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Under semi-weak and weak compatibility conditions of bimodules, we establish necessary and sufficient conditions of Gorenstein-projective modules over rings of Morita contexts with one bimodule homomorphism zero. This extends greatly the results on triangular matrix Artin algebras and on Artin algebras of Morita contexts with two bimodule homomorphisms zero in the literature, where only sufficient conditions are given under a strong assumption of compatibility of bimodules. An application is provided to describe Gorenstein-projective modules over noncommutative tensor products arising from Morita contexts. Our results are proved under a general setting of noetherian rings and modules instead of Artin algebras and modules.

We establish relations between Frobenius parts and between flat-dominant dimensions of algebras linked by Frobenius bimodules. This is motivated by the Nakayama conjecture and an approach of Martinez-Villa to the Auslander-Reiten conjecture on stable equivalences. We show that the Frobenius parts of Frobenius extensions are again Frobenius extensions. Furthermore, let Aand Bbe finite-dimensional algebras over a field k, and let domdim(AX) stand for the dominant dimension of an A-module X. If BMAis a Frobenius bimodule, then domdim(A) ⩽ domdim(BM) and domdim(B) ⩽ domdim(AHomB(M, B)). In particular, if B⊆ Ais a left-split (or right-split) Frobenius extension, then domdim(A) = domdim(B). These results are applied to calculate flat-dominant dimensions of a number of algebras: shew group algebras, stably equivalent algebras, trivial extensions and Markov extensions. We also prove that the universal (quantised) enveloping algebras of semisimple Lie algebras are QF-3 rings in the sense of Morita.

Abstract Suppose k is a field. Let A and B be two finite dimensional k-algebras such that there is a stable equivalence of Morita type between A and B. In this paper, we prove that (1) if A and B are representation-finite then their Auslander algebras are stably equivalent of Morita type; (2) The n-th Hochschild homology groups of A and B are isomorphic for all n≥1. A new proof is also provided for Hochschild cohomology groups of self-injective algebras under a stable equivalence of Morita type.

Double centralizer properties play a central role in many parts of algebraic Lie theory. Soergel's double centralizer theorem relates the principal block of the Bernstein–Gelfand–Gelfand category O of a semisimple complex Lie algebra with the coinvariant algebra (i.e., the cohomology algebra of the corresponding flag manifold). Schur–Weyl duality relates the representation theories of general linear and symmetric groups in defining characteristic, or (via the quantized version) in nondefining characteristic. In this paper we exhibit algebraic structures behind these double centralizer properties. We show that the finite dimensional algebras relevant in this context have dominant dimension at least two with respect to some projective–injective or tilting modules. General arguments which combine methods from ring theory (QF-3 rings and dominant dimension) with tools from representation theory (approximations, tilting modules) then imply the validity of these double centralizer properties as well as new ones. In contrast to the traditional proofs (e.g., by the fundamental theorems of invariant theory) no computations are necessary.

The partition algebra P(q) is a generalization both of the Brauer algebra and the Temperley–Lieb algebra for q-state n-site Potts models, underpining their transfer matrix formulation on the arbitrary transverse lattices. We prove that for arbitrary field k and any element q∈ k the partition algebra P(q) is always cellular in the sense of Graham and Lehrer. Thus the representation theory of P(q) can be determined by applying the developed general representation theory on cellular algebras and symmetric groups. Our result also provides an explicit structure of P(q) for arbitrary field and implies the well-known fact that the Brauer algebra D(q) and the Temperley–Lieb algebra TL(q) are cellular.

The partition algebra P(q) is a generalization both of the Brauer algebra and the Temperley–Lieb algebra for q-state n-site Potts models, underpining their transfer matrix formulation on the arbitrary transverse lattices. We prove that for arbitrary field k and any element q∈ k the partition algebra P(q) is always cellular in the sense of Graham and Lehrer. Thus the representation theory of P(q) can be determined by applying the developed general representation theory on cellular algebras and symmetric groups. Our result also provides an explicit structure of P(q) for arbitrary field and implies the well-known fact that the Brauer algebra D(q) and the Temperley–Lieb algebra TL(q) are cellular.

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Abstract: For a finite dimensional algebra with triangular decomposition, a new kind of modules is defined, the twisting modules. Using the structure of these modules, for some algebras with vanishing radical cube the characteristic tilting module is described and its endomorphism ring is computed. This covers both Temperley-Lieb algebras and q-Schur algebras of finite representation type. Our basic tool is a new symmetry condition, stronger than the symmetry provided by the existence of a triangular decomposition in general.

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