Related papers: Some remarks on Nichols algebras

An equilevel algebra is a subalgebra of the space of smooth functions $f: M \to {\mathbb R}$ distinguished in this space by finitely many linear conditions of the type $f(x_i) = f(\tilde x_i)$, $x_i \neq \tilde x_i \in M$, or approximated…

All finite dimensional Nichols algebras with diagonal type of connected finite dimensional Yetter-Drinfeld modules over finite cyclic group $\mathbb Z_n$ are found. It is proved that finite dimensional Nichols algebra over $\mathbb Z_2$ is…

Yangian-like algebras, associated with current R-matrices, different from the Yang ones, are introduced. These algebras are of two types. The so-called braided Yangians are close to the Reflection Equation algebras, arising from involutive…

We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, $\operatorname{GKdim}$ for short, through the study of Nichols algebras over abelian groups. We deal first with braided vector spaces over $\mathbb…

The classification of Nichols algebras is an essential step in the classification theory of pointed Hopf algebras by lifting method of N. Andruskiewitsch and H.-J. Schneider. Arithmetic root systems are invariants of Nichols algebras of…

Algebraic relations that characterize quantum statistics (Bose-Einstein statistic, Fermi-Dirac statistic, supersymmetry, parastatistic, anyonic statistic, ...) are reformulated herein in terms of a new algebraic structure, which we call…

The problem of the classification of the extensions of the Virasoro algebra is discussed. It is shown that all $H$-reduced $\hat{\cal G}_{r}$-current algebras belong to one of the following basic algebraic structures: local quadratic…

The quiver Hopf algebras are classified by means of ramification systems with irreducible representations. This leads to the classification of Nichols algebras over group algebras and pointed Hopf algebras of type one.

We consider an interesting class of braidings defined by a combinatorial property in an earlier paper. We show that it consists exactly of those braidings that come from certain Yetter-Drinfeld module structures over pointed Hopf algebras…

We introduce the notion of a braided Lie algebra consisting of a finite-dimensional vector space $\CL$ equipped with a bracket $[\ ,\ ]:\CL\tens\CL\to \CL$ and a Yang-Baxter operator $\Psi:\CL\tens\CL\to \CL\tens\CL$ obeying some axioms. We…

This is an invited contribution to the 2nd edition of the Encyclopedia of Mathematical Physics. We review the following algebraic structures which appear in two-dimensional conformal field theory (CFT): The symmetries of two-dimensional…

The boson-fermion correspondences are an important phenomena on the intersection of several areas in mathematical physics: representation theory, vertex algebras and conformal field theory, integrable systems, number theory, cohomology. Two…

There is a very general picture emerging that conjecturally describes what happens to the representation theory of a vertex algebra $\mathcal{V}$ if we pass to the kernel $\mathcal{W}$ of a set of screening operators. Namely, the screening…

A vertex algebra is an algebraic counterpart of a two-dimensional conformal field theory. We give a new definition of a vertex algebra which includes chiral algebras as a special case, but allows for fields which are neither meromorphic nor…

In this paper we push forward results on the invariant ${\cal F}$-module of a virtual knot investigated by the first named author where ${\cal F}$ is the algebra with two invertible generators $A,B$ and one relation…

This is a survey of general aspects of the theory of braided Hopf algebras with emphasis on a special class of braided graded Hopf algebras named tobas. The interest on tobas arises from problems of classification of pointed Hopf algebras.…

The interaction of various algebraic structures describing fusion, braiding and group symmetries in quantum projective field theory is an object of an investigation in the paper. Structures of projective Zamolodchikov al- gebras, their…

In GT/0006019 oriented quantum algebras were motivated and introduced in a natural categorical setting. Invariants of knots and links can be computed from oriented quantum algebras, and this includes the Reshetikhin-Turaev theory for Ribbon…

The Cayley-Hamilton-Newton theorem - which underlies the Newton identities and the Cayley-Hamilton identity - is reviewed, first, for the classical matrices with commuting entries, second, for two q-matrix algebras, the RTT-algebra and the…

It is an important open problem whether the dimension of the Nichols algebra B(O,\rho) is finite when O is the class of the transpositions and \rho is the sign representation, with m>= 6. In the present paper, we discard most of the other…