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European Mathematical Society Publishing House
2024-03-29 12:25:53
3
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=QT&vol=7&iss=3&update_since=2024-03-29
Quantum Topology
Quantum Topol.
QT
1663-487X
1664-073X
General
10.4171/QT
http://www.ems-ph.org/doi/10.4171/QT
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
7
2016
3
Skein and cluster algebras of marked surfaces
Greg
Muller
University of Michigan, ANN ARBOR, UNITED STATES
Cluster algebra, quantum cluster algebra, skein algebra, quantum torus, triangulation of surfaces
This paper considers several algebras associated to an oriented surface $\Sigma$ with a finite set of marked points on its boundary. The first is the skein algebra $\mathsf {Sk}_q(\Sigma)$, which is spanned by links in the surface which are allowed to have endpoints at the marked points, modulo several locally defined relations. The product is given by superposition of links. A basis of this algebra is given, as well as several algebraic results. When $\Sigma $ is triangulable, a quantum cluster algebra $\mathcal A_q(\Sigma)$ and quantum upper cluster algebra $\mathcal U_q(\Sigma)$ can be defined. These are algebras coming from the triangulations of $\Sigma$ and the elementary moves between them. Cluster algebras have been a subject of significant recent interest, due in part to their extraordinary positivity and Laurent properties. Natural inclusions $\mathcal A_q(\Sigma) \subseteq \mathsf {Sk}_q^o (\Sigma) \subseteq \mathcal U_q(\Sigma)$ are shown, where $\mathsf {Sk}_q^o(\Sigma)$ is a certain Ore localization of $\mathsf {Sk}_q(\Sigma)$. When $\Sigma$ has at least two marked points in each component, these inclusions are strengthened to equality, exhibiting a quantum cluster structure on $\mathsf {Sk}_q^o(\Sigma)$. The method for proving these equalities has the potential to show $\mathcal A_q = \mathcal U_q$ for other classes of cluster algebras. As a demonstration of this fact, a new proof is given that $\mathcal A_q = \mathcal U_q$ for acyclic cluster algebras.
Commutative rings and algebras
General
Associative rings and algebras
Manifolds and cell complexes
435
503
10.4171/QT/79
http://www.ems-ph.org/doi/10.4171/QT/79
The symplectic properties of the PGL($n,\mathbb C$)-gluing equations
Stavros
Garoufalidis
Georgia Institute of Technology, ATLANTA, UNITED STATES
Christian
Zickert
University of Maryland, COLLEGE PARK, UNITED STATES
ideal triangulations, generalized gluing equations, PGL($n,\mathbb C$)-gluing equations, shape coordinates, symplectic properties, Neumann-Zagier equations
In [12] we studied PGL($n,\mathbb C$)-representations of a 3-manifold via a generalization of Thurston’s gluing equations. Neumann has proved some symplectic properties of Thurston’s gluing equations that play an important role in recent developments of exact and perturbative Chern–Simons theory. In this paper, we prove similar symplectic properties of the PGL($n,\mathbb C$)-gluing equations for all ideal triangulations of compact oriented 3-manifolds.
Manifolds and cell complexes
Algebraic topology
505
551
10.4171/QT/80
http://www.ems-ph.org/doi/10.4171/QT/80
Quantum shuffles and quantum supergroups of basic type
Sean
Clark
Northeastern University, BOSTON, UNITED STATES
David
Hill
Washington State University, VANCOUVER, UNITED STATES
Weiqiang
Wang
University of Virginia, CHARLOTTESVILLE, UNITED STATES
Shuffle algebras, quantum supergroups, PBW bases, canonical bases
We initiate the study of several distinguished bases for the positive half of a quantum supergroup $U_q$ associated to a general super Cartan datum $(\mathrm{I}, (\cdot,\cdot))$ of basic type inside a quantum shuffle superalgebra. The combinatorics of words for an arbitrary total ordering on $\mathrm{I}$ is developed in connection with the root system associated to $\mathrm{I}$. The monomial, Lyndon, and PBW bases of $U_q$ are constructed, and moreover, a direct proof of the orthogonality of the PBW basis is provided within the framework of quantum shuffles. Consequently, the canonical basis is constructed for $U_q$ associated to the standard super Cartan datum of type $\mathfrak{gl}(n|1)$, $\mathfrak{osp}(1|2n)$, or $\mathfrak{osp}(2|2n)$ or an arbitrary non-super Cartan datum. In the non-super case, this refines Leclerc's work and provides a new self-contained construction of canonical bases. The canonical bases of $U_q$, of its polynomial modules, as well as of Kac modules in the case of quantum $\mathfrak{gl}(2|1)$ are explicitly worked out.
Nonassociative rings and algebras
Associative rings and algebras
553
638
10.4171/QT/81
http://www.ems-ph.org/doi/10.4171/QT/81