Narrow Results

We investigate non-semisimple modular categories with an eye towards a structure theory, low-rank classification and applications to low dimensional topology and topological physics. We aim to extend the well-understood theory of semisimple modular categories to the non-semisimple case by using representations of factorizable ribbon Hopf algebras as a case study. We focus on the Cohen–Westreich modular data, which is obtained from the Lyubashenko–Majid modular representation restricted to the Higman ideal of a factorizable ribbon Hopf algebra. The Cohen–Westreich $S$-matrix diagonalizes the mixed fusion rules and reduces to the usual $S$-matrix for semisimple modular categories. The paper includes detailed studies on small quantum groups $U_{q}sl(2)$ and the Drinfeld doubles of Nichols Hopf algebras, especially the $\mathrm{\text{SL}}(2,\mathbb{Z})$-representation on their centers, Cohen–Westreich modular data, and the congruence kernel theorem’s validity.

An alternative construction of the invariant of homology three spheres valued in a completion of an integral polynomial ring associated to each quantized complex simple Lie algebra by Habiro and Lê [Unified quantum invariants for integral homology spheres associated with simple Lie algebras, Geom. Topol.20 (2016) 2687–2835, doi:10.2140/gt.2016.20.2687, arXiv:1503:03549v1] is given.

For a quantum group, we study those right coideal subalgebras, for which all irreducible representations are one-dimensional. If a right coideal subalgebra is maximal with this property, then we call it a Borel subalgebra.

Besides the positive part of the quantum group and its reflections, we find new unfamiliar Borel subalgebras, for example, ones containing copies of the quantum Weyl algebra. Given a Borel subalgebra, we study its induced (Verma-)modules and prove among others that they have all irreducible finite-dimensional modules as quotients. We give two structural conjectures involving the associated graded right coideal subalgebra, which we prove in certain cases. In particular, they predict the shape of all triangular Borel subalgebras. As examples, we determine all Borel subalgebras of $U_q({𝔰l}_2)$ and $U_q({𝔰l}_3)$ and discuss the induced modules.

In this paper, we calculate the combinatorial rank of the positive part $u_q^{+}(𝔤)$ of the multiparameter version of the small Lusztig quantum group, where $𝔤$ is a simple Lie algebra of type $F_4$. For this purpose, we also present the basis and relations of the quantum algebra, calculate the coproduct of its PBW-generators and determine all skew-primitive elements of this quantum group. In conclusion, we obtain that the combinatorial rank of $u_q^{+}(F_4)$ is 4.

We consider the algebra of N×N matrices as a reduced quantum plane on which a finite-dimensional quantum group ℋ acts. This quantum group is a quotient of , q being an Nth root of unity. Most of the time we shall take N=3; in that case dim(ℋ)=27. We recall the properties of this action and introduce a differential calculus for this algebra: it is a quotient of the Wess–Zumino complex. The quantum group ℋ also acts on the corresponding differential algebra and we study its decomposition in terms of the representation theory of ℋ. We also investigate the properties of connections, in the sense of non commutative geometry, that are taken as 1-forms belonging to this differential algebra. By tensoring this differential calculus with usual forms over space-time, one can construct generalized connections with covariance properties with respect to the usual Lorentz group and with respect to a finite-dimensional quantum group.

We study the "quantized calculus" corresponding to the algebraic ideas related to "twisted cyclic cohomology" introduced in [12]. With very similar definitions and techniques as those used in [9], we define and study "twisted entire cyclic cohomology" and the "twisted Chern character" associated with an appropriate operator theoretic data called "twisted spectral data", which consists of a spectral triple in the conventional sense of noncommutative geometry [1] and an additional positive operator having some specified properties. Furthermore, it is shown that given a spectral triple (in the conventional sense) which is equivariant under the (co-) action of a compact matrix pseudogroup, it is possible to obtain a canonical twisted spectral data and hence the corresponding (twisted) Chern character, which will be invariant (in the usual sense) under the (co-)action of the pseudogroup, in contrast to the fact that the Chern character coming from the conventional noncommutative geometry need not to be invariant. In the last section, we also try to detail out some remarks made in [3], in the context of a new definition of invariance satisfied by the conventional (untwisted) cyclic cocycles when lifted to an appropriate larger algebra.

We study the spectral geometry of the quantum projective plane , a deformation of the complex projective plane ℂP², the simplest example of spin^c manifold which is not spin. In particular, we construct a Dirac operator D which gives a 0⁺-summable triple, equivariant under U_q(su(3)). The square of D is a central element for which left and right actions on spinors coincide, a fact that is exploited to compute explicitly its spectrum.

We study a three-dimensional differential calculus on the standard Podleś quantum two-sphere , coming from the Woronowicz 4D₊ differential calculus on the quantum group SU_q(2). We use a frame bundle approach to give an explicit description of and its associated spin geometry in terms of a natural spectral triple over . We equip this spectral triple with a real structure for which the commutant property and the first order condition are satisfied up to infinitesimals of arbitrary order.

