We show that the category of representations of the Euclidean group of orientation-preserving isometries of two-dimensional Euclidean space is equivalent to the category of representations of the preprojective algebra of type A∞. We also consider the moduli space of representations of the Euclidean group along with a set of generators. We show that these moduli spaces are quiver varieties of th...

LetG be a finite p-group,K a field of characteristic p, and J the radical of the group algebra K[G]. We study modular representations using some new results of the theory of extensions of modules. More precisely, we describe the K[G]-modules M such that J2M = 0 and give some properties and isomorphism invariants which allow us to compute the number of isomorphism classes of K[G]-modulesM such t...

During the 2004-2005 academic year the VIGRE algebra research group at the University of Georgia computed the complexities of certain Specht modules S for the symmetric group Σd, using the computer algebra program Magma. The complexity of an indecomposable module does not exceed the p-rank of the defect group of its block. The Georgia group conjectured that, generically, the complexity of a Spe...

The principle “Every result in classical homological algebra should have a counterpart in Gorenstein homological algebra” was given by Henrik Holm. There is a remarkable body of evidence supporting this claim. Perhaps one of the most glaring exceptions is provided by the fact that tensor products of Gorenstein projective modules need not be Gorenstein projective, even over Gorenstein rings. So ...

Let G be an adjoint simple algebraic group of inner type. We express the Chow motive (with integral coefficients) of some anisotropic projective G-homogeneous varieties in terms of motives of simpler G-homogeneous varieties, namely, those that correspond to maximal parabolic subgroups of G. We decompose the motive of a generalized Severi-Brauer variety SB2(A), where A is a division algebra of d...

Let (G, S) be a finitely generated Coxeter group, such that the Coxeter system is indecomposable and the canonical bilinear form is indefinite but non-degenerate. We show that the reduced C∗-algebra of G is simple with unique normalised trace. For an arbitrary finitely generated Coxeter group we prove the validity of a Haagerup inequality: There exist constants C > 0 and Λ ∈ N such that for a f...

Let A and B be two Artin algebras with no semisimple summands. Suppose that there is a stable equivalence α between A and B such that α is induced by exact functors. We present a nice correspondence between indecomposable modules over A and B. As a consequence, we have the following: (1) If A is a self-injective algebra, then so is B; (2) If A and B are finite dimensional algebras over an algeb...

We compare by a very elementary approach the second adjoint and trivial Leibniz cohomology spaces of a Lie algebra to the usual ones. Examples are given of coupled cocycles. Some properties are deduced as to Leibniz deformations. We also consider the class of Lie algebras for which the Koszul 3-form is zero, and prove that it contains all quotients of Borel subalgebras, or of their nilradicals,...

Using a result of T. P. Bidigare [Bidigare, 1997], we identity the descent algebra Σk(W ) (over a field k) of a finite Coxeter group W with a subalgebra of kF , an algebra built from the hyperplane arrangement associated to W . Specifically, Σk(W ) is anti-isomorphic to the W -invariant subalgebra (kF) . We use this identification and results about kF to study Σk(W ). We construct a complete sy...

A hica is a highest weight, homogeneous, indecomposable, Calabi-Yau category of dimension 0. A hica has length l if its objects have Loewy length l and smaller. We classify hicas of length ≤ 4, up to equivalence, and study their properties. Over a fixed field F , we prove that hicas of length 4 are in one-one correspondence with bipartite graphs. We prove that an algebra AΓ controlling the hica...