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We give a decomposition of the equivariant Kasparov category for discrete quantum group with torsions. As an outcome, we show that the crossed product by a discrete quantum group in a certain class preserves the UCT. We then show that quasidiagonality of a reduced C*-algebra of a countable discrete quantum group $\Gamma$ implies that $\Gamma$ is amenable, and deduce from the work of Tikuisis, White and Winter, and the results in the first part of the paper, the converse (i.e. the quantum Rosenberg Conjecture) for a large class of countable discrete unimodular quantum groups. We also note that the unimodularity is a necessary condition., Comment: 15 pages, v2 corrects a few minor points. The final version of the paper will appear in the Proceedings of the American Mathematical Society

We study the “geometric Ricci curvature lower bound”, introduced previously by Junge, Li and LaRacuente, for a variety of examples including group von Neumann algebras, free orthogonal quantum groups [Formula: see text], [Formula: see text]-deformed Gaussian algebras and quantum tori. In particular, we show that Laplace operator on [Formula: see text] admits a factorization through the Laplace–Beltrami operator on the classical orthogonal group, which establishes the first connection between these two operators. Based on a non-negative curvature condition, we obtain the completely bounded version of the modified log-Sobolev inequalities for the corresponding quantum Markov semigroups on the examples mentioned above. We also prove that the “geometric Ricci curvature lower bound” is stable under tensor products and amalgamated free products. As an application, we obtain a sharp Ricci curvature lower bound for word-length semigroups on free group factors.

In this paper, we compute the derivations of the positive part of the two-parameter quantum group of type G2 by embedding it into a quantum torus. We also show that the first Hochschild cohomology group of this algebra is a two-dimensional vector space over the complex field.

In this article, we obtain a basis for the center of the q-Schur algebra U(2,r) in terms of its normalized basis.

Classical symmetric pairs consist of a symmetrizable Kac-Moody algebra $\mathfrak{g}$, together with its subalgebra of fixed points under an involutive automorphism of the second kind. Quantum group analogs of this construction, known as quantum symmetric pairs, replace the fixed point Lie subalgebras by one-sided coideal subalgebras of the quantized enveloping algebra $U_q(\mathfrak{g})$. We provide a complete presentation by generators and relations for these quantum symmetric pair coideal subalgebras. These relations are of inhomogeneous $q$-Serre type and are valid without restrictions on the generalized Cartan matrix. We draw special attention to the split case, where the quantum symmetric pair coideal subalgebras are generalized $q$-Onsager algebras., Comment: 51 pages

In this paper, we construct a class of new modules for the quantum group U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) which are free of rank 1 when restricted to C ⁢ [ K ± 1 ] \mathbb{C}[K^{\pm 1}] . The irreducibility of these modules and submodule structure for reducible ones are determined. It is proved that any C ⁢ [ K ± 1 ] \mathbb{C}[K^{\pm 1}] -free U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) -module of rank 1 is isomorphic to one of the modules we constructed, and their isomorphism classes are obtained. We also investigate the tensor products of the C ⁢ [ K ± 1 ] \mathbb{C}[K^{\pm 1}] -free modules with finite-dimensional simple modules over U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) , and for the generic cases, we obtain direct sum decomposition formulas for them, which are similar to the well-known Clebsch–Gordan formula for tensor products between finite-dimensional weight modules over U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) .

In this note we compute the centers of the categories of tilting modules for G=SL2 in prime characteristic, of tilting modules for the corresponding quantum group at a complex root of unity, and of projective GgT-modules when g=1,2., Comment: 18 pages, some figures, revised version, to appear in Glasg. Math. J., comments welcome

We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. Our main results are that the 3-category of rigid tensor categories with enough compact projectives is 2-dualizable, that the 4-category of rigid braided tensor categories with enough compact projectives is 3-dualizable, and that (in characteristic zero) the 4-category of braided fusion categories is 4-dualizable. Via the cobordism hypothesis, this produces respectively 2, 3 and 4-dimensional framed local topological field theories. In particular, we produce a framed 3-dimensional local TFT attached to the category of representations of a quantum group at any value of $q$., Comment: Minor updates and edits; final version

