Given a finite dimensional algebra A of finite global dimension, we consider the trivial extension of A by the A − A-bimodule ⊕i≥2 Ext 2 A(DA,A), which we call the higher relation bimodule. We first give a recipe allowing to construct the quiver of this trivial extension in case A is a string algebra and then apply it to prove that, if A is gentle, then the tensor algebra of the higher relation...

It is well-known that if T is a Dm–Dn bimodule map on the m×n complex matrices, then T is a Schur multiplier and ‖T‖cb = ‖T‖. If n = 2 and T is merely assumed to be a right D2-module map, then we show that ‖T‖cb = ‖T‖. However, this property fails if m ≥ 2 and n ≥ 3. For m ≥ 2 and n = 3, 4 or n ≥ m2 we give examples of maps T attaining the supremum C(m,n) = sup{‖T‖cb : T a right Dn-module map o...

Let T,U and V be rings with identity and M be a unitary (T,U)-bimodule, N be a unitary (U, V )bimodule, D be a unitary (T, V )-bimodule . We characterize homomorphisms and isomorphisms of the generalized matrix ring Γ = ( T M D 0 U N 0 0 V )

We compare two different types of mapping class invariants: the Hochschild homology an A∞ bimodule coming from bordered Heegaard Floer and fixed point cohomology. first compute invariants their in genus case. then resulting computations to cohomology make a conjecture that are isomorphic. also discuss construction map potentially giving isomorphism. It comes as open-closed context surface viewe...

For a maximal coaction δ of a discrete group G on a C-algebra A and a normal subgroup N of G, there are at least three natural A ×δ G δ̂| N − A ×δ| G/N imprimitivity bimodules: Mansfield’s bimodule Y G G/N(A); the bimodule assembled by Ng from Green’s A ×δ G δ̂ G ׈̂ δ| G/N − A ×δ G δ̂| N imprimitivity bimodule X N (A ×δ G) and Katayama duality; and the bimodule assembled from X G N (A ×δ G) and th...

We give an algorithmic way to construct a free bimodule resolution of an algebra admitting a Gröbner base. It enables us to compute the Hochschild (co)homology of the algebra. Let A be a finitely generated algebra over a commutative ring K with a (possibly infinite) Gröbner base G on a free algebra F , that is, A is the quotient F/I(G) with the ideal I(G) of F generated by G. Given a Gröbner ba...

Let A and B be C∗-algebras. We consider a noncommutative convexity in Hilbert A -B-bimodules, called A -B-convexity, as a generalization of C∗-convexity in C∗-algebras. We show that if X is a Hilbert A -B-bimodule, then Mn(X ) is a Hilbert Mn(A )-Mn(B)-bimodule and apply it to show that the closed unit ball of every Hilbert A -B-bimodule is A -B-convex. Some properties of this kind of convexity...

A connection-like objects, termed hom-connections are defined in the realm of non-commutative geometry. The definition is based on the use of homomorphisms rather than tensor products. It is shown that hom-connections arise naturally from (strong) connections in non-commutative principal bundles. The induction procedure of hom-connections via a map of differential graded algebras or a different...