We present an account of the ADHM construction of instantons on Euclidean space-time ℝ⁴ from the point of view of noncommutative geometry. We recall the main ingredients of the classical construction in a coordinate algebra format, which we then deform using a cocycle twisting procedure to obtain a method for constructing families of instantons on noncommutative space-time, parametrized by solutions to an appropriate set of ADHM equations. We illustrate the noncommutative construction in two special cases: the Moyal–Groenewold plane and the Connes–Landi plane .

We describe Laplacian operators on the quantum group SU_q(2) equipped with the four-dimensional bicovariant differential calculus of Woronowicz as well as on the quantum homogeneous space with the restricted left covariant three-dimensional differential calculus. This is done by giving a family of Hodge dualities on both the exterior algebras of SU_q(2) and . We also study gauged Laplacian operators acting on sections of line bundles over the quantum sphere.

We collect and systematize general definitions and facts on the application of quantum groups to the construction of functional relations in the theory of integrable systems. As an example, we reconsider the case of the quantum group related to the six-vertex model. We prove the full set of the functional relations in the form independent of the representation of the quantum group in the quantum space and specialize them to the case of the six-vertex model.

We define concepts of amenability and co-amenability for locally compact quantum groups in the sense of J. Kustermans and S. Vaes. Co-amenability of a lcqg (locally compact quantum group) is proved to be equivalent to a series of statements, all of which imply amenability of the dual lcqg. Further, it is shown that if a lcqg is amenable, then its universal dual lcqg is nuclear. We also define and study amenability and weak containment concepts for representations and corepresentations of lcqg's.

We study analytic properties of spectral zeta functions associated to actions of the quantum group SU_q(2) such as Z(s, SU_q(2)), the zeta function corresponding to the regular representation introduced in [15]. As an application, we show the special value ζ(3) of the Riemann zeta function ζ(s) is given in terms of the classical limit of Z(s, SU_q(2)). We further discuss a spectral zeta function associated with the so-called model of the representations of and show a presence of its series of "trivial" zeros, which is noteworthy.

We observe that the von Neumann envelop of (i.e. the smallest von Neumann algebra that contains) the quantum algebra of functions on the normalizer of the group SU(1,1)≅SL(2,ℝ) in via deformation quantization contains the von Neumann algebraic quantum normalizer of SU(1,1) in the frame work of Waronowicz–Korogodsky, see ([8, Introduction and Sec. 1, Definition 1]), i.e. the C*-envelop or von Neumann envelop (W*-envelop) of the algebraic Hopf algebra. We then use the technique of reduction to the maximal subgroup to compute the K-theory, the periodic cyclic homology and the corresponding Chern–Connes character.

In this paper we study the Fock representation of a certain *-algebra which appears naturally in the framework of quantum group theory. It is also a generalization of the twisted CCR-algebra introduced by Pusz and Woronowicz. We prove that the Fock representation is a faithful irreducible representation of the algebra by bounded operators in a Hilbert space, and, moreover, it is the only (up to unitary equivalence) representation possessing these properties.

In this paper we shall find a matrix realization of the quantum group 𝔤_p,q introduced in [B. L. Feigin et al., Nucl. Phys. B757 (2006) 303–343]. For this purpose, we construct all primitive idempotents and a basis of 𝔤_p,q. We determine the action of elements of the basis on indecomposable projective modules, that gives rise to a matrix realization of 𝔤_p,q. By using this result, we obtain a basis of the space of symmetric linear functions on 𝔤_p1,p2 and express the symmetric linear functions obtained by the left integral, the balancing element, and the center of 𝔤_p,q in terms of this basis.

We give a unified construction of quantum groups, q-boson algebras and quantized Weyl algebras and an action of quantum groups on quantized Weyl algebras. This enables us to give a conceptual proof of the semi-simplicity of the category introduced by Nakashima and the classification of all simple objects in it.

We determine the K-theory of the C*-algebra C(SU_-1(2)), describe its spectrum, and compute its Haar state. As an application, we use Blanchard's technique in order to prove the existence of a continuous C*-bundle over [-1, 0) whose fiber at q is isomorphic to C(SU_q(2)).

We show that the rigid $C^{\ast }$-tensor categories of finite-dimensional type 1 unitary representations of the quantum groups $U_q({𝔤}_2)$ corresponding to the exceptional Lie group $G_2$ for positive $q\ne 1$ have property (T).

We study the cut-off phenomenon for random walks on free unitary quantum groups coming from quantum conjugacy classes of classical reflections. We obtain in particular a quantum analogue of the result of U. Porod concerning certain mixtures of reflections. We also study random walks on quantum reflection groups and more generally on free wreath products of finite groups by quantum permutation groups.