In this paper we study weighted versions of Fourier algebras of compact quantum groups. We focus on the spectral aspects of these Banach algebras in two different ways. We first investigate their Gelfand spectrum, which shows a connection to the maximal classical closed subgroup and its complexification. Secondly, we study specific finite dimensional representations coming from the complexification of the underlying quantum group. We demonstrate that the weighted Fourier algebras can detect the complexification structure in the special case of $SU_q(2)$, whose complexification is the quantum Lorentz group $SL_q(2,\mathbb{C})$., Comment: Another substantial updates. 29 pages

The generalized quantum group U ( ϵ ) of type A is an affine analogue of quantum group associated to a general linear Lie superalgebra gl M | N . We prove that there exists a unique R matrix on the tensor product of fundamental type representations of U ( ϵ ) for arbitrary parameter sequence ϵ corresponding to a non-conjugate Borel subalgebra of gl M | N . We give an explicit description of its spectral decomposition, and then as an application, construct a family of finite-dimensional irreducible U ( ϵ ) -modules which have subspaces isomorphic to the Kirillov-Reshetikhin modules of usual affine type A M − 1 ( 1 ) or A N − 1 ( 1 ) .

In this two-part paper, we introduce a $$p$$ -adic analytic analogue of Backelin and Kremnizer’s construction of the quantum flag variety of a semisimple algebraic group, when $$q$$ is not a root of unity and $$\vert q-1\vert

We construct canonical strongly bicovariant differential graded algebra structures on all four flavours of cross product Hopf algebras, namely double cross products $A\hookrightarrow A\bowtie H\hookleftarrow H$, double cross coproducts $A\twoheadleftarrow A {\blacktriangleright\!\!\blacktriangleleft} H\twoheadrightarrow H$, biproducts $A{\buildrel\hookrightarrow\over \twoheadleftarrow}A{\cdot\kern-.33em\triangleright\!\!\!, Comment: 36 pages latex, no figures

We analyze families of Markov chains that arise from decomposing tensor products of irreducible representations. This illuminates the Burnside-Brauer theorem for building irreducible representations, the McKay correspondence, and Pitman's 2 M − X theorem. The chains are explicitly diagonalizable, and we use the eigenvalues/eigenvectors to give sharp rates of convergence for the associated random walks. For modular representations, the chains are not reversible, and the analytical details are surprisingly intricate. In the quantum group case, the chains fail to be diagonalizable, but a novel analysis using generalized eigenvectors proves successful.

In this paper the authors investigate the q-Schur algebras of type B that were constructed earlier using coideal subalgebras for the quantum group of type Α. The authors present a coordinate algebra type construction that allows us to realize these q-Schur algebras as the duals of the dth graded components of certain graded coalgebras. Under suitable conditions an isomorphism theorem is proved that demonstrates that the representation theory reduces to the q-Schur algebra of type Α. This enables the authors to address the questions of cellularity, quasi-hereditariness and representation type of these algebras. Later it is shown that these algebras realize the 1-faithful quasi hereditary covers of the Hecke algebras of type Β. As a further consequence, the authors demonstrate that these algebras are Morita equivalent to the category 𝒪 for rational Cherednik algebras for the Weyl group of type Β. In particular, we have introduced a Schur-type functor that identifies the type Β Knizhnik–Zamolodchikov functor.

We give the analogue for Hopf algebras of the polyuble Lie bialgebra construction by Fock and Rosli. By applying this construction to the Drinfeld-Jimbo quantum group, we obtain a deformation quantization $\mathbb{C}_\hslash[(N \backslash G)^m]$ of a Poisson structure $\pi^{(m)}$ on products $(N \backslash G)^m$ of principal affine spaces of a connected and simply connected complex semisimple Lie group $G$. The Poisson structure $\pi^{(m)}$ descends to a Poisson structure $\pi_m$ on products $(B \backslash G)^m$ of the flag variety of $G$ which was introduced and studied by the Lu and the author. Any ample line bundle on $(B \backslash G)^m$ inherits a natural flat Poisson connection, and the corresponding graded Poisson algebra is quantized to a subalgebra of $\mathbb{C}_\hslash[(N \backslash G)^m]$. We define the notion of a strongly coisotropic subalgebra in a Hopf algebra, and explain how strong coisotropicity guarantees that any homogeneous coordinate ring of a homogeneous space of a Poisson Lie group can be quantized in the sense of Ciccoli, Fioresi, and Gavarini., Comment: 28 pages, to be published in J. Noncommut. Geom

