量子代数

Let $\mathbb{F}$ denote an algebraically closed field with characteristic $0$, and let $q$ denote a nonzero scalar in $\mathbb{F}$ that is not a root of unity. Let $\mathbb{Z}_4$ denote the cyclic group of order $4$. Let $\square_q$ denote…

The purpose of this work is to illustrate in a family of interesting examples how to study the representation theory of vertex operator superalgebras by combining the theory of vertex algebra extensions and modular forms. Let…

We explore questions of projectivity and tensor products of modules for finite dimensional Hopf algebras. We construct many classes of examples in which tensor powers of nonprojective modules are projective and tensor products of modules in…

Consider a group $\Gamma$ acting on a formal (Fedosov) deformation quantization $\mathbb{A}_\hbar(M)$ of a symplectic manifold $(M,\omega)$. This canonically induces an action of $\Gamma$ by symplectomorphisms on $M$. We examine the reverse…

We consider $C^*$-algebra $Isom_{q_{ij}}$ generated by $n$ isometries $a_1, \ldots, a_n$ satisfying the relations $a_i^* a_j = q_{ij} a_j a_i^*$ with $\max |q_{ij}| < 1$. This $C^*$-algebra is shown to be nuclear. We prove that the Fock…

The simple finite-dimensional modules over the Drinfeld double of the bosonization of the Nichols algebra $\mathfrak{ufo}(7)$ are classified.

The problem of constructing all the non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation recently has been reduced to the problem of describing all the left braces. In particular, the classification of all finite…

We define the universal quantum group $\mathcal{H}$ that preserves a pair of Hopf comodule maps, whose underlying vector space maps are preregular forms defined on dual vector spaces. This generalizes the construction of Bichon and…

We study spin and super-modular categories systematically as inspired by fermionic topological phases of matter, which are always fermion parity enriched and modelled by spin TQFTs at low energy. We formulate a $16$-fold way conjecture for…

We consider the (finite-dimensional) small quantum group $\bar{U}_q sl(2)$ at $q=i$. We show that $\bar{U}_i sl(2)$ does not allow for an R-matrix, even though $U \otimes V \cong V \otimes U$ holds for all finite-dimensional representations…

In this article, we prove that the full automorphism group of the Z_2-orbifold of the Barnes-Wall lattice vertex operator algebra of central charge 32 has the shape $2^{27}.E_6(2)$. In order to identify the group structure, we introduce a…

We give an interpretation of Yetter's Invariant of manifolds $M$ in terms of the homotopy type of the function space $TOP(M,B(G))$, where $G$ is a crossed module and $B(G)$ is its classifying space. From this formulation, there follows that…

We analyse the possibility of defining complex valued Knot invariants associated with infinite dimensional unitary representations of $SL(2,R)$ and the Lorentz Group taking as starting point the Kontsevich Integral and the notion of…

We analyse the perturbative expansion of the knot invariants defined from the unitary representations of the Quantum Lorentz Group in two different ways, namely using the Kontsevich Integral and weight systems, and the $R$-matrix in the…

After a thorough treatment of all algebraic structures involved, we address two dimensional holonomy operators with values in crossed modules of Hopf algebras and in crossed modules of associative algebras (called here crossed modules of…

We study the (compact) quantum subgroups of the compact quantum group $SU_{-1}(3)$: we show that any non-classical such quantum subgroup is a twist of a compact subgroup of SU(3) or is isomorphic to a quantum subgroup of $U_{-1}(2)$.

We realise Heckenberger and Kolb's canonical calculus on quantum projective (n-1)-space as the restriction of a distinguished quotient of the standard bicovariant calculus for Cq[SUn]. We introduce a calculus on the quantum (2n-1)-sphere in…

We study $\mathbb Z_2$-graded Poisson structures defined on $\mathbb Z_2$-graded commutative polynomial algebras. In small dimensional cases, we exhibit classifications of such Poisson structures, obtain the associated Poisson $\mathbb…

In this paper we study the family of prime irreducible representations of quantum affine $\lie{sl}_{n+1}$ which arise from the work of D. Hernandez and B. Leclerc. These representations can also be described as follows: the highest weight…

We construct an extended oriented $(2+\epsilon)$-dimensional topological field theory, the character field theory $X_G$ attached to a affine algebraic group in characteristic zero, which calculates the homology of character varieties of…