Exponent matrices appeared in the study of tiled orders over discrete valuation rings. Many properties of such orders are formulated using this notion. We think that such matrices are of interest in them own right, in particular, it is convenient to write finite partially ordered sets (posets) and finite metric spaces as special exponent matrices. Note that when we defined a quiver Q(E) of a re...

Let (A, G) be a C*-dynamical system with G discrete. In this paper we investigate the ideal structure of the reduced crossed product C*-algebra and in particular we determine sufficient – and in some cases also necessary – conditions for A to separate the ideals in A⋊r G. When A separates the ideals in A ⋊r G, then there is a one-to-one correspondence between the ideals in A⋊r G and the invaria...

We describe obstructions to a direct limit preserving right exact functor between categories of quasi-coherent sheaves on schemes being isomorphic to tensoring with a bimodule. When the domain scheme is affine, all obstructions vanish and we recover the Eilenberg-Watts Theorem. We use our description of these obstructions to prove that if a direct limit preserving right exact functor F from a s...

A triangular matrix algebra over a field k is defined by a triplet (R, S, M) where R and S are k-algebras and RMS is an SR-bimodule. We show that if R, S and M are finite dimensional and the global dimensions of R and S are finite, then the triangular matrix algebra corresponding to (R, S, M) is derived equivalent to the one corresponding to (S, R, DM), where DM = Homk(M, k) is the dual of M , ...

Homotopy Quantum Field Theories (HQFTs) were introduced by the second author to extend the ideas and methods of Topological Quantum Field Theories to closed d-manifolds endowed with extra structure in the form of homotopy classes of maps into a given ‘target’ space, B. For d = 1, classifications of HQFTs in terms of algebraic structures are known when B is a K(G, 1) and also when it is simply c...

In this article, we will prove that the subsectors of α-induced sectors for M ⋊ Ĝ ⊃ M forms a modular category, where M ⋊ Ĝ is the crossed product of M by the group dual Ĝ of a finite group G. In fact, we will prove that it is equivalent to Müger’s crossed product. By using this identification, we will exhibit an orbifold aspect of the quantum double of ∆(not necessarily non-degenerate) obtaine...

Homotopy Quantum Field Theories (HQFTs) were introduced by the second author to extend the ideas and methods of Topological Quantum Field Theories to closed d-manifolds endowed with extra structure in the form of homotopy classes of maps into a given 'target' space, B. For d = 1, classifications of HQFTs in terms of algebraic structures are known when B is a K(G, 1) and also when it is simply c...

The Gasperini syndrome is a very rare brainstem disease characterized by the typical combination of ipsilateral lesions of the cranial nerves V-VII and dissociated contralateral hemihypesthesia, whereas both contralateral and ipsilateral hypacusis was described. Since the first description in 1912, only a few cases of this crossed brainstem syndrome were published so far. Pontine infarction and...

Beliakova-Putyra-Wehrli studied various kinds of traces, in relation to annular Khovanov homology. In particular, a graded algebra and bimodule over it, they associate quantum Hochschild homology the with coefficients bimodule, use this obtain deformation link. A spectral refinement resulting invariant was recently given by Akhmechet-Krushkal-Willis. short note we observe that is composition tw...

We prove the extended Hilbert’s Nullstellensatz in the context of Hu-Liu polynomial trirings. Which kinds of noncommutative rings are suitable for extending algebra geometry? Different attempts have been made to answer this question, but a satisfactory answer is still in hiding. My attempt at answering this question comes from the trivial extension of a ring by a bimodule over the ring. The tri...