In this paper, we give a Grobner-Shirshov basis of quantum group of type ℂ3 by using the Ringel-Hall algebra approach. For this, first we compute all skew-commutator relations between the isoclasses of indecomposable reprersentations of Ringel-Hall algebras of type ℂ3 by using an “inductive” method. Precisely, we do not use the traditional way of computing the skew-commutative relations, that is first compute all Hall polynomials then compute the corresponding skew-commutator relations; contrarily, we compute the “easier” skew-commutator relations which corresponding to those exact sequences with middile term indecomposable or the split exact sequences first, then “inductive” others from these “easier” ones and this in turn gives Hall polynomials as a byproduct. Then we prove that the set of these relations is closed under composition. So they constitutes a minimal Grobner-Shirshov basis of the positive part of quantum group of type ℂ3. Dually, we get a Grobner-Shirshov basis of the negative part of quantum group of type C3. And finally we give a Grobner-Shirshov basis for the whole quantum group of type ℂ3.

We propose a definition of partition quantum spaces. They are given by universal $C^*$-algebras whose relations come from partitions of sets. We ask for the maximal compact matrix quantum group acting on them. We show how those fit into the setting of easy quantum groups: Our approach yields spaces these groups are acting on. In a way, our partition quantum spaces arise as the first $d$ columns of easy quantum groups. However, we define them as universal $C^*$-algebras rather than as $C^*$-subalgebras of easy quantum groups. We also investigate the minimal number $d$ needed to recover an easy quantum group as the quantum symmetry group of a partition quantum space. In the free unitary case, $d$ takes the values one or two., Comment: 35 pages version 2: Notation changes

Let $\mathbb{G}$ be a locally compact quantum group and let $I$ be a closed ideal of $L^{1}(\mathbb{G})$ with $y|_{I}\neq 0$ for some $y\in \text{sp}(L^{1}(\mathbb{G}))$. In this paper, we give a characterization for compactness of $\mathbb{G}$ in terms of the existence of a weakly compact left or right multiplier $T$ on $I$ with $T(f)(y|_{I})\neq 0$ for some $f\in I$. Using this, we prove that $I$ is an ideal in its second dual if and only if $\mathbb{G}$ is compact. We also study Arens regularity of $I$ whenever it has a bounded left approximate identity. Finally, we obtain some characterizations for amenability of $\mathbb{G}$ in terms of the existence of some $I$-module homomorphisms on $I^{\ast \ast }$ and on $I^{\ast }$.

Let $H$ be the Hopf $C^*$-algebra of continuous functions on a (locally) compact quantum group of either reduced or full type. We show that endomorphisms of $H$ that respect its right regular comodule structure are translations by elements of largest classical subgroup of $G$. Furthermore, we show that for compact $G$ such an endomorphism is automatically an automorphism regardless of the quantum group norm on the $C^*$-algebra $H$; this answers a question of Piotr M. Hajac., Comment: 8 pages + references; changes reflecting referee suggestions: proof of Theorem 2.13 changed so as to appeal to reference [6]

The second author constructed a topological ribbon Hopf algebra from the unrolled quantum group associated with the super Lie algebra $\mathfrak{sl}(2|1)$. We generalize this fact to the context of unrolled quantum groups and construct the associated topological ribbon Hopf algebras. Then we use such an algebra, the discrete Fourier transforms, a symmetrized graded integral and a modified trace to define a modified graded Hennings invariant. Finally, we use the notion of a modified integral to extend this invariant to empty manifolds and show that it recovers the CGP-invariant., 54 pages, 42 figures

