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 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.     

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.     

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

Partially supported by NSF grant #DMS-8807336     

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:

Capelli identities for reductive dual pairs

Analogues of the Capelli identity are given for every irreducible reductive dual pair in the complex symplectic group and the complex orthogonal group. They describe a natural correspondence between the “centers” of the two universal enveloping algebras and an algebra of invariant operators.     

Quantum immanants and higher Capelli identities

We consider some remarkable central elements of the universal enveloping algebraU(gl(n)) which we call quantum immanants. We express them in terms of generatorsE _ij ofU(gl(n)) and as differential operators on the space of matrices These expressions are a direct generalization of the classical Capelli identities. They result in many nontrivial properties of quantum immanants. The author is supported by the International Science Foundation and the Russian Fundamental Research Foundation.