Coxeter Transformations in Quantum Groups

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

Ringel–Hall algebras and Lusztig's symmetries

Quantum groups have been studied intensively for the last two decades from various points of view. The underlying mathematical structure is that of an algebra with a coproduct. Compact quantum groups admit Haar measures. However, if we want to have a Haar measure also in the noncompact case, we are forced to work with algebras without identity, and the notion of a coproduct has to be adapted. These considerations lead to the theory of multiplier Hopf algebras, which provides the mathematical tool for studying noncompact quantum groups with Haar measures. I will concentrate on the *-algebra case and assume positivity of the invariant integral. Doing so, I create an algebraic framework that serves as a model for the operator algebra approach to quantum groups. Indeed, the theory of locally compact quantum groups can be seen as the topological version of the theory of quantum groups as they are developed here in a purely algebraic context.

We introduce the notion of integrable modules over $\imath $ quantum groups (a.k.a. quantum symmetric pair coideal subalgebras). After determining a presentation of such modules, we prove that each integrable module over a quantum group is integrable when restricted to an $\imath $ quantum group. As an application, we show that the space of matrix coefficients of all simple integrable modules over an $\imath $ quantum group of finite type with specific parameters coincides with Bao-Song’s coordinate ring of the $\imath $ quantum group.

It is known that Green's formula over finite fields gives rise to the comultiplications of Ringel–Hall algebras and quantum groups (see [Invent. Math. 120 (1995), 361–377], see also [J. Amer. Math. Soc. 4 (1991), 365–421]). In this chapter, we prove a projective version of Green's formula in a geometric way. Then following the method of Hubery in [Hubery, Acyclic cluster algebras via Ringel–Hall algebras (preprint)], we apply this formula to proving Caldero–Keller's multiplication formula for acyclic cluster algebras of arbitrary type.

We show that the operation of taking generic extensions provides the set of isomorphism classes of representations of a quiver of Dynkin type with a monoid structure. Its monoid ring is isomorphic to the specialization at q = 0 q=0 of Ringel’s Hall algebra. This provides the latter algebra with a multiplicatively closed basis. Using a crystal-type basis for a two-parameter quantum group, this multiplicative basis is related to Lusztig’s canonical basis.

Editorial

Generalized tilting functors and Ringel–Hall algebras

Let $\mathsf{u}_q(\mathfrak{g})$ be the small quantum group associated with a complex semisimple Lie algebra $\mathfrak{g}$ and a primitive root of unity q, satisfying certain restrictions. We establish the equivalence between three different actions of $\mathfrak{g}$ on the center of $\mathsf{u}_q(\mathfrak{g})$ and on the higher derived center of $\mathsf{u}_q(\mathfrak{g})$. Based on the triviality of this action for $\mathfrak{g} = \mathfrak{sl}_2, \mathfrak{sl}_3, \mathfrak{sl}_4$, we conjecture that, in finite type A, central elements of the small quantum group $\mathsf{u}_q(\mathfrak{sl}_n)$ arise as the restriction of central elements in the big quantum group $\mathsf{U}_q(\mathfrak{sl}_n)$. We also study the role of an ideal $\mathsf{z}_{\mathrm{Hig}}$ known as the Higman ideal in the center of $\mathsf{u}_q(\mathfrak{g})$. We show that it coincides with the intersection of the Harish-Chandra center and its Fourier transform, and compute the dimension of $\mathsf{z}_{\mathrm{Hig}}$ in type A. As an illustration we provide a detailed explicit description of the derived center of $\mathsf{u}_q(\mathfrak{sl}_2)$ and its various symmetries.

For quantum symmetric pairs $(\textbf {U}, \textbf {U}^\imath )$ of Kac–Moody type, we construct $\imath$-canonical bases for the highest weight integrable $\textbf U$-modules and their tensor products regarded as $\textbf {U}^\imath$-modules, as well as an $\imath$-canonical basis for the modified form of the $\imath$-quantum group $\textbf {U}^\imath$. A key new ingredient is a family of explicit elements called $\imath$-divided powers, which are shown to generate the integral form of $\dot {\textbf {U}}^\imath$. We prove a conjecture of Balagovic–Kolb, removing a major technical assumption in the theory of quantum symmetric pairs. Even for quantum symmetric pairs of finite type, our new approach simplifies and strengthens the integrality of quasi-$K$-matrix and the constructions of $\imath$-canonical bases, by avoiding a case-by-case rank-one analysis and removing the strong constraints on the parameters in a previous work.