In [arXiv:1912.02063], we constructed 3-dimensional Topological Quantum Field Theories (TQFTs) using not necessarily semisimple modular categories. Here, we study projective representations of mapping class groups of surfaces defined by these TQFTs, and we express the action of a set of generators through the algebraic data of the underlying modular category $\mathcal{C}$. This allows us to prove that the projective representations induced from the non-semisimple TQFTs of [arXiv:1912.02063] are equivalent to those obtained by Lyubashenko via generators and relations in [arXiv:hep-th/9405167]. Finally, we show that, when $\mathcal{C}$ is the category of finite-dimensional representations of the small quantum group of $\mathfrak{sl}_2$, the action of all Dehn twists for surfaces without marked points has infinite order., 41 pages, minor corrections, Section 2.4 and Appendix C added

We prove that a finite braided tensor category A is invertible in the Morita 4-category BrTens of braided tensor categories if, and only if, it is non-degenerate. This includes the case of semisimple modular tensor categories, but also non-semisimple examples such as categories of representations of the small quantum group at good roots of unity. Via the cobordism hypothesis, we obtain new invertible 4-dimensional framed topological field theories, which we regard as a non-semisimple framed version of the Crane-Yetter-Kauffman invariants, after Freed--Teleman and Walker's construction in the semisimple case. More generally, we characterize invertibility for E_1- and E_2-algebras in an arbitrary symmetric monoidal oo-category, and we conjecture a similar characterization of invertible E_n-algebras for any n. Finally, we propose the Picard group of BrTens as a generalization of the Witt group of non-degenerate braided fusion categories, and pose a number of open questions about it.

Kitaev's lattice models are usually defined as representations of the Drinfeld quantum double $D(H)=H\bowtie H^{*\text{op}} $, as an example of a double cross product quantum group. We propose a new version based instead on $M(H)=H^{\text{cop}}\blacktriangleright\!\!\!\triangleleft H$ as an example of Majid's bicrossproduct quantum group, related by semidualisation or `quantum Born reciprocity' to $D(H)$. Given a finite-dimensional Hopf algebra $H$, we show that a quadrangulated oriented surface defines a representation of the bicrossproduct quantum group $H^{\text{cop}}\blacktriangleright\!\!\!\triangleleft H$. Even though the bicrossproduct has a more complicated and entangled coproduct, the construction of this new model is relatively natural as it relies on the use of the covariant Hopf algebra actions. Working locally, we obtain an exactly solvable Hamiltonian for the model and provide a definition of the ground state in terms of a tensor network representation., 34 pages

In this paper we generalize cellular algebras by allowing different partial orderings relative to fixed idempotents. For these relative cellular algebras we classify and construct simple modules, and we obtain other characterizations in analogy to cellular algebras. We also give several examples of algebras that are relative cellular, but not cellular. Most prominently, the restricted enveloping algebra and the small quantum group for $\mathfrak{sl}_{2}$, and an annular version of arc algebras., Comment: 39 pages, many figures, revised version, to appear in Transform. Groups, comments welcome

A quantum covering group is an algebra with parameters $q$ and $\pi$ subject to $\pi^2=1$ and it admits an integral form; it specializes to the usual quantum group at $\pi=1$ and to a quantum supergroup of anisotropic type at $\pi=-1$. In this paper we establish the Frobenius-Lusztig homomorphism and Lusztig-Steinberg tensor product theorem in the setting of quantum covering groups at roots of 1. The specialization of these constructions at $\pi=1$ recovers Lusztig's constructions for quantum groups at roots of 1., Comment: v2, 21 pages, mild corrections, Lett. Math. Phys. (to appear)

We show that, for a given compact or discrete quantum group G, the class of actions of G on C*-algebras is first-order axiomatizable in the logic for metric structures. As an application, we extend the notion of Rokhlin property for G-C*-algebra, introduced by Barlak, Szabó, and Voigt in the case when G is second countable and coexact, to an arbitrary compact quantum group G. All the the preservations and rigidity results for Rokhlin actions of second countable coexact compact quantum groups obtained by Barlak, Szabó, and Voigt are shown to hold in this general context. As a further application, we extend the notion of equivariant order zero dimension for equivariant *-homomorphisms, introduced in the classical setting by the first and third authors, to actions of compact quantum groups. This allows us to define the Rokhlin dimension of an action of a compact quantum group on a C*-algebra, recovering the Rokhlin property as Rokhlin dimension zero. We conclude by establishing a preservation result for finite nuclear dimension and finite decomposition rank when passing to fixed point algebras and crossed products by compact quantum group actions with finite Rokhlin dimension.

