Linear coactions of discrete quantum groups on the circle

Abstract Since the establishment of the quantum Schur–Weyl duality in Jimbo (Lett. Math. Phys. 11 , 247–252, 1986), the duality pair $$(\textbf{U}(\mathfrak {gl}_n),\varvec{\mathcal {H}}(\mathfrak S_r))$$ ( U ( gl n ) , H ( S r ) ) of type A has been extended to the duality pairs $$(\textbf{U}^\jmath (n),\varvec{\mathcal {H}}(B_r))$$ ( U ȷ ( n ) , H ( B r ) ) and $$(\textbf{U}^\imath (n),\varvec{\mathcal {H}}(C_r))$$ ( U ı ( n ) , H ( C r ) ) in the Hecke algebra series in Bao and Wang (Astérisque 402 , vii+134, 2018), where $$\textbf{U}^\jmath (n),\textbf{U}^\imath (n)$$ U ȷ ( n ) , U ı ( n ) are i -quantum groups arising from certain quantum symmetric pairs. The quantum Schur algebra associated with the pair $$(\textbf{U}(\mathfrak {gl}_n),\varvec{\mathcal {H}}(\mathfrak S_r))$$ ( U ( gl n ) , H ( S r ) ) has a nice and simple presentation; see Doty and Giaquinto (2002). In this paper, we tackle the presentation problem for the i -quantum Schur algebras associated with the duality pair $$(\textbf{U}^\imath (n),\varvec{\mathcal {H}}(C_r))$$ ( U ı ( n ) , H ( C r ) ) . Such a q -Schur algebra is called the hyperoctahedral q -Schur algebras in Green (J. Algebra 192 , 418–438, 1997). See Bhattacharya (2026) for the $$\textbf{U}^\jmath (n)$$ U ȷ ( n ) case. Building on the explicit epimorphism $$\phi _{n,r}^\imath $$ ϕ n , r ı from the i -quantum group $$\textbf{U}^\imath (n)$$ U ı ( n ) to the hyperoctahedral q -Schur algebras $$\mathcal {S}^\imath (n,r)$$ S ı ( n , r ) (see Du and Wu, Pacific J. Math. 320 (1), 61–101, 2022), we compute the kernel of $$\phi _{n,r}^\imath $$ ϕ n , r ı in terms of generators. This results in a presentation for $$\mathcal {S}^\imath (n,r)$$ S ı ( n , r ) with defining relations which include not only the Doty–Giaquinto’s diagonal relations but also some tridiagonal relations.

Category <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"> <mml:mi mathvariant="script">O</mml:mi> </mml:math> for hybrid quantum groups and non-commutative Springer resolutions

Quantum group decision making based on regret theory and grey relational degree considering psychological preference

A quantum group decision-making model for patient-capital project selection integrating cumulative prospect theory under linear Diophantine fuzzy uncertainty

We construct an action of the positive Witt algebra on the categorified quantum group associated to a simply-laced Lie algebra. In the type-A case, we show that this action induces an action of the positive Witt algebra on \mathfrak{gl}_{n} -foams, recovering the action of Qi, Robert, Sussan, and Wagner. We also show that this construction is compatible with the trace decategorification, inducing the action of the positive Witt algebra on the current algebra.

Abstract We introduce a framework to define coalgebra and bialgebra structures on two-dimensional (2D) square lattices, extending the algebraic theory of Hopf algebras and quantum groups beyond the one-dimensional (1D) setting. Our construction is based on defining 2D coproducts through horizontal and vertical maps that satisfy compatibility and associativity conditions, enabling the consistent growth of vector spaces over lattice sites. We present several examples of 2D bialgebras, including group-like and Lie algebra-inspired constructions and a quasi-1D coproduct instance that is applicable to Taft-Hopf algebras and to quantum groups. The approach is further applied to the quantum group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mo stretchy="false">[</mml:mo> <mml:mi>s</mml:mi> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> </mml:math> , for which we construct 2D generalizations of its generators, analyze q -deformed singlet states, and derive a 2D R-matrix satisfying an intertwining relation in the semiclassical limit. Additionally, we show how tensor network states, particularly projected entangled pair states, naturally induce 2D coalgebra structures when supplemented with appropriate boundary conditions. Our results establish a local and algebraically consistent method to embed quantum group symmetries into higher-dimensional lattice systems, potentially connecting to the emerging theory of fusion 2-categories and categorical symmetries in quantum many-body physics.

