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For finite groups , and the right -action on by group automorphisms, the non-balanced quantum double is defined as the crossed product . We firstly prove that is a finite-dimensional Hopf -algebra. For any subgroup of , can be defined as a Hopf -subalgebra of in the natural way. Then there is a conditonal expectation from onto and the index is . Moreover, we prove that an associated natural inclusion of non-balanced quantum doubles is the crossed product by the group algebra....
By analogy with the projective, injective and flat modules, in this paper we study some properties of -Gorenstein projective, injective and flat modules and discuss some connections between -Gorenstein injective and -Gorenstein flat modules. We also investigate some connections between -Gorenstein projective, injective and flat modules of change of rings.
We describe the stable module categories of the self-injective finite-dimensional algebras of finite representation type over an algebraically closed field which are Calabi-Yau (in the sense of Kontsevich).
For a finite Coxeter group and a Coxeter element of ; the -Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of . Its maximal cones are naturally
indexed by the -sortable elements of . The main result of this paper is that the known bijection cl between -sortable elements and -clusters induces a combinatorial isomorphism of fans. In particular, the -Cambrian fan is combinatorially isomorphic to the normal fan of the generalized
associahedron for . The rays...
A multiplicative functional on a graded connected Hopf algebra is called the character. Every character decomposes uniquely as a product of an even character and an odd character. We apply the character theory of combinatorial Hopf algebras to the Hopf algebra of simple graphs. We derive explicit formulas for the canonical characters on simple graphs in terms of coefficients of the chromatic symmetric function of a graph and of canonical characters on quasi-symmetric functions. These formulas and...
The Cartan matrix of a finite dimensional algebra A is an important combinatorial invariant reflecting frequently structural properties of the algebra and its module category. For example, one of the important features of the modular representation theory of finite groups is the nonsingularity of Cartan matrices of the associated group algebras (Brauer’s theorem). Recently, the class of all tame selfinjective algebras having simply connected Galois coverings and the stable Auslander-Reiten quiver...
Let G be a group, R a G-graded ring and X a right G-set. We study functors between categories of modules graded by G-sets, continuing the work of [M]. As an application we obtain generalizations of Cohen-Montgomery Duality Theorems by categorical methods. Then we study when some functors introduced in [M] (which generalize some functors ocurring in [D1], [D2] and [NRV]) are separable. Finally we obtain an application to the study of the weak dimension of a group graded ring.
We exhibit a monoidal structure on the category of finite sets indexed by P-trees for a finitary polynomial endofunctor P. This structure categorifies the monoid scheme (over Spec ℕ) whose semiring of functions is (a P-version of) the Connes-Kreimer bialgebra H of rooted trees (a Hopf algebra after base change to ℤ and collapsing H 0). The monoidal structure is itself given by a polynomial functor, represented by three easily described set maps; we show that these maps are the same as those occurring...
We develop a diagrammatic categorification of the polynomial ring ℤ[x]. Our categorification satisfies a version of Bernstein-Gelfand-Gelfand reciprocity property with the indecomposable projective modules corresponding to xⁿ and standard modules to (x-1)ⁿ in the Grothendieck ring.
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