We give a partial solution to a long-standing open problem in the theory of quantum groups, namely we prove that all finite-dimensional representations of a wide class of locally compact quantum groups factor through matrix quantum groups (Admissibility Conjecture for quantum group representations). We use this to study Kazhdan's Property (T) for quantum groups with non-trivial scaling group, strengthening and generalising some of the earlier results obtained by Fima, Kyed and So{\l}tan, Chen and Ng, Daws, Skalski and Viselter, and Brannan and Kerr. Our main results are: (i) All finite-dimensional unitary representations of locally compact quantum groups which are either unimodular or arise through a special bicrossed product construction are admissible. (ii) A generalisation of a theorem of Wang which characterises Property (T) in terms of isolation of finite-dimensional irreducible representations in the spectrum. (iii) A very short proof of the fact that quantum groups with Property (T) are unimodular. (iv) A generalisation of a quantum version of a theorem of Bekka--Valette proven earlier for quantum groups with trivial scaling group, which characterises Property (T) in terms of non-existence of almost invariant vectors for weakly mixing representations. (v) A generalisation of a quantum version of Kerr-Pichot theorem, proven earlier for quantum groups with trivial scaling group, which characterises Property (T) in terms of denseness properties of weakly mixing representations., Comment: Update to final version

In this paper, we study the representations of a class of small nonstandard quantum group X¯q(A1), over which the isomorphism classes of all indecomposable modules are classified, and the decomposi...

In this letter we prove that the unrolled small quantum group, appearing in quantum topology, is a Hopf subalgebra of Lusztig’s quantum group of divided powers. We do so by writing down non-obvious primitive elements with the correct adjoint action. As application, we explain how this gives a realization of the unrolled quantum group as operators on a conformal field theory and match some calculations on this side. In particular, our results explain a prominent weight shift that appears in Feigin and Tipunin (Logarithmic CFTs connected with simple Lie algebras, preprint, 2010. arXiv:1002.5047 ). Our result extends to other Nichols algebras of diagonal type, including super-Lie algebras.

The notion of multiple Ore extension is introduced as a natural generalization of Ore extensions and double Ore extensions. For a PBW-deformation B q ( sl ( n + 1 , C ) ) of type A n quantum group , we explicitly obtain the commutation relations of its root vectors , then show that it can be realized via a series of multiple Ore extensions, which we call a ladder Ore extension of type ( 1 , 2 , ⋯ , n ) . Moreover, we analyze the quantum algebras B q ( g ) with g of type D 4 , B 2 and G 2 and give some examples and counterexamples that can be realized by a ladder Ore extension.

Let g be a symmetrisable Kac-Moody algebra, and U_h(g) the corresponding quantum group. We showed in arXiv:1610.09744 and arXiv:1610.09741 that the braided quasi-Coxeter structure on integrable, category O representations of U_h(g) which underlies the R-matrix actions arising from the Levi subalgebras of U_h(g) and the quantum Weyl group action of the generalised braid group B_g can be transferred to integrable, category O representations of g. We prove in this paper that, up to unique equivalence, there is a unique such structure on the latter category with prescribed restriction functors, R--matrices, and local monodromies. This extends, simplifies and strengthens a similar result of the second author valid when g is semisimple, and is used in arXiv:1512.03041 to describe the monodromy of the rational Casimir connection of g in terms of the quantum Weyl group operators of U_h(g). Our main tool is a refinement of Enriquez's universal algebras, which is adapted to the PROP describing a Lie bialgebra graded by the non-negative roots of g., Expanded Introduction and Sec. 5 to discuss convolution product (5.11), cosimplicial structure on basis elements (5.13) and module structure on coinvariants (5.15). Minor revisions in Sec. 7.1 (gradings), 7.4 (deformation DY modules), 9.7 (exposition), 15.7 (Drinfeld double) and 15.15 (rigidity for diagrammatic KM algebras). Final version, to appear in Adv. Math. 81 pages