Abstract We consider a 2-homogeneous bipartite distance-regular graph $$\Gamma $$ Γ with diameter $$D \ge 3$$ D ≥ 3 . We assume that $$\Gamma $$ Γ is not a hypercube nor a cycle. We fix a Q -polynomial ordering of the primitive idempotents of $$\Gamma $$ Γ . This Q -polynomial ordering is described using a nonzero parameter $$q \in \mathbb {C}$$ q ∈ C that is not a root of unity. We investigate $$\Gamma $$ Γ using an $$S_3$$ S 3 -symmetric approach. In this approach one considers $$V^{\otimes 3} = V \otimes V \otimes V$$ V ⊗ 3 = V ⊗ V ⊗ V where V is the standard module of $$\Gamma $$ Γ . We construct a subspace $$\Lambda $$ Λ of $$V^{\otimes 3}$$ V ⊗ 3 that has dimension $$\left( {\begin{array}{c}D+3\\ 3\end{array}}\right) $$ D + 3 3 , together with six linear maps from $$\Lambda $$ Λ to $$\Lambda $$ Λ . Using these maps we turn $$\Lambda $$ Λ into an irreducible module for the nonstandard quantum group $$U^\prime _q(\mathfrak {so}_6)$$ U q ′ ( so 6 ) introduced by Gavrilik and Klimyk in 1991.

We show that the quantum invariants arising from typical representations of the quantum group U h (𝔰𝔩(2|1)) are q-holonomic. In particular, this implies the existence of an underlying field theory for which this family of invariants are partition functions.

In this article, we investigate the Green ring of the small quantum group [Formula: see text], where [Formula: see text] is a root of unity of order [Formula: see text] and [Formula: see text] is even. We describe the structures of the Green rings by generators with relations.

Abstract We consider quantum group representations Rep ⁡ ( G q ) \operatorname{Rep}(G_{q}) for a semisimple algebraic group 𝐺 at a complex root of unity 𝑞. Here we allow 𝑞 to be of any order. We first show that the Tannakian center in Rep ⁡ ( G q ) \operatorname{Rep}(G_{q}) is calculated via a twisting of Lusztig’s quantum Frobenius functor Rep ⁡ ( G ̌ ) → Rep ⁡ ( G q ) \operatorname{Rep}(\check{G})\to\operatorname{Rep}(G_{q}) , where G ̌ \check{G} is a dual group to 𝐺. We then consider the associated fiber category Vect ⊗ Rep ⁡ ( G ̌ ) Rep ⁡ ( G q ) \mathrm{Vect}\otimes_{\operatorname{Rep}{(\check{G})}}\operatorname{Rep}(G_{q}) over B ⁢ G ̌ B\check{G} , and show that this fiber is a finite, integral braided tensor category. Furthermore, when 𝐺 is simply connected and 𝑞 is of even order, the fiber in question is shown to be a modular tensor category. Finally, we exhibit a finite-dimensional quasitriangular quasi-Hopf algebra (also known as small quantum group) whose representations recover the tensor category Vect ⊗ Rep ⁡ ( G ̌ ) Rep ⁡ ( G q ) \mathrm{Vect}\otimes_{\operatorname{Rep}{(\check{G})}}\operatorname{Rep}(G_{q}) , and we describe the representation theory of this algebra in detail. At particular pairings of 𝐺 and 𝑞, our quasi-Hopf algebra is identified with Lusztig’s original finite-dimensional Hopf algebra from the ’90s. This work completes the author’s project from [C. Negron, Log-modular quantum groups at even roots of unity and the quantum Frobenius I, Comm. Math. Phys. 382 (2021), 2, 773–814].

A bstract Double-scaled SYK (DSSYK) is known to have an underlying quantum group theoretical description. We precisely pinpoint the quantum group structure, improving upon earlier work in the literature. This allows us to utilize this framework for bulk gravitational applications. We explain bulk discretization in DSSYK from the underlying irreducibility of the representations. We derive trumpet and brane amplitudes using character insertions of the quantum group, simplifying earlier calculations. Most importantly, we factorize the bulk Hilbert space dual to DSSYK in the quantum group description using a complete set of edge degrees of freedom living at a bulk entangling surface. An analogous treatment for $$ \mathcal{N}=1 $$ N = 1 DSSYK is provided in the same quantum group theoretical framework.

In the paper, we show some properties of (reduced) stated S L ( n ) \mathrm {SL}(n) -skein algebras related to their centers for essentially bordered punctured bordered surfaces, especially their centers, finitely generation over their centers, and their PI-degrees. The proofs are based on the quantum trace maps, embeddings of (reduced) stated S L ( n ) \mathrm {SL}(n) -skein algebras into quantum tori appearing in higher Teichmüller theory. Thanks to the Unicity theorem by Brown and Goodearl [ Lectures on algebraic quantum groups , Birkhäuser Verlag, Basel, 2002] and Frohman, Kania-Bartoszynska, and Lê [Invent. Math. 215 (2019), pp. 609–650], we can understand the representation theory of (reduced) stated S L ( n ) \mathrm {SL}(n) -skein algebras. Moreover, the applications are beyond low-dimensional topology. For example, we can access the representation theory of unrestricted quantum moduli algebras, and that of quantum higher cluster algebras potentially.

