Let $\mathfrak{g}$ be a complex semisimple Lie algebra, $\mathfrak{b}$ a Borel subalgebra, and $\mathfrak{h}\subset\mathfrak{b}$ a Cartan subalgebra. Let V be a finite dimensional simple $U(\mathfrak{g})$ module. Based on a principal s-triple (e,h,f) and following work of Kostant, Brylinski (J Amer Math Soc 2(3):517–533, 1989) defined a filtration $\mathcal{F}_e$ for all weight subspaces V

_μ
of V and calculated the dimensions of the graded subspaces for μ dominant. In Joseph et al. (J Amer Math Soc 13(4):945–970, 2000) these dimensions were calculated for all μ. Let δM(0) be the finite dual of the Verma module of highest weight 0. It identifies with the global functions on a Weyl group translate of the open Bruhat cell and so inherits a natural degree filtration. On the other hand, up to an appropriate shift of weights, there is a unique $U(\mathfrak{b})$ module embedding of V into δM(0) and so the degree filtration induces a further filtration $\mathcal{F}$ on each weight subspace V

_μ
. A casual reading of Joseph et al. (J Amer Math Soc 13(4):945–970, 2000) might lead one to believe that $\mathcal{F}$ and $\mathcal{F}_e$ coincide. However this is quite false. Rather one should view $\mathcal{F}_e$ as coming from a left action of $U(\mathfrak{n})$ and then there is a second (Brylinski-Kostant) filtration $\mathcal{F}’_e$ coming from a right action. It is $\mathcal{F}’_e$ which may coincide with $\mathcal{F}$. In this paper the above claim is made precise. First it is noted that $\mathcal{F}$ is itself not canonical, but depends on a choice of variables. Then it is shown that a particular choice can be made to ensure that $\mathcal{F}=\mathcal{F}’_e$. An explicit form for the unique left $U(\mathfrak{b})$ module embedding $V\hookrightarrow\delta M(0)$ is given using the right action of $U(\mathfrak{n})$. This is used to give a purely algebraic proof of Brylinski’s main result in Brylinski (J Amer Math Soc 2(3):517–533, 1989) which is much simpler than Joseph et al. (J Amer Math Soc 13(4):945–970, 2000). It is noted that the dimensions of the graded subspaces of $\rm{gr}_{\mathcal{F}_e} V_{\!\mu}$ and $\rm{gr}_{\mathcal{F}’_e} V_{\!\mu}$ can be very different. Their interrelation may involve the Kashiwara involution. Indeed a combinatorial formula for multiplicities in tensor products involving crystal bases and the Kashiwara involution is recovered. Though the dimensions for the graded subspaces of $\rm{gr}_{\mathcal{F}’_e} V_{\!\mu}$ are determined by polynomial degree, their values remain unknown.

DOI: 10.1007/s10468-009-9159-0
Online Date: 3/27/2009
Print publication date: 10/1/2009
View article on SpringerLink

It is well known that the torsion part of any finitely generated module over the formal power series ring K[[X]] is a direct summand. In fact, K[[X]] is an algebra dual to the divided power coalgebra over K and the torsion part of any K[[X]]-module actually identifies with the rational part of that module. More generally, for a certain general enough class of coalgebras—those having only finite dimensional subcomodules—we see that the above phenomenon is preserved: the set of torsion elements of any C
^*-module is exactly the rational submodule. With this starting point in mind, given a coalgebra C we investigate when the rational submodule of any finitely generated left C
^*-module is a direct summand. We prove various properties of coalgebras C having this splitting property. Just like in the K[[X]] case, we see that standard examples of coalgebras with this property are the chain coalgebras which are coalgebras whose lattice of left (or equivalently, right, two-sided) coideals form a chain. We give some representation theoretic characterizations of chain coalgebras, which turn out to make a left-right symmetric concept. In fact, in the main result of this paper we characterize the colocal coalgebras where this splitting property holds non-trivially (i.e. infinite dimensional coalgebras) as being exactly the chain coalgebras. This characterizes the cocommutative coalgebras of this kind. Furthermore, we give characterizations of chain coalgebras in particular cases and construct various and general classes of examples of coalgebras with this splitting property.

DOI: 10.1007/s10468-009-9144-7
Online Date: 3/26/2009
Print publication date: 10/1/2009
View article on SpringerLink

In a recent paper (Böhm and Stefan, Commun Math Phys 282:239–286, 2008), we gave a general construction of a para-cocyclic structure on a cosimplex, associated to a so called admissible septuple—consisting of two categories, three functors and two natural transformations, subject to compatibility relations. The main examples of such admissible septuples were induced by algebra homomorphisms. In this note we provide more general examples coming from appropriate (‘locally braided’) morphisms of monads.