We initiate the study of a new notion of duality defined with respect to the module Haagerup tensor product. This notion not only recovers the standard operator space dual for Hilbert $C^*$-modules, it also captures quantum group duality in a fundamental way. We compute the so-called Haagerup dual for various operator algebras arising from $\ell^p$ spaces. In particular, we show that the dual of $\ell^1$ under any operator space structure is $\min\ell^\infty$. In the setting of abstract harmonic analysis we generalize a result of Varopolous by showing that $C(\mathbb{G})$ is an operator algebra under convolution for any compact Kac algebra $\mathbb{G}$. We then prove that the corresponding Haagerup dual $C(\mathbb{G})^h=\ell^\infty(\widehat{\mathbb{G}})$, whenever $\widehat{\mathbb{G}}$ is weakly amenable. Our techniques comprise a mixture of quantum group theory and the geometry of operator space tensor products., Comment: 18 pages

The quantum co-ordinate algebra $A_{q}(\mathfrak{g})$ associated to a Kac-Moody Lie algebra $\mathfrak{g}$ forms a Hopf algebra whose comodules are precisely the $U_{q}(\mathfrak{g})$ modules in the BGG category $\mathcal{O}_{\mathfrak{g}}$. In this paper we investigate whether an analogous result is true when $q=0$. We classify crystal bases as coalgebras over a comonadic functor on the category of pointed sets and encode the monoidal structure of crystals into a bicomonadic structure. In doing this we prove that there is no coalgebra in the category of pointed sets whose comodules are equivalent to crystal bases. We then construct a bialgebra over $\mathbb{Z}$ whose based comodules are equivalent to crystals, which we conjecture is linked to Lusztig's quantum group at $v = \infty$.

Sekine quantum groups are a family of finite quantum groups. The main result of this paper is to compute all the idempotent states on Sekine quantum groups, which completes the work of Franz and Skalski. This is achieved by solving a complicated system of equations using linear algebra and basic number theory. From this we discover a new class of non-Haar idempotent states. The order structure of the idempotent states on Sekine quantum groups is also discussed. Finally we give a sufficient condition for the convolution powers of states on Sekine quantum group to converge., 18 pages, 2 figures

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 Dubois-Violette (2013), where the target of these comodule maps are the ground field. We also recover the quantum groups introduced by Dubois-Violette and Launer (1990), by Takeuchi (1990), by Artin, Schelter, and Tate (1991), and by Mrozinski (2014), via our construction. As a consequence, we obtain an explicit presentation of a universal quantum group that coacts simultaneously on a pair of $N$-Koszul Artin-Schelter regular algebras with arbitrary quantum determinant., v2: To appear in Journal of Noncommutative Geometry

This paper studies the “reduction mod p” method, which constructs large classes of representations for a semisimple algebraic group G from representations for the corresponding Lusztig quantum group U ζ at a p r -th root of unity. The G-modules arising in this way include the Weyl modules, the induced modules, and various reduced versions of these modules. We present a relation between Ext G n ( V , W ) and Ext U ζ n ( V ′ , W ′ ) , when V , W are obtained from V ′ , W ′ by reduction mod p. Since the dimensions of Ext n -spaces for U ζ -modules are known in many cases, our result guarantees the existence of many new extension classes and homomorphisms between rational G-modules. One application is a new proof of James Franklin's result on certain homomorphisms between two Weyl modules. We also provide some examples which show that the p-th root of unity case and a general p r -th root of unity case are essentially different.

Let k be an arbitrary field and let q ∈ k ∖ { 0 } . In this paper we use the known tilting theory for the quantum group U q ( s l 2 ) to obtain the dimensions of simple modules for the Temperley–Lieb algebras T L n ( q + q − 1 ) and related algebras over k. Our main result is an algorithm which calculates the dimensions of simple modules for these algebras. We take advantage of the fact that T L n ( q + q − 1 ) is isomorphic to the endomorphism ring of the n'th tensor power of the natural 2-dimensional module for the quantum group for s l 2 . This algorithm is easy when the characteristic is 0 and more involved in positive characteristic. We point out that our results for the Temperley–Lieb algebras contain a complete description of the simple modules for the Jones quotient algebras. Moreover, we illustrate how the same results lead to corresponding information about simple modules for the BMW-algebras with special parameters and other algebras closely related with endomorphism algebras of families of tilting modules for U q ( s l 2 ) .

