3,184 research outputs found
Regions of linearity, Lusztig cones and canonical basis elements for the quantized enveloping algebra of type A_4
Let U_q be the quantum group associated to a Lie algebra g of rank n. The
negative part U^- of U has a canonical basis B with favourable properties,
introduced by Kashiwara and Lusztig. The approaches of Kashiwara and Lusztig
lead to a set of alternative parametrizations of the canonical basis, one for
each reduced expression for the longest word in the Weyl group of g. We show
that if g is of type A_4 there are close relationships between the Lusztig
cones, canonical basis elements and the regions of linearity of
reparametrization functions arising from the above parametrizations. A graph
can be defined on the set of simplicial regions of linearity with respect to
adjacency, and we further show that this graph is isomorphic to the graph with
vertices given by the reduced expressions of the longest word of the Weyl group
modulo commutation and edges given by long braid relations.
Keywords: Quantum group, Lie algebra, Canonical basis, Tight monomials, Weyl
group, Piecewise-linear functions.Comment: 61 pages, 17 figures, uses picte
A geometric description of the m-cluster categories of type D_n
We show that the m-cluster category of type D_n is equivalent to a certain
geometrically-defined category of arcs in a punctured regular nm-m+1-gon. This
generalises a result of Schiffler for m=1. We use the notion of the mth power
of a translation quiver to realise the m-cluster category in terms of the
cluster category.Comment: 14 pages, 11 figure
Coloured quivers for rigid objects and partial triangulations: The unpunctured case
We associate a coloured quiver to a rigid object in a Hom-finite
2-Calabi--Yau triangulated category and to a partial triangulation on a marked
(unpunctured) Riemann surface. We show that, in the case where the category is
the generalised cluster category associated to a surface, the coloured quivers
coincide. We also show that compatible notions of mutation can be defined and
give an explicit description in the case of a disk. A partial description is
given in the general 2-Calabi-Yau case. We show further that Iyama-Yoshino
reduction can be interpreted as cutting along an arc in the surface.Comment: 29 pages, 17 figures. Discussion in Section 6 clarified and expanded.
Some minor corrections, clarification of notatio
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