A bstract We develop a general formalism to describe the Renormalization Group Flow of Schur indices and fusion algebras of BPS line defects in four-dimensional 𝒩 = 2 Supersymmetric Quantum Field Theories. The formalism includes and extends known results about the Seiberg-Witten description of these structures. Another application of the formalism is to describe the spectrum of BPS partices of 𝒩 = 2 gauge theories with matter in terms of the spectrum of pure 𝒩 = 2 gauge theories. Applications to the theory of quantum groups and to the quantization of cluster varieties are also discussed.

Quantum metrics from length functions on quantum groups

Mančinska and Roberson [FOCS&amp;apos;20] showed that two graphs are quantum isomorphic if and only if they admit the same number of homomorphisms from any planar graph. Atserias et al. [JCTB&amp;apos;19] proved that quantum isomorphism is undecidable in general, which motivates the study of its relaxations. In the classical setting, Roberson and Seppelt [ICALP&amp;apos;23] characterized the feasibility of each level of the Lasserre hierarchy of semidefinite programming relaxations of graph isomorphism in terms of equality of homomorphism counts from an appropriate graph class. The NPA hierarchy, a noncommutative generalization of the Lasserre hierarchy, provides a sequence of semidefinite programming relaxations for quantum isomorphism. In the quantum setting, we show that the feasibility of each level of the NPA hierarchy for quantum isomorphism is equivalent to equality of homomorphism counts from an appropriate class of planar graphs. Combining this characterization with the convergence of the NPA hierarchy, and noting that the union of these classes is the set of all planar graphs, we obtain a new proof of the result of Mančinska and Roberson [FOCS&amp;apos;20] that avoids the use of quantum groups. Moreover, this homomorphism indistinguishability characterization also yields a randomized polynomial-time algorithm deciding exact feasibility of each fixed level of the NPA hierarchy of SDP relaxations for quantum isomorphism.

We give a new proof for the description of the blocks in the category of representations of a reductive algebraic group G over a field of positive characteristic ℓ (originally due to Donkin), by working in the Satake category of the Langlands dual group and applying Smith–Treumann theory as developed by Riche and Williamson. On the representation theoretic side, our methods enable us to give a bound for the length of a minimum chain linking two weights in the same block, and to give a new proof for the block decomposition of a quantum group at an ℓ th root of unity.

Abstract Suppose a Lie group G acts on a vertex algebra $$\mathcal {V}$$ V . In this article we construct a vertex algebra $${\tilde{V}}$$ V ~ , which is an extension of $$\mathcal {V}$$ V by a big central vertex subalgebra identified with the algebra of functionals on the space of regular $$\mathfrak {g}$$ g -connections $$(\textrm{d}+A)$$ ( d + A ) . The category of representations of $${\tilde{\mathcal {V}}}$$ V ~ fibres over the set of connections, and the fibres should be viewed as $$(\textrm{d}+A)$$ ( d + A ) -twisted modules of $$\mathcal {V}$$ V , generalizing the familiar notion of g -twisted modules. In fact, another application of our result is that it proposes an explicit definition of $$(\textrm{d}+A)$$ ( d + A ) -twisted modules of $$\mathcal {V}$$ V in terms of a twisted commutator formula, and we feel that this subject should be pursued further. Vertex algebras with big centers appear in practice as critical level or large level limits of vertex algebras. In particular, we have in mind limits of the generalized quantum Langlands kernel, in which case G is the Langland dual and $$\mathcal {V}$$ V is conjecturally the Feigin-Tipunin vertex algebra and the extension $${\tilde{\mathcal {V}}}$$ V ~ is conjecturally related to the Kac-DeConcini-Procesi quantum group with big center. With the current article, we can give a uniform and independent construction of these limits.

Abstract Given an action by a finite quantum group $\mathbb {G}$ on a von Neumann algebra M , we prove that a number of familiar $W^*$ properties are equivalent for M and the fixed-point algebra $M^{\mathbb {G}}$ (i.e., hold or not simultaneously for the two algebras); these include being hyperfinite, atomic, diffuse, and of type I , $II$ , or $III$ . Moreover, in all cases, the canonical central projections of M and $M^{\mathbb {G}}$ cutting out the summand with the respective property coincide. The result generalizes its classical- $\mathbb {G}$ analog due to Jones–Takesaki.

We revisit the quantum de Finetti theorem. We state and prove a couple of variants thereof. In parallel, we introduce an operator version of the Martin boundary on quantum groups and prove generalizations of Biane’s theorem. Our proof of the de Finetti theorem is new in the sense that it is based on an analogy with the theory of operator-valued Martin boundary that we introduce.