Let $k$ be a field and let $H$ denote a pointed Hopf $k$-algebra with antipode $S$. We are interested in determining the order of $S$. Building on the work done by Taft and Wilson $[7]$, we define an invariant for $H$, denoted $m_{H}$, and prove that the value of this invariant is connected to the order of $S$. In the case where $\operatorname{char}k=0$, it is shown that if $S$ has finite order then it is either the identity or has order $2m_{H}$. If in addition $H$ is assumed to be coradically graded, it is shown that the order of $S$ is finite if and only if $m_{H}$ is finite. We also consider the case where $\operatorname{char}k=p>0$, generalising the results of $[7]$ to the infinite-dimensional setting.

Let $\mathbb{G}$ be a free (unitary or orthogonal) quantum group. We prove that for any non-amenable subfactor $N\subset L^\infty(\mathbb{G})$, which is an image of a faithful normal conditional expectation, and for any $\sigma$-finite factor $B$, the tensor product $N \otimes B$ has no Cartan subalgebras. This generalizes our previous work that provides the same result when $B$ is finite. In the proof, we establish Ozawa--Popa and Popa--Vaes's weakly compact action on the continuous core of $N \otimes B$ as the one relative to B, by using an operator valued weight to B and the central weak amenability of $\mathbb{G}$., Comment: 28 pages, final version

We find a dual version of a previous double-bosonisation theorem whereby each finite-dimensional braided-Hopf algebra B in the category of comodules of a coquasitriangular Hopf algebra A has an associated coquasitriangular Hopf algebra . As an application we find new generators for c q [ S L 2 ] reduced at q a primitive odd root of unity with the remarkable property that their monomials are essentially a dual basis to the standard PBW basis of the reduced Drinfeld–Jimbo quantum enveloping algebra u q ( s l 2 ) . Our methods apply in principle for general c q [ G ] as we demonstrate for c q [ S L 3 ] at certain odd roots of unity.

In this paper, we compute explicitly the Khovanov polynomials (using the computer program from katlas.org) for the two simplest families of the satellite knots, which are the twisted Whitehead doubles and the two-strand cables. We find that a quantum group decomposition for the HOMFLY polynomial of a satellite knot can be extended to the Khovanov polynomial, whose quantum group properties are not manifest. Namely, the Khovanov polynomial of a twisted Whitehead double or two-strand cable (the two simplest satellite families) can be presented as a naively deformed linear combination of the pattern and companion invariants. For a given companion, the satellite polynomial “smoothly” depends on the pattern but for the “jump” at one critical point defined by the [Formula: see text]-invariant of the companion knot. A similar phenomenon is known for the knot Floer homology and [Formula: see text]-invariant for the same kind of satellites.

The real forms of complex groups (or algebras) are important in physics and mathematics. The Lie group SL2,C is one of these important groups. There are real forms of the classical Lie group SL2,C and the quantum group SL2,C in the literature. Inspired by this, in our study, we obtain the real forms of the fractional supergroups shown with A3NSL2,C, for the non-trivial N = 1 and N = 2 cases, that is, the real forms of the fractional supergroups A31SL2,C and A32SL2,C.