DOI: 10.1007/s10468-009-9160-7
Online Date: 3/26/2009
Print publication date: 10/1/2009
View article on SpringerLink

We introduce the notion of ‘bar category’ by which we mean a monoidal category equipped with additional structure formalising the notion of complex conjugation. Examples of our theory include bimodules over a *-algebra, modules over a conventional *-Hopf algebra and modules over a more general object which we call a ‘quasi-*-Hopf algebra’ and for which examples include the standard quantum groups $u_q(\mathfrak{g})$ at q a root of unity (these are well-known not to be usual *-Hopf algebras). We also provide examples of strictly quasiassociative bar categories, including modules over ‘*-quasiHopf algebras’ and a construction based on finite subgroups H ⊂ G of a finite group. Inside a bar category one has natural notions of ‘⋆-algebra’ and ‘unitary object’ therefore extending these concepts to a variety of new situations. We study braidings and duals in bar categories and ⋆-braided groups (Hopf algebras) in braided-bar categories. Examples include the transmutation B(H) of a quasitriangular *-Hopf algebra and the quantum plane ${\mathbb C}_q^2$ at certain roots of unity q in the bar category of $\widetilde{u_q(su_2)}$-modules. We use our methods to provide a natural quasi-associative C
^*-algebra structure on the octonions ${\mathbb O}$ and on a coset example. In the Appendix we extend the Tannaka-Krein reconstruction theory to bar categories in relation to *-Hopf algebras.

DOI: 10.1007/s10468-009-9141-x
Online Date: 3/24/2009
Print publication date: 10/1/2009
View article on SpringerLink

DOI: 10.1007/s10468-009-9162-5
Online Date: 3/24/2009
Print publication date: 10/1/2009
View article on SpringerLink

In this paper we introduce the notions of connected, 0-connected and strictly graded coalgebra in the framework of an abelian monoidal category $ \mathcal{M} $ and we investigate the relations between these concepts. We recover several results, involving these notions, which are well known in the case when $ \mathcal{M} $ is the category of vector spaces over a field K. In particular we characterize when a 0-connected graded bialgebra is a bialgebra of type one.

DOI: 10.1007/s10468-009-9147-4
Online Date: 3/21/2009
Print publication date: 10/1/2009
View article on SpringerLink

At the beginning of this note the $\mathcal{G}$-covers, $\mathcal{G}$ being a hereditary class of modules, are characterized as that for which the homomorphisms into $\mathcal{G}$-precovers are injective as well as that for which the homomorphisms from $\mathcal{G}$-precovers are surjective. The next part studies the (pre)covers of (relatively) injective modules and some relations between the (relative) injectivity of modules and their (pre)covers.

DOI: 10.1007/s10468-009-9146-5
Online Date: 3/21/2009
Print publication date: 10/1/2009
View article on SpringerLink

A rotation in a Euclidean space V is an orthogonal map δ ∈ O(V) which acts locally as a plane rotation with some fixed angle a(δ) ∈ [0,π]. We give a classification of all finite-dimensional representations of the real algebra $\mathbb{R}\left\langle X,Y\right\rangle$ that are given by rotations, up to orthogonal isomorphism.

DOI: 10.1007/s10468-009-9156-3
Online Date: 3/17/2009
Print publication date: 10/1/2009
View article on SpringerLink

This is a companion note to “Hochschild cohomology and Atiyah classes” by Damien Calaque and the author. Using elementary methods we compute the Kontsevich weight of a wheel with spokes pointing outward. The result is in terms of modified Bernoulli numbers. The same result had been obtained earlier by Torossian (unpublished) and also recently by Thomas Willwacher using more advanced methods.

DOI: 10.1007/s10468-009-9161-6
Online Date: 3/13/2009
Print publication date: 10/1/2009
View article on SpringerLink

Let R be a left Noetherian ring, S a right Noetherian ring and
_R

U a generalized tilting module with S = End(
_R

U). We give some equivalent conditions that the injective dimension of U

_S
is finite implies that of
_R

U is also finite. As an application, under the assumption that the injective dimensions of
_R

U and U

_S
are finite, we construct a hereditary and complete cotorsion theory by some subcategories associated with
_R

U.

DOI: 10.1007/s10468-009-9157-2
Online Date: 3/6/2009
Print publication date: 10/1/2009
View article on SpringerLink