Let $\Sigma$ be a finite type surface, and $G$ a complex algebraic simple Lie group with Lie algebra $\mathfrak{g}$. The quantum moduli algebra of $(\Sigma,G)$ is a quantization of the ring of functions of $X_G(\Sigma)$, the variety of $G$-characters of $\pi_1(\Sigma)$, introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche in the mid '90s. It can be realized as the invariant subalgebra of so-called graph algebras, which are $U_q(\mathfrak{g})$-module-algebras associated to graphs on $\Sigma$, where $U_q(\mathfrak{g})$ is the quantum group corresponding to $G$. We study the structure of the quantum moduli algebra in the case where $\Sigma$ is a sphere with $n+1$ open disks removed, $n\geq 1$, using the graph algebra of the "daisy" graph on $\Sigma$ to make computations easier. We provide new results that hold for arbitrary $G$ and generic $q$, and develop the theory in the case where $q=\epsilon$, a primitive root of unity of odd order, and $G={\rm SL}(2,{\mathbb C})$. In such a situation we introduce a Frobenius morphism that provides a natural identification of the center of the daisy graph algebra with a finite extension of the coordinate ring $\mathcal{O}(G^n)$. We extend the quantum coadjoint action of De-Concini-Kac-Procesi to the daisy graph algebra, and show that the associated Poisson structure on the center corresponds by the Frobenius morphism to the Fock-Rosly Poisson structure on $\mathcal{O}(G^n)$. We show that the set of fixed elements of the center under the quantum coadjoint action is a finite extension of ${\mathbb C}[X_G(\Sigma)]$ endowed with the Atiyah-Bott-Goldman Poisson structure. Finally, by using Wilson loop operators we identify the Kauffman bracket skein algebra $K_{\zeta}(\Sigma)$ at $\zeta:={\rm i}\epsilon^{1/2}$ with this quantum moduli algebra specialized at $q=\epsilon$.

On the Vassiliev knot invariants

Let $\mathfrak {g}\neq \mathfrak {s}\mathfrak {o}_{8}$ be a simple Lie algebra of type $A,D,E$ with $\widehat {\mathfrak {g}}$ the corresponding affine Kac–Moody algebra and $\mathfrak {n}_{+}\subset \widehat {\mathfrak {g}}$ is the standard positive nilpotent subalgebra. Given $\mathfrak {n}_{+}$ as above, we provide an infinite collection of cyclic finite abelian subgroups of $SL_{3}(\mathbb {C})$ with the following properties. Let $G$ be any group in the collection, $Y=G\operatorname {-}\mbox {Hilb}(\mathbb {C}^{3})$, and $\Psi : D^{b}_{G}(Coh(\mathbb {C}^{3}))\rightarrow D^{b}(Coh(Y))$ the derived equivalence of Bridgeland, King, and Reid. We present an (explicitly described) subset of objects in $Coh_{G}(\mathbb {C}^{3})$, s.t. the Hall algebra generated by their images under $\Psi $ is isomorphic to $U(\mathfrak {n}_{+})$. In case the field $\Bbbk $ (in place of $\mathbb {C}$) is finite and $\mbox {char}(\Bbbk )$ is coprime with the order of $G$, we establish isomorphisms of the corresponding “counting” Ringel–Hall algebras and the specializations of quantized universal enveloping algebras $U_{v}(\mathfrak {n}_{+})$ at $v=\sqrt {|\Bbbk |}$.

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.

Let g be a finite-dimensional complex simple Lie algebra and be the finitedimensional hereditary algebra associated to g.Let U + r,s (g) (respectively U ≥0 r,s (g)) denote the two-parameter quantized enveloping algebra of the positive maximal nilpotent (respectively Borel) Lie subalgebra of g.We study the two-parameter quantized enveloping algebras U + r,s (g) and U ≥0 r,s (g) using the approach of Ringel-Hall algebras.First of all, we show that U + r,s (g) is isomorphic to a certain two-parameter twisted Ringel-Hall algebra H r,s ( ), which generalizes a result of Reineke.Based on detailed computations in H r,s ( ), we show that H r,s ( ) can be presented as an iterated skew polynomial ring.As an result, we obtain a PBW-basis for H r,s ( ), which can be further used to construct a PBW-basis for the two-parameter quantized enveloping algebra U r,s (g).We also show that all prime ideals of U + r,s (g) are completely prime under some mild conditions on the parameters r, s.Second, we study the two-parameter extended Ringel-Hall algebra H r,s ( ).In particular, we define a Hopf algebra structure on H r,s ( ); and we prove that U ≥0 r,s (g) is isomorphic as a Hopf algebra to the two-parameter extended Ringel-Hall algebra H r,s ( ).