We construct quadratic quantum algebra based on the dynamical RLL-relation for the quantum $R$-matrix related to $SL(NM)$-bundles with nontrivial characteristic class over elliptic curve. This $R$-matrix generalizes simultaneously the elliptic nondynamical Baxter--Belavin and the dynamical Felder $R$-matrices,and the obtained quadratic relations generalize both -- the Sklyanin algebra and the relations in the Felder-Tarasov-Varchenko elliptic quantum group, which are reproduced in the particular cases $M=1$ and $N=1$ respectively., 10 pages

We study the tensor product decomposition of the split real quantum group $${\mathcal {U}}_{q\widetilde{q}}(\mathfrak {sl}(2,\mathbb {R}))$$ from the perspective of finite-dimensional representation theory of compact quantum groups. It is known that the class of positive representations of $${\mathcal {U}}_{q\widetilde{q}}(\mathfrak {sl}(2,\mathbb {R}))$$ is closed under taking tensor product. In this paper, we show that one can derive the corresponding Hilbert space decomposition, given explicitly by quantum dilogarithm transformations, from the Clebsch–Gordan coefficients of the tensor product decomposition of finite-dimensional representations of the compact quantum group $${\mathcal {U}}_q(\mathfrak {sl}_2)$$ by solving certain functional equations arising from analytic continuation and using normalization arising from tensor products of canonical basis.

that: "some of the me-thods used in Galois theory andformulated in the Hopf algebrasto Chase and Sweedler to defineterms of Hopf algebras whichdon 't formulate a corresponding theory of descent" . The pur-pose of this notes is to develop this theory, using asGalois definition of object, the one given by Chase andSweedler in

A classical theorem of Frucht states that any finite group appears as the automorphism group of a finite graph. In the quantum setting the problem is to understand the structure of the compact quantum groups which can appear as quantum automorphism groups of finite graphs. We discuss here this question, notably with a number of negative results., Comment: 41 pages; v3 further revisions, to appear in Glasg. Math. J

We exploit the correspondence between twisted homology and quantum group to construct an algebra explanation of the open string kinematic numerator. In this setting the representation depends on string modes, and therefore the cohomology content of the numerator, as well as the location of the punctures. We show that quantum group root system thus identified helps determine the Casimir appears in the Knizhnik-Zamolodchikov connection, which can be used to relate representations associated with different puncture locations., Comment: v2: Published version, clarifying comments and reference added, 32 pages, 8 figures

Let (A,&Delta, ) be a weak multiplier Hopf algebra. It is a pair of a non-degenerate algebra A, with or without identity, and a coproduct &Delta, A⟶M(A&otimes, A), satisfying certain properties. In this paper, we continue the study of these objects and construct new examples. A symmetric pair of the source and target maps &epsilon, s and &epsilon, t are studied, and their symmetric pair of images, the source algebra and the target algebra &epsilon, s(A) and &epsilon, t(A), are also investigated. We show that the canonical idempotent E (which is eventually &Delta, (1)) belongs to the multiplier algebra M(B&otimes, C), where (B=&epsilon, s(A), C=&epsilon, t(A)) is the symmetric pair of source algebra and target algebra, and also that E is a separability idempotent (as studied). If the weak multiplier Hopf algebra is regular, then also E is a regular separability idempotent. We also see how, for any weak multiplier Hopf algebra (A,&Delta, ), it is possible to make C&otimes, B (with B and C as above) into a new weak multiplier Hopf algebra. In a sense, it forgets the &rsquo, Hopf algebra part&rsquo, of the original weak multiplier Hopf algebra and only remembers symmetric pair of the source and target algebras. It is in turn generalized to the case of any symmetric pair of non-degenerate algebras B and C with a separability idempotent E&isin, M(B&otimes, C). We get another example using this theory associated to any discrete quantum group. Finally, we also consider the well-known &rsquo, quantization&rsquo, of the groupoid that comes from an action of a group on a set. All these constructions provide interesting new examples of weak multiplier Hopf algebras (that are not weak Hopf algebras introduced).

Chern–Simons theory provides a natural framework to construct a variety of knot invariants. The calculation of colored HOMFLY-PT polynomials of knots using SU(N) Chern–Simons theory requires the knowledge of 6j-symbols for the quantum group \(U_q(\frak {sl}_N)\) which are not known for arbitrary representation. Interestingly, our conjectured formula for superpolynomials (categorification of colored HOMFLY-PT polynomials) of twist knots led to deducing closed form expression for these symbols for a class of multiplicity-free \(U_q(\frak {sl}_N)\) representation. Using the twist knot superpolynomials, we compute the classical and quantum super-A-polynomials and test the categorified version of the quantum volume conjecture.