The graded Hecke algebra for a finite Weyl group is intimately related to the geometry of the Springer correspondence. A construction of Drinfeld produces an analogue of a graded Hecke algebra for any finite subgroup of GL(V ). This paper classifies all the algebras obtained by applying Drinfeld’s construction to complex reflection groups. By giving explicit (though nontrivial) isomorphisms, we...

Let T (X) be the tensor bialgebra over an alphabet X. It is a graded connected cocommutative bialgebra, canonically isomorphic to the envelopping bialgebra of the free Lie algebra over X, Lie(X). The subalgebra of its convolution algebra generated by the projections arising from the graduation is also an algebra for the composition of morphisms and is anti-isomorphic as such with the direct sum...

In this paper, we present an isomorphism between the ring of general polynomials over a division algebra D with center F and the group ring of the free monoid with [D : F ] variables over D. Using this isomorphism, we define the characteristic polynomial of any matrix over any division algebra, i.e., a general polynomial with one variable over the algebra whose roots are precisely the left eige...

We compute the monoid V (LK(E)) of isomorphism classes of finitely generated projective modules over certain graph algebras LK(E), and we show that this monoid satisfies the refinement property and separative cancellation. We also show that there is a natural isomorphism between the lattice of graded ideals of LK(E) and the lattice of order-ideals of V (LK(E)). When K is the field C of complex ...

Let G be a split reductive algebraic group over a non-archimedean local field. We study the representation theory of a central extension G̃ of G by a cyclic group of order n, under some mild tameness assumptions on n. In particular, we focus our attention on the development of the theory of principal series representations for G̃ and its applications to the study of Hecke algebras via a Satake is...

Commutative H *-algebra is characterized in terms of idempotents. Here we offer three characterizations. 1. Introduction. In the past, the author used commuting idempotents to characterize continuous functions defined on a certain space [3, 4]. For example , it was shown in [3] that a certain Banach algebra is isometrically isomorphic to the space C(S) of all continuous complex-valued functions...

We prove a Thom isomorphism theorem for differential forms in the setting of transverse Lie algebra actions on foliated manifolds and vector bundles.

We study the equivalence problem of cubic forms. We lower bound its complexity by that of F-algebra isomorphism problem and hence by the graph isomorphism problem (for all fields F). For finite fields we upper bound the complexity of cubic forms by NP∩coAM. We also study the cubic forms obtained from F-algebras and show that they are regular and indecomposable.

One of the basic properties of the Clifford algebra gives an explicit basis for it in terms of a basis of the underlying vector space (Theorem 1 below), and another one provides a vector space isomorphism between the Clifford algebra and the exterior algebra of the same vector space (the so-called Chevalley map, Theorem 2 below). While both of these properties appear in standard literature such...