33 research outputs found
The cohomology ring of the 12-dimensional Fomin-Kirillov algebra
The -dimensional Fomin-Kirillov algebra is defined as the
quadratic algebra with generators , and which satisfy the relations
and . By a result of A. Milinski and H.-J.
Schneider, this algebra is isomorphic to the Nichols algebra associated to the
Yetter-Drinfeld module , over the symmetric group ,
corresponding to the conjugacy class of all transpositions and the sign
representation. Exploiting this identification, we compute the cohomology ring
, showing that it is a polynomial ring with
coefficients in the symmetric braided algebra of . As an application we also
compute the cohomology rings of the bosonization and
of its dual, which are -dimensional ordinary Hopf algebras.Comment: v3: Final version, accepted for publication in Advances in
Mathematic
Verma and simple modules for quantum groups at non-abelian groups
The Drinfeld double D of the bosonization of a finite-dimensional Nichols
algebra B(V) over a finite non-abelian group G is called a quantum group at a
non-abelian group. We introduce Verma modules over such a quantum group D and
prove that a Verma module has simple head and simple socle. This provides two
bijective correspondences between the set of simple modules over D and the set
of simple modules over the Drinfeld double D(G). As an example, we describe the
lattice of submodules of the Verma modules over the quantum group at the
symmetric group S3 attached to the 12-dimensional Fomin-Kirillov algebra,
computing all the simple modules and calculating their dimensions.Comment: 29 pages, 4 figures v2: final version. Main changes: Theorem 5 is new
and Sections 4.3, 4.4, 4.5 and 4.5 were improve
Representations of copointed Hopf algebras arising from the tetrahedron rack
We study the copointed Hopf algebras attached to the Nichols algebra of the
affine rack \Aff(\F_4,\omega), also known as tetrahedron rack, and the
2-cocycle -1. We investigate the so-called Verma modules and classify all the
simple modules. We conclude that these algebras are of wild representation type
and not quasitriangular, also we analyze when these are spherical
On the representation theory of the Drinfeld Double of the Fomin-Kirillov Algebra FK 3
Let D be the Drinfeld double of FK3#S3 . We have described the simple D-modules in Pogorelsky and Vay (Adv. Math. 301, 423-457, 2016). In the present work, we describe the indecomposable summands of the tensor products between them. We classify the extensions of the simple modules and show that D is of wild representation type. We also investigate the projective modules and their tensor products.Fil: Pogorelsky, Barbara. Universidade Federal do Rio Grande do Sul; BrasilFil: Vay, Cristian Damian. Universidad Nacional de Córdoba; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentin
On a family of Hopf algebras of dimension 72
We investigate a family of Hopf algebras of dimension 72 whose coradical is isomorphic to the algebra of functions on S3. We determine the lattice of submodules of the so-called Verma modules and as a consequence we classify all simple modules. We show that these Hopf algebras are unimodular (as well as their duals) but not quasitriangular; also, they are cocycle deformations of each other.Fil: Andruskiewitsch, Nicolas. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; ArgentinaFil: Vay, Cristian Damian. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentin