Related papers: Webs for the Quantum Orthogonal Group
We generalize categories of spatial partitions in the sense of C\'ebron-Weber by introducing new base partitions. This allows us to construct additional examples of free orthogonal quantum groups but yields the same class of spatial…
This note presents an elementary iterative construction of the generators for the complex $K$-groups $K_i(C(\SM^d))$ of the $d$-dimensional spheres. These generators are explicitly given as the restrictions of Dirac or Weyl Hamiltonians to…
Consider a compact connected Lie group $G$ and the corresponding Lie algebra $\cal L$. Let $\{X_1,...,X_m\}$ be a set of generators for the Lie algebra $\cal L$. We prove that $G$ is uniformly finitely generated by $\{X_1,...,X_m\}$. This…
This paper is an adaptation of a chapter from an upcoming monograph on noncommutative geometry and quantum groups. We present examples of non compact quantum groups which are deformations of low dimensional Lie groups. The paper is of…
Group representations play a central role in theoretical physics. In particular, in quantum mechanics unitary --- or, in general, projective unitary --- representations implement the action of an abstract symmetry group on physical states…
In this paper, we construct an infinite presentation of the Torelli subgroup of the mapping class group of a surface whose generators consist of the set of all "separating twists", all "bounding pair maps", and all "commutators of simply…
We describe the generators and prove a number of relations for the construction of a planar algebra from the restricted quantum group $\bar{U}_{q}(\mathfrak{sl}_{2})$. This is a diagrammatic description of…
In Part I of this series we presented the general ideas of applying group-algebraic methods for describing quantum systems. The treatment was there very "ascetic" in that only the structure of a locally compact topological group was used.…
We give generators for a certain complex hyperbolic braid group. That is, we remove a hyperplane arrangement from complex hyperbolic $13$-space, take the quotient of the remaining space by a discrete group, and find generators for the…
The aim of this paper is to survey some aspects of mapping class groups with focus on their finite dimensional representations arising in topological quantum field theory.
An idea to present a classical Lie group of positive dimension by generators and relations sounds dubious, but happens to be fruitful. The isometry groups of classical geometries admit elegant and useful presentations by generators and…
A notion of a quantum automorphism group of a finite quantum group, generalising that of a classical automorphism group of a finite group, is proposed and a corresponding existence result proved.
We introduce a $Z_3$-graded version of exterior (Grassmann) algebra with two generators and using this object we obtain a new $Z_3$-graded quantum group denoted by $O(\widetilde{GL}_q(2))$. We also discuss some properties of ${…
The structure and representations of the quantum general linear supergroup GLq(m|n) are studied systematically by investigating the Hopf superalgebra Gq of its representative functions. Gq is factorized into $Gq^{\pi} Gq^{\bar\pi}$, and a…
For the quantum group $GL_{p,q}(2)$ and the corresponding quantum algebra $U_{p,q}(gl(2))$ Fronsdal and Galindo explicitly constructed the so-called universal $T$-matrix. In a previous paper we showed how this universal $T$-matrix can be…
Gaussian unitaries, generated by quadratic Hamiltonians, are fundamental in quantum optics and continuous-variable computing. Their structures correspond to symplectic (bosons) and orthogonal (fermions) groups, but physical realizations…
We construct a braid group action on quantum covering groups. We further use this action to construct a PBW basis for the positive half in finite type which is pairwise-orthogonal under the inner product. This braid group action is induced…
We find all unitary representations of the quantum "ax+b" group. It turns out that this quantum group is selfdual in the sense that all unitary representations are 'numbered' by elements of the same group. Moreover, we discover the formula…
We introduce quiver representation-valued invariants of oriented virtual knots and links associated to a choice of finite virtual biquandle, abelian group, set of virtual Boltzmann weights, commutative unital ring and set of virtual…
We characterise the elements of the (maximum) idempotent generated subsemigroup of the Kauffman monoid in terms of combinatorial data associated to certain normal forms. We also calculate the smallest size of a generating set and idempotent…