arXiv subjects

Ivan Chi-Ho Ip

Publications and source records attributed to Ivan Chi-Ho Ip.

At least 19 recordsLinked to original sources

arXiv2026

On the expansion formulas of cluster varieties from surfaces and their combinatorial properties

This paper explores the cluster algebra structure of the moduli space $\mathscr{A}_{\mathrm{SL}_{n+1},\mathbb{S}}$ of twisted $\mathrm{SL}_{n+1}$-local systems on a surface. We derive general recurrence relations for cluster variables arising from flips of a triangulation, corresponding to specific sequences of mutations. Our approach is grounded in a detailed combinatorial analysis over the standard $n$-triangulated $m$-gon (with explicit calculations for $n=1,2$). As a generalization, the non-simply-laced $G_2$ type is also considered. We prove the "well-triangulated" property for cluster mutations under flips, which provides a combinatorial framework for understanding the stability and transformation rules of these cluster algebra structures, and compute the monomial counts for the cluster expansion formula.

math.CO↗

arXiv2023

On the polynomiality conjecture of cluster realization of quantum groups

In this paper, we give a sufficient and necessary condition for a regular element of a quantum cluster algebra $\mathcal{O}_q(\mathcal{X})$ to be universally polynomial. This resolves several conjectures by the first author on the polynomiality of the cluster realization of quantum group generators in different families of positive representations.

math.QA↗

arXiv2022

Quantum cluster mutations and reduced word graphs

We give an algebraic proof of the independence of Coxeter moves involved in the construction of positive representations of split-real quantum groups, thus completing a gap in the original construction. To do this, we propose a new quantized version of Lusztig's Injectivity Lemma in the language of quantum cluster algebra, the proof of which by Tits' Lemma reduces to calculations involving sequences of Coxeter moves forming rank 3 cycles. We give a new, constructive proof of Tits' Lemma, and provide the required explicit computation of the quantum cluster mutations under these rank 3 cycles using certain cluster algebraic tricks via universally Laurent polynomials.

math.QA↗

arXiv2022

Positive Representations with Zero Casimirs

In this paper, we construct a new family of generalization of the positive representations of split-real quantum groups based on the degeneration of the Casimir operators acting as zero on some Hilbert spaces. It is motivated by a new observation arising from modifying the representation in the simplest case of $\mathcal{U}_q(\mathfrak{sl}(2,\mathbb{R}))$ compatible with Faddeev's modular double, while having a surprising tensor product decomposition. For higher rank, the representations are obtained by the polarization of Chevalley generators of $\mathcal{U}_q(\mathfrak{g})$ in a new realization as universally Laurent polynomials of a certain skew-symmetrizable quantum cluster algebra. We also calculate explicitly the Casimir actions of the maximal $A_{n-1}$ degenerate representations of $\mathcal{U}_q(\mathfrak{g}_\mathbb{R})$ for general Lie types based on the complexification of the central parameters.

math.QA↗

arXiv2020

Parabolic Positive Representations of $\mathcal{U}_q(\mathfrak{g}_\mathbb{R})$

We construct a new family of irreducible representations of $\mathcal{U}_q(\mathfrak{g}_\mathbb{R})$ and its modular double by quantizing the classical parabolic induction corresponding to arbitrary parabolic subgroups, such that the generators of $\mathcal{U}_q(\mathfrak{g}_\mathbb{R})$ act by positive self-adjoint operators on a Hilbert space. This generalizes the well-established positive representations which corresponds to induction by the minimal parabolic (i.e. Borel) subgroup. We also study in detail the special case of type $A_n$ acting on $L^2(\mathbb{R}^n)$ with minimal functional dimension, and establish the properties of its central characters and universal $\mathcal{R}$ operator. We construct a positive version of the evaluation module of the affine quantum group $\mathcal{U}_q(\widehat{\mathfrak{sl}}_{n+1})$ modeled over this minimal positive representation of type $A_n$.

math.QA↗

arXiv2020

Positive Representations of Split Real Simply-laced Quantum Groups

We construct the positive principal series representations for $\mathcal{U}_q(\mathfrak{g}_\mathbb{R})$ where $\mathfrak{g}$ is of simply-laced type, parametrized by $\mathbb{R}_{\geq 0}^r$ where $r$ is the rank of $\mathfrak{g}$. We describe explicitly the actions of the generators in the positive representations as positive essentially self-adjoint operators on a Hilbert space, and prove the transcendental relations between the generators of the modular double. We define the modified quantum group $\mathbf{U}_{\mathfrak{q}\tilde{\mathfrak{q}}}(\mathfrak{g}_\mathbb{R})$ of the modular double and show that the representations of both parts of the modular double commute weakly with each other, there is an embedding into a quantum torus algebra, and the commutant contains its Langlands dual.

math.RT↗

arXiv2017

Cluster realization of positive representations of split real quantum Borel subalgebra

In our previous work, we studied the positive representations of split real quantum groups $\mathcal{U}_{q\tilde{q}}(\mathfrak{g}_\mathbb{R})$ restricted to its Borel part, and showed that they are closed under taking tensor products. However, the tensor product decomposition was only constructed abstractly using the GNS-representation of a $C^*$-algebraic version of the Drinfeld-Jimbo quantum groups. In this paper, using the recently discovered cluster realization of quantum groups, we write down the decomposition explicitly by realizing it as a sequence of cluster mutations in the corresponding quiver diagram representing the tensor product.

math.QA↗

arXiv2017

Cluster Realization of $U_q(\mathfrak{g})$ and Factorization of the Universal $R$-Matrix

For each simple Lie algebra $\mathfrak{g}$, we construct an algebra embedding of the quantum group $U_q(\mathfrak{g})$ into certain quantum torus algebra $D_\mathfrak{g}$ via the positive representations of split real quantum group. The quivers corresponding to $D_\mathfrak{g}$ is obtained from amalgamation of two basic quivers, where each of them is mutation equivalent to the cluster structure of the moduli space of framed $G$-local system on a disk with 3 marked points when $G$ is of classical type. We derive a factorization of the universal $R$-matrix into quantum dilogarithms of cluster variables, and show that conjugation by the $R$-matrix corresponds to a sequence of quiver mutations which produces the half-Dehn twist rotating one puncture about the other in a twice punctured disk.

math.QA↗

arXiv2016

On tensor products of positive representations of split real quantum Borel subalgebra $U_{q\tilde{q}}(b_R)$

We studied the positive representations $P_λ$ of split real quantum groups $U_{q\tilde{q}}(g_R)$ restricted to the Borel subalgebra $U_{q\tilde{q}}(b_R)$. We proved that the restriction is independent of the parameter $λ$. Furthermore, we prove that it can be constructed from the GNS-representation of the multiplier Hopf algebra $U_{q\tilde{q}}^{C^*}(b_R)$ constructed earlier, which enables us to decompose their tensor product using the theory of the "multiplicative unitary". This will be an essential ingredient in the construction of quantum higher Teichmüller theory from the perspective of representation theory, generalizing earlier work by Frenkel-Kim.

math.RT↗

arXiv2016

Positive representations of non-simply-laced split real quantum groups

We construct the positive principal series representations for $U_q(g_R)$ where $g$ is of type $B_n$, $C_n$, $F_4$ or $G_2$, parametrized by $R^r$ where $r$ is the rank of $g$. We show that under the representations, the generators of the Langlands dual group $U_{\tilde{q}}({}^Lg_R)$ are related to the generators of $U_q(g_R)$ by the transcendental relations. We define the modified quantum group of the modular double and show that the representations of both parts of the modular double commute with each other, and there is an embedding into the $q$-tori polynomials.

math.QA↗

arXiv2015

Positive Casimir and Central Characters of Split Real Quantum Groups

We describe the generalized Casimir operators and their actions on the positive representations $P_λ$ of the modular double of split real quantum groups $U_{q\tilde{q}}(g_R)$. We introduce the notion of virtual highest and lowest weights, and show that the central characters admit positive values for all parameters $λ$. We show that their image defines a semi-algebraic region bounded by real points of the discriminant variety independent of $q$, and we discuss explicit examples in the lower rank cases.

