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The Lattice Structure of Connection Preserving Deformations for q-Painlev\'e Equations I
We wish to explore a link between the Lax integrability of the -Painlev\'e
equations and the symmetries of the -Painlev\'e equations. We shall
demonstrate that the connection preserving deformations that give rise to the
-Painlev\'e equations may be thought of as elements of the groups of
Schlesinger transformations of their associated linear problems. These groups
admit a very natural lattice structure. Each Schlesinger transformation induces
a B\"acklund transformation of the -Painlev\'e equation. Each translational
B\"acklund transformation may be lifted to the level of the associated linear
problem, effectively showing that each translational B\"acklund transformation
admits a Lax pair. We will demonstrate this framework for the -Painlev\'e
equations up to and including -
Representations and cohomology for Frobenius-Lusztig kernels
Let be the quantum group (Lusztig form) associated to the simple
Lie algebra , with parameter specialized to an -th
root of unity in a field of characteristic . In this paper we study
certain finite-dimensional normal Hopf subalgebras of ,
called Frobenius-Lusztig kernels, which generalize the Frobenius kernels
of an algebraic group . When , the algebras studied here reduce to the
small quantum group introduced by Lusztig. We classify the irreducible
-modules and discuss their characters. We then study the
cohomology rings for the Frobenius-Lusztig kernels and for certain nilpotent
and Borel subalgebras corresponding to unipotent and Borel subgroups of . We
prove that the cohomology ring for the first Frobenius-Lusztig kernel is
finitely-generated when \g has type or , and that the cohomology rings
for the nilpotent and Borel subalgebras are finitely-generated in general.Comment: 26 pages. Incorrect references fixe
Corrigendum to "On injective modules and support varieties for the small quantum group"
The proof of Theorem 5.12 in [C.M. Drupieski, On injective modules and
support varieties for the small quantum group, Int. Math. Res. Not. 2011
(2011), 2263-2294] does not make sense as written because the algebra
need not be a Hopf subalgebra of
unless is a simple root. This note describes
how the proof should be modified to work around this fact.Comment: 2 pages; Corrigendum to [C.M. Drupieski, On injective modules and
support varieties for the small quantum group, Int. Math. Res. Not. 2011
(2011), 2263-2294
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