Spherical objects of the multiplicity free Brauer tree algebra with two edges

We explicitly describe all spherical objects, as well as tilting objects, of the multiplicity free Brauer tree algebra with two edges.     

Finitary Incidence Algebras

It is well known that incidence algebras can be defined only for locally finite partially ordered sets (Doubilet et al., 1972 Doubilet , P. , Rota , G.-C. , Stanley , R. ( 1972 ). On the foundations of combinatorial theory (VI): The idea of generating function . In: Proc. of the Sixth Berkely Symp. on Math. Stat. and Probab . v. II , Univ. of Calif. Press , pp. 267 – 318 . [Google Scholar]; Stanley 1986 Stanley , R. P. ( 1986 ). Enumerative Combinatorics . v. 1 . Monterey, CA : Wadsworth &; Brooks/Cole .[Crossref] , [Google Scholar]). At the same time, for example, the poset of cells of a noncompact cell partition of a topological space is not locally finite. On the other hand, some operations, such as the order sum and the order product (Stanley, 1986 Stanley , R. P. ( 1986 ). Enumerative Combinatorics . v. 1 . Monterey, CA : Wadsworth &; Brooks/Cole .[Crossref] , [Google Scholar]), do not save the locally finiteness. So it is natural to try to generalize the concept of incidence algebra.

In this article, we consider the functions in two variables on an arbitrary poset (finitary series), for which the convolution operation is defined. We obtain the generalization of incidence algebra—finitary incidence algebra and describe its properties: invertibility, the Jackobson radical, idempotents, regular elements. As a consequence a positive solution of the isomorphism problem for such algebras is obtained.     

Maschke Type Theorems for Nonassociative Hopf Algebras

In this article, we give necessary and sufficient conditions for a possibly nonassociative comodule algebra over a nonassociative Hopf algebra to have a total integral, thus extending the classical theory developed by Doi in the associative setting. Also, from this result we deduce a version of Maschke's Theorems and the consequent characterization of projectives for (H, B)-Hopf triples associated with a nonassociative Hopf algebra H and a nonassociative right H-comodule algebra B.     

Some Results Obtained by Application of the LLT Algorithm

We determine the v-decomposition numbers d _μλ(v) for μ a partition with at most three parts. We use this information to compute the composition factors of the Specht modules of the Hecke algebra ?₀ = ?_{?, ω}( _n ) which correspond to partitions with at most three parts.     

Covering morphisms

Let (R, m) be a two-dimensional regular local ring and let A be a finitely generated torsion-free R-module. If A is a complete module, then Dan Katz and Vijay Kodiyalam show A satisfies five conditions. They ask whether these five conditions are equivalent without assuming A to be complete. In a previous paper we determined all implications among these five conditions with one exception. In this paper, with an additional hypothesis on the units of R, we resolve the remaining case.     

Exponentiation of infinite dimensional Z-graded lie algebras

In this article we give a new technique for exponentiating infinite dimensional graded representations of graded Lie algebras that allows for the exponentiation of some non-locally nilpotent elements. Our technique is to naturally extend the representation of the Lie algebra g on the space V naturally to a representation on a subspace £ of the dual space V ^*. After introducing the technique, we prove that it enables the exponentiation of all elements of free Lie Algebras and afhne Kac-Moody Lie algebras.     

Dixmier's Problem 6 for the Weyl Algebra (The Generic Type Problem)

ABSTRACT

In Dixmier (1968 Dixmier , J. ( 1968 ). Sur les algèbres de Weyl . Bull. Soc. Math. France 96 : 209 – 242 . [CSA] [Crossref] , [Google Scholar]), the author posed six problems for the Weyl algebra A ₁ over a field K of characteristic zero. Problems 3, 6, and 5 were solved respectively by Joseph (1975 Joseph , A. ( 1975 ). The Weyl algebra—semisimple and nilpotent elements . Amer. J. Math. 97 ( 3 ): 597 – 615 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]) and Bavula (2005a Bavula , V. V. ( 2005a ). Dixmier's Problem 5 for the Weyl algebra . J. Algebra 283 ( 2 ): 604 – 621 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]). Problems 1, 2, and 4 are still open. In this article a short proof is given to Dixmier's problem 6 for the ring of differential operators 𝒟 (X) on a smooth irreducible algebraic curve X. It is proven that, for a given maximal commutative subalgebra C of 𝒟 (X), (almost) all noncentral elements of it have the same type, more precisely, have exactly one of the following types: (i) strongly nilpotent; (ii) weakly nilpotent; (iii) generic; (iv) generic, except for a subset K*a + K of strongly semi-simple elements; (iv) generic, except for a subset K*a + K of weakly semi-simple elements, where K* := K\{0}. The same results are true for other popular algebras.