Related papers: The Split-Algebras and Non-compact Hopf Maps

For a noncommutative configuration space whose coordinate algebra is the universal enveloping algebra of a finite dimensional Lie algebra, it is known how to introduce an extension playing the role of the corresponding noncommutative phase…

Let $\mathbb{k}$ be an algebraically closed field. We give a complete classification of non-connected pointed Hopf algebras of dimension $16$ with char$\,\mathbb{k}=2$ that are generated by group-like elements and skew-primitive elements.…

Any finite-dimensional quasitriangular Hopf algebra $H$ can be formally extended to a ribbon Hopf algebra $\tilde H$ of twice the dimension. We investigate this extension and its representations. We show that every indecomposable $H$-module…

A non-commutative, planar, Hopf algebra of rooted trees was proposed in L. Foissy, Bull. Sci. Math. 126 (2002) 193-239. In this paper we propose such a non-commutative Hopf algebra for graphs. In order to define a non-commutative product we…

In this paper, we construct a bialgebraic and further a Hopf algebraic structure on top of subgraphs of a given graph. Further, we give the dual structure of this Hopf algebraic structure. We study the algebra morphisms induced by graph…

The aim of this paper is to show that there is a Hopf structure of the parabosonic and parafermionic algebras and this Hopf structure can generate the well known Hopf algebraic structure of the Lie algebras, through a realization of Lie…

For a given group $G$, we construct an invariant of flat $G$-connections on 4-manifolds from a finite type involutory quasitriangular Hopf $G$-algebra. Hopf $G$-algebras are generalizations of Hopf algebras, equipped with gradings by $G$.…

The zx-calculus and related theories are based on so-called interacting Frobenius algebras, where a pair of dagger-special commutative Frobenius algebras jointly form a pair of Hopf algebras. In this setting we introduce a generalisation of…

We here give polynomial realizations of various Hopf algebras or bialgebras on Feynman graphs, graphs, posets or quasi-posets, that it to say injections of these objects into polynomial algebras generated by an alphabet. The alphabet here…

We introduce a new Hopf algebra that operates on pairs of finite interval partitions and permutations of equal length. This algebra captures vincular patterns, which involve specifying both the permutation patterns and the consecutive…

The aim of this paper is to study quasitriangular structures on a class of semisimple Hopf algebras constructed through abelian extensions of $Z_2$ for an abelian group $G$. We prove that there are only two forms of them. Using such…

We introduce a natural Hopf algebra structure on the space of noncommutative symmetric functions which was recently studied as a vector space by Rosas and Sagan. The bases for this algebra are indexed by set partitions. We show that there…

In this paper we describe the right-sided combinatorial Hopf structure of three Hopf algebras appearing in the context of renormalization in quantum field theory: the non-commutative version of the Fa\`a di Bruno Hopf algebra, the…

The Kuperberg invariant is a topological invariant of closed 3-manifolds based on finite-dimensional Hopf algebras. In this paper, we initiate the program of constructing 4-manifold invariants in the spirit of Kuperberg's 3-manifold…

The representations of some Hopf algebras have curious behavior: Nonprojective modules may have projective tensor powers, and the variety of a tensor product of modules may not be contained in the intersection of their varieties. We explain…

Based on invariant algebras, we introduce representations$^{6-th}$ of Lie algebras and representations$^{< 4-th>}$ of Leibniz algebras, give the extended P-B-W Theorems in the context of the new representations of Lie algebras and Leibniz…

In this paper, we present an approach to the definition of multiparameter quantum groups by studying Hopf algebras with triangular decomposition. Classifying all of these Hopf algebras which are of what we call weakly separable type over a…

We classify and construct irreducible completely splittable representations of affine and finite Hecke-Clifford algebras over an algebraically closed field of characteristic not equal to 2.

The level of a module over a differential graded algebra measures the number of steps required to build the module in an appropriate triangulated category. Based on this notion, we introduce a new homotopy invariant of spaces over a fixed…

We replace the group of group-like elements of the quantized enveloping algebra $U_q({\frak{g}})$ of a finite dimensional semisimple Lie algebra ${\frak g}$ by some regular monoid and get the weak Hopf algebra ${\frak{w}}_q^{\sf d}({\frak…