For a two-parameter family of lower triangular matrices with entries involving Jacobi polynomials an explicit inverse is given, with entries involving a sum of two Jacobi polynomials. The formula simplifies in the Gegenbauer case and then one choice of the parameter solves an open problem in a recent paper by Koelink, van Pruijssen & Román. The twoparameter family is closely related to two two-parameter groups of lower triangular matrices, of which we also give the explicit generators. Another family of pairs of mutually inverse lower triangular matrices with entries involving Jacobi polynomials, unrelated to the family just mentioned, was given by J. Koekoek & R. Koekoek (1999). We show that this last family is a limit case of a pair of connection relations between Askey-Wilson polynomials having one of their four parameters in common.
A main contribution of this paper is the explicit construction of comparison morphisms between the standard bar resolution and Bardzell's minimal resolution for truncated quiver algebras over arbitrary fields (TQA's). As a direct application we describe explicitly the Yoneda product and derive several results on the structure of the cohomology ring of TQA's over a field of characteristic zero. For instance, we show that the product of odd degree cohomology classes is always zero. We prove that TQA's associated with quivers with no cycles or with neither sinks nor sources have trivial cohomology rings. On the other side we exhibit a fundamental example of a TQA with nontrivial cohomology ring. Finally, for truncated polynomial algebras in one variable, we construct explicit cohomology classes in the bar resolution and give a full description of their cohomology ring.
Let G o be a semisimple Lie group and let K o denote a maximal compact subgroup of G o . Let U(g) be the complex universal enveloping algebra of G o and let U(g) K denote the centralizer of K o in U(g). Also let P :be the projection map corresponding to the direct sum U(g) = (U (k) ⊗ U(a)) ⊕ U(g)n associated to an Iwasawa decomposition of G o adapted to K o . In this paper we give a characterization of the image of U(g) K under the injective antihomomorphism P : U(g) K −→ U(k) ⊗ U(a), considered by Lepowsky in [J. Lepowsky, Algebraic results on representations of semisimple Lie groups, Trans. Amer. Math. Soc. 176 (1973) 1-44], when G o = Sp(n, 1).
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.