer a construction of the homomorphism based solely on the Harish-Chandra's radial component maps. Thus we provide a geometric interpretation of the -shift.

     

On loop extensions satisfying one single identity and cohomology of loops

In this paper are defined cohomology-like groups that classify loop extensions satisfying a given identity in three variables for association identities and in two variables for the case of commutativity. It is considered a large amount of identities. These groups generalize those defined in Nishigori [3 Nishigori, N. (1963). On loop extensions of groups and M-cohomology groups I. J. Sci. Hiroshima Univ. Ser. A-I 27:151–165. [Google Scholar]] and of Kenneth and Leedham-Green [2 Kenneth, W. J., Leedham-Green, Ch. R. (1990). Loop cohomology. Czec. Math. J. 40(2):182–194.[Web of Science ®] , [Google Scholar]]. It is computed the number of metacyclic extensions for trivial action of the quotient on the kernel in one particular case for left Bol loops and in general for commutative loops.