Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userNoncommutative marked surfaces
164
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The aim of the paper is to attach a noncommutative cluster-like structure to each marked surface $Sigma$. This is a noncommutative algebra ${mathcal A}_Sigma$ generated by noncommutative geodesics between marked points subject to certain triangle relations and noncommutative analogues of Ptolemy-Plucker relations. It turns out that the algebra ${mathcal A}_Sigma$ exhibits a noncommutative Laurent Phenomenon with respect to any triangulation of $Sigma$, which confirms its cluster nature. As a surprising byproduct, we obtain a new topological invariant of $Sigma$, which is a free or a 1-relator group easily computable in terms of any triangulation of $Sigma$. Another application is the proof of Laurentness and positivity of certain discrete noncommutative integrable systems.
rate research
Read More
The goal of this paper is to introduce and study noncommutative Catalan numbers $C_n$ which belong to the free Laurent polynomial algebra in $n$ generators. Our noncommutative numbers admit interesting (commutative and noncommutative) specializations, one of them related to Garsia-Haiman $(q,t
In this paper, we prove that a non-semisimple Hopf algebra H of dimension 4p with p an odd prime over an algebraically closed field of characteristic zero is pointed provided H contains more than two group-like elements. In particular, we prove that non-semisimple Hopf algebras of dimensions 20, 28 and 44 are pointed or their duals are pointed, and this completes the classification of Hopf algebras in these dimensions.
In this paper we introduce a trace-like invariant for the irreducible representations of a finite dimensional complex Hopf algebra H. We do so by considering the trace of the map induced by the antipode S on the endomorphisms End(V) of a self-dual module V. We also compute the values of this trace for the representations of two non-semisimple Hopf algebras: u_q(sl_2) and D(H_n(q)), the Drinfeld double of the Taft algebra.
We derive a formula for the trace of the antipode on endomorphism algebras of simple self-dual modules of nilpotent liftings of quantum planes. We show that the trace is equal to the quantum dimension of the module up to a nonzero scalar depending on the simple module.
We introduce a new filtration on Hopf algebras, the standard filtration, generalizing the coradical filtration. Its zeroth term, called the Hopf coradical, is the subalgebra generated by the coradical. We give a structure theorem: any Hopf algebra with injective antipode is a deformation of the bosonization of the Hopf coradical by its diagram, a connected graded Hopf algebra in the category of Yetter-Drinfeld modules over the latter. We discuss the steps needed to classify Hopf algebras in suitable classes accordingly. For the class of co-Frobenius Hopf algebras, we prove that a Hopf algebra is co-Frobenius if and only if its Hopf coradical is so and the diagram is finite dimensional. We also prove that the standard filtration of such Hopf algebras is finite. Finally, we show that extensions of co-Frobenius (resp. cosemisimple) Hopf algebras are co-Frobenius (resp. cosemisimple).
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
