Left-right noncommutative Poisson algebras

The notions of left-right noncommutative Poisson algebra (NP^lr-algebra) and left-right algebra with bracket AWB^lr are introduced. These algebras are special cases of NLP-algebras and algebras with bracket AWB, respectively, studied earlier. An NP^lr-algebra is a noncommutative analogue of the classical Poisson algebra. Properties of these new algebras are studied. In the categories AWB^lr and NP^lr-algebras the notions of actions, representations, centers, actors and crossed modules are described as special cases of the corresponding wellknown notions in categories of groups with operations. The cohomologies of NP^lr-algebras and AWB^lr (resp. of NP^r-algebras and AWB^r) are defined and the relations between them and the Hochschild, Quillen and Leibniz cohomologies are detected. The cases P is a free AWB^r, the Hochschild or/and Leibniz cohomological dimension of P is ≤ n are considered separately, exhibiting interesting possibilities of representations of the new cohomologies by the well-known ones and relations between the corresponding cohomological dimensions.     

Derivations of noncommutative Arens algebras

The present paper deals with derivations of noncommutative Arens algebras. We prove that every derivation of an Arens algebra associated with a von Neumann algebra and a faithful normal finite trace is inner. In particular, each derivation on such algebras is automatically continuous in the natural topology, and in the commutative case, even for semi-finite traces, all derivations are identically zero. At the same time, the existence of noninner derivations is proved for noncommutative Arens algebras with a semi-finite but nonfinite trace.     

Binomial free resolutions for normal toric surfaces

We construct the minimal free resolution of the residue field over a normal toric surface.

     

Derivations of noncommutative Banach algebras II

Supported in part by a grant from the Ministry of Science of Slovenia.     

On triangular algebras with noncommutative diagonals

We construct a triangular algebra whose diagonals form a noncommutative algebra and its lattice of invariant projections contains only two nontrivial projections. Moreover we prove that our triangular algebra is maximal.     

Cellular algebras arising from Hecke algebras of type H_n

We study a finite-dimensional quotient of the Hecke algebra of type for general n, using a calculus of diagrams. This provides a basis of monomials in a certain set of generators. Using this, we prove a conjecture of C.K. Fan about the semisimplicity of the quotient algebra. We also discuss the cellular structure of the algebra, with certain restrictions on the ground ring. Received February 24, 1997; in final form May 9, 1997     

Cellular resolutions of cointerval ideals

Minimal cellular resolutions of the edge ideals of cointerval hypergraphs are constructed. This class of d-uniform hypergraphs coincides with the complements of interval graphs (for the case d?=?2), and strictly contains the class of ‘strongly stable’ hypergraphs corresponding to pure shifted simplicial complexes. The polyhedral complexes supporting the resolutions are described as certain spaces of directed graph homomorphisms, and are realized as subcomplexes of mixed subdivisions of the Minkowski sums of simplices. Resolutions of more general hypergraphs are obtained by considering decompositions into cointerval hypergraphs.