Algebras and Representation Theory 

About: Algebras and Representation Theory is an academic journal published by Springer Science+Business Media. The journal publishes majorly in the area(s): Hopf algebra & Lie algebra. It has an ISSN identifier of 1386-923X. Over the lifetime, 1295 publications have been published receiving 13468 citations.

Nonstable K -theory for Graph Algebras

Abstract: We compute the monoid V(L K (E)) of isomorphism classes of finitely generated projective modules over certain graph algebras L K (E), and we show that this monoid satisfies the refinement property and separative cancellation. We also show that there is a natural isomorphism between the lattice of graded ideals of L K (E) and the lattice of order-ideals of V(L K (E)). When K is the field $\mathbb C$ of complex numbers, the algebra $L_{\mathbb C}(E)$ is a dense subalgebra of the graph C *-algebra C *(E), and we show that the inclusion map induces an isomorphism between the corresponding monoids. As a consequence, the graph C*-algebra of any row-finite graph turns out to satisfy the stable weak cancellation property.

Representations of Hom-Lie Algebras

Abstract: In this paper, we study representations of hom-Lie algebras. In particular, the adjoint representation and the trivial representation of hom-Lie algebras are studied in detail. Derivations, deformations, central extensions and derivation extensions of hom-Lie algebras are also studied as an application.

Periodic Algebras which are Almost Koszul

Abstract: The preprojective algebra and the trivial extension algebra of a Dynkin quiver (in bipartite orientation) are very close to being a Koszul dual pair of algebras. In this case the usual duality theory may be adapted to show that each algebra has a periodic bimodule resolution built using the other algebra and some extra data: an algebra automorphism. A general theory of such ‘almost Koszul’ algebras is developed and other examples are given.

The Structure of Corings: Induction Functors, Maschke-Type Theorem, and Frobenius and Galois-Type Properties

Abstract: Given a ring A and an A-coring C, we study when the forgetful functor from the category of right C-comodules to the category of right A-modules and its right adjoint −⊗ A C are separable. We then proceed to study when the induction functor −⊗ A C is also the left adjoint of the forgetful functor. This question is closely related to the problem when A→ A Hom(C,A) is a Frobenius extension. We introduce the notion of a Galois coring and analyse when the tensor functor over the subring of A fixed under the coaction of C is an equivalence. We also comment on possible dualisation of the notion of a coring.

Quivers with Relations and Cluster Tilted Algebras

Abstract: Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. To a cluster algebra of simply laced Dynkin type one can associate the cluster category. Any cluster of the cluster algebra corresponds to a tilting object in the cluster category. The cluster tilted algebra is the algebra of endomorphisms of that tilting object. Viewing the cluster tilted algebra as a path algebra of a quiver with relations, we prove in this paper that the quiver of the cluster tilted algebra is equal to the cluster diagram. We study also the relations. As an application of these results, we answer several conjectures on the connection between cluster algebras and quiver representations.