math.RT↗

arXiv2015

Q-operator and fusion relations for $C^{(2)}_q(2)$

The construction of the Q-operator for twisted affine superalgebra $C^{(2)}_q(2)$ is given. It is shown that the corresponding prefundamental representations give rise to evaluation modules some of which do not have a classical limit, which nevertheless appear to be a necessary part of fusion relations.

math.QA↗

arXiv2014

Quantum Dilogarithm Identities at Root of Unity

We study the root of unity degeneration of cluster algebras and quantum dilogarithm identities. We prove identities for the cyclic dilogarithm associated with a mutation sequence of a quiver, and as a consequence new identities for the non-compact quantum dilogarithm at $b=1$.

math.QA↗

arXiv2014

Positive representations, multiplier Hopf algebra, and continuous canonical basis

We introduce the language of multiplier Hopf algebra in the context of positive representations of split real quantum groups, and discuss its applications with a continuous version of Lusztig-Kashiwara's canonical basis, which may provide a key to prove the closure of the positive representations under tensor products, and harmonic analysis of quantized algebra of functions in the sense of locally compact quantum groups.

math.RT↗

arXiv2013

Supersymmetry and the Modular Double

A counterpart of the modular double for quantum superalgebra $\cU_q(\osp(1|2))$ is constructed by means of supersymmetric quantum mechanics. We also construct the $R$-matrix operator acting in the corresponding representations, which is expressed via quantum dilogarithm.

math.RT↗

arXiv2012

Positive Representations of Split Real Quantum Groups: The Universal R Operator

The universal $R$ operator for the positive representations of split real quantum groups is computed, generalizing the formula of compact quantum groups $U_q(g)$ by Kirillov-Reshetikhin and Levendorski\uı-Soibelman, and the formula in the case of $U_{q\tilde{q}}(sl(2,R))$ by Faddeev, Kashaev and Bytsko-Teschner. Several new functional relations of the quantum dilogarithm are obtained, generalizing the quantum exponential relations and the pentagon relations. The quantum Weyl element and Lusztig's isomorphism in the positive setting are also studied in detail. Finally we introduce a $C^*$-algebraic version of the split real quantum group in the language of multiplier Hopf algebras, and consequently the definition of $R$ is made rigorous as the canonical element of the Drinfeld's double $\textbf{U}$ of certain multiplier Hopf algebra $\textbf{Ub}$. Moreover a ribbon structure is introduced for an extension of $\textbf{U}$.

math.QA↗

arXiv2012

The Classical Limit of Representation Theory of the Quantum Plane

We showed that there is a complete analogue of a representation of the quantum plane B_q where |q|=1, with the classical ax+b group. We showed that the Fourier Transform of the representation of B_q on H=L^2(R) has a limit (in the dual co-representation) towards the Mellin transform of the unitary representation of the ax+b group, and furthermore the intertwiners of the tensor products representation has a limit towards the intertwiners of the Mellin transform of the classical ax+b representation. We also wrote explicitly the multiplicative unitary defining the quantum ax+b semigroup and showed that it defines the co-representation that is dual to the representation of B_q above, and also correspond precisely to the classical family of unitary representation of the ax+b group.

math.RT↗

arXiv2012

Gauss-Lusztig Decomposition for $GL_q^+(N,R)$ and Representation by q-Tori

We found an explicit construction of a representation of the positive quantum group $GL_q^+(N,\R)$ and its modular double $GL_{q\til[q]}^+(N,\R)$ by positive essentially self-adjoint operators. Generalizing Lusztig's parametrization, we found a Gauss type decomposition for the totally positive quantum group $GL_q^+(N,\R)$ for $|q|=1$, parametrized by the standard decomposition of the longest element $w_0\in W=S_{N-1}$. Under this parametrization, we found explicitly the relations between the standard quantum variables, the relations between the quantum cluster variables, and realizing them using non-compact generators of the $q$-tori $uv=q^2 vu$ by positive essentially self-adjoint operators. The modular double arises naturally from the transcendental relations, and an $L^2(GL_{q\til[q]}^+(N,\R))$ space in the von Neumann setting can also be defined.

math.QA↗