Roots of Unity Specializations Advance Understanding of Quantum Graph Algebras for Genus Surfaces

The mathematical structures underlying quantum graph algebras are now yielding to closer scrutiny, with implications for areas ranging from theoretical physics to computer science. Stéphane Baseilhac of IMAG, Univ Montpellier, CNRS, Matthieu Faitg from Univ Toulouse, CNRS, IMT, and Philippe Roche of IMAG, Univ Montpellier, CNRS, alongside their colleagues, have been investigating the behaviour of these algebras when specialised at roots of unity. Their work focuses on algebras linked to complex semi-simple algebraic groups and compact oriented surfaces, revealing properties of central localisations and invariant subalgebras. This research is significant because it demonstrates these localisations are central simple algebras with precisely determined properties, and establishes the integral closure of their centres, furthering understanding of their complex structure.

Their work focuses on algebras linked to complex semi-simple algebraic groups and compact oriented surfaces, revealing properties of central localisations and invariant subalgebras. This research is significant because it demonstrates these localisations are central simple algebras with precisely determined properties, and establishes the integral closure of their centres, furthering understanding of their complex structure.

The research utilises techniques from Hopf algebra theory, quantum group representation theory, and invariant theory to achieve these results. Special attention is given to the interplay between the coadjoint action and the resulting central localisations. The study builds upon existing knowledge of simple Lie algebras and their parameterisations with q and qD, utilising the specialisation of A-modules to facilitate the analysis of invariant elements.

Quantum Group Structures and Graph Algebra Specialisation Researchers

Organization of a domain is presented, followed by general facts on graph algebras. Section 3 details the definition of Lg,n(H), the quantum moment map, and L1,0(H) alongside the Heisenberg double. The Alekseev morphism and its representations are also explored within this section. Preliminaries on quantum groups are then outlined, beginning with the quantum enveloping algebra U Λ q (g), followed by discussion of the quantum function algebra Oq(G), integral forms of U Λ q (g) and Oq(G), and the specialization of Uq(g). The quantum Frobenius morphism and specialisation of Oq(G) are subsequently examined, concluding with a consideration of quasitriangular structures at roots of unity.

Quantum graph algebras at roots of unity form the basis of section 5, which covers the specialization of quantum graph algebras and the modified Alekseev morphism bΦg,n. The quantum moment map and invariant elements are then analysed, leading to structure results for Lε g,n. Further investigation extends to the PI degree of Q(Luε g,n). Appendices provide supplementary material, including the quantum Killing form and Φ0,1, alongside definitions relating to Lusztig’s modified quantum group and canonical basis. Proofs for Lemma 4.8 and the injectivity of ΦεD 1,0 are also included, culminating in a demonstration that Lε g,n is a domain.

Let G be a simply-connected complex semi-simple algebraic group, and g its Lie algebra. Let Σ◦ g,n be a compact oriented surface of genus g, with n punctures and one boundary component. The quantum graph algebra Lg,n is a quantization of the algebra of functions on the variety of representations HomGrp π1(Σ◦ g,n), G endowed with the Fock-Rosly Poisson structure, initially introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche, and actively studied in connection with non-semisimple TQFTs.

Quantum Graph Algebra Centrality and PI Degrees Scientists

Scientists have achieved a significant breakthrough in the study of quantum graph algebras, specifically their specializations at roots of unity of odd order. The research focuses on algebras associated with simply-connected complex semi-simple algebraic groups and compact oriented surfaces with genus g, n punctures, and one boundary component. The central localizations of both the quantum graph algebra and its subalgebra of invariant elements are central simple algebras, possessing PI degrees. The team meticulously described the centers of these algebras, demonstrating they are integrally closed rings.

Measurements confirm the existence of a surjective morphism from the algebra Lε g,n to a related quantum graph algebra, Lg,n(uε), associated with a small quantum group uε. This connection is particularly noteworthy as Lg,n(uε) is known to be the algebra of endomorphisms of the state space for Kerler, Lyubashenko’s TQFT, suggesting potential implications for quantum topology. Results demonstrate that for the case of g = slm+1 and n = 0, the PI-degree of Lε g,0 could be derived using previously established isomorphisms between the quantum graph algebra and the corresponding skein algebra. The study provides a uniform approach applicable to all complex simple Lie algebras, avoiding the need for explicit generator and relation presentations.

This allows for the application of tools from invariant theory to investigate subalgebras of invariant elements, such as their centers. Further experiments established that the algebras Lε g,n and Luε g,n are domains, meaning they do not contain zero divisors, and their central localizations are indeed central simple algebras. Precise calculations were performed to determine their PI-degrees, and a complete description of their centers was achieved, confirming their integral closure. The breakthrough delivers a powerful new framework for studying representations of these algebras when specialized to roots of unity, with potential applications in TQFTs and the broader field of quantum topology.

Central Simple Algebras and PI Degree Determination

The authors have established that the central localizations of certain graph algebras, specifically those associated with simply-connected complex semi-simple algebraic groups and compact oriented surfaces, possess a defined algebraic structure. They demonstrate these localizations, and subalgebras invariant under coadjoint action, are central simple algebras with precisely determined PI degrees. Furthermore, the researchers characterise the centres of these algebras, proving they constitute integrally closed rings, a significant property in algebraic geometry. This work advances understanding of the algebraic structures arising from specializations at roots of unity, particularly concerning the behaviour of centralizers within these algebras.

The authors show how elements commute under specific conditions, leading to insights into invariance properties and the structure of centralizing subalgebras. They also establish a connection between the centralizer of a particular element and the centralizer of an iterated coproduct, refining existing results in the field. The authors acknowledge a limitation stemming from the fact that certain operations are not defined on the entirety of the relevant algebraic structures, necessitating the introduction of auxiliary constructions to achieve their results. Future research may focus on exploring the implications of these findings for other algebraic settings and potentially relaxing the current restrictions on the order of the root of unity. The demonstrated Noetherian properties of the algebras and their centres open avenues for further investigation into their module theory and representation theory.