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The Drinfel'd double D(A) of a finite-dimensional Hopf algebra A is a Hopf algebraic counterpart of the monoidal center construction. Majid introduced an important representation of the Drinfel'd double, which he called the Schr\"odinger…
For each $\ell\geq 1$ and $\lambda,\mu\in\Bbbk$, we study the representations of a family of pointed Hopf algebras $\mathcal{A}_{\lambda,\mu}$. These arise as Hopf cocycle deformations of the graded algebra $\mathcal{FK}_3\#\Bbbk…
We investigate the Drinfel'd doubles $D(\Lambda_{n,d})$ of a certain family of Hopf algebras. We determine their simple modules and their indecomposable projective modules, and we obtain a presentation by quiver and relations of these…
A contractive $n$-tuple $A=(A_1,...,A_n)$ has a minimal joint isometric dilation $S=(S_1,...,S_n)$ where the $S_i$'s are isometries with pairwise orthogonal ranges. This determines a representation of the Cuntz-Toeplitz algebra. When $A$…
We study the diagram alphabet of knot moves associated with the character rings of certain matrix groups. The primary object is the Hopf algebra Char-GL of characters of the finite dimensional polynomial representations of the complex group…
We construct finite-dimensional Hopf algebras whose coradical is the group algebra of a central extension of an abelian group. They fall into families associated to a semisimple Lie algebra together with a Dynkin diagram automorphism. We…
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 semisimple bismash product Hopf algebra $J_n=k^{S_{n-1}}\#kC_n$ for an algebraically closed field $k$ is constructed using the matched pair actions of $C_n$ and $S_{n-1}$ on each other. In this work, we reinterpret these actions and use…
We study a probability measure on integral dominant weights in the decomposition of $N$-th tensor power of spinor representation of the Lie algebra $so(2n+1)$. The probability of the dominant weight $\lambda$ is defined as the ratio of the…
We parameterize the finite-dimensional irreducible representations of a class of pointed Hopf algebras over an algebraically closed field of characteristic zero by dominant characters. The Hopf algebras we are considering arise in the work…
This talk discusses two topological features in non-abelian gauge theories, related by the notion of abelian projection and the Hopf invariant. Minimising the energy of the non-linear sigma model with a Skyrme-like term (the Faddeev-Niemi…
We introduce a natural generalization of the definition of a symmetric Hopf algebroid, internal to any symmetric monoidal category with coequalizers that commute with the monoidal product. Motivation for this is the study of Heisenberg…
In a previous paper we prove that any semisimple triangular Hopf algebra A over an algebraically closed field of characteristic 0 (say the field of complex numbers C) is obtained from a finite group after twisting the ordinary…
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…
We discuss an object from algebraic topology, Hopf invariant, and reinterpret it in terms of the $\phi$-mapping topological current theory. The main purpose in this paper is to present a new theoretical framework which can directly give the…
In this paper, we establish a connection between evolution algebras of dimension two and Hopf algebras, via the algebraic group of automorphisms of an evolution algebra. Initially, we describe the Hopf algebra associated with the…
The universal invariant with respect to a given ribbon Hopf algebra is a tangle invariant that dominates all the Reshetikhin-Turaev invariants built from the representation theory of the algebra. We construct a canonical strict monoidal…
In [1] a new notion of Hopf algebroid has been introduced. It was shown to be inequivalent to the structure introduced under the same name in [17]. We review this new notion of Hopf algebroid. We prove that two Hopf algebroids are…
We show that indecomposable exact module categories over the category Rep H of representations of a finite-dimensional Hopf algebra H are classified by left comodule algebras, H-simple from the right and with trivial coinvariants, up to…
We classify $n$-representation infinite algebras $\Lambda$ of type \~A. This type is defined by requiring that $\Lambda$ has higher preprojective algebra $\Pi_{n+1}(\Lambda) \simeq k[x_1, \ldots, x_{n+1}] \ast G$, where $G \leq…