Publication

Quantum loop groups for symmetric Cartan matrices

2026

Abstract

We introduce a quantum loop group associated to a general symmetric Cartan matrix, by imposing just enough relations between the usual generators {e_{i,k},f_{i,k}}_{i\in I,,k\in\mathbb{Z}} in order for the natural Hopf pairing between the positive and negative halves of the quantum loop group to be perfect. As an application, we describe the localized 𝐾-theoretic Hall algebra of any quiver without loops, endowed with a particularly important \mathbb{C}^{*} action.

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2026

Abstract

Official source

Related concepts (18)

Loop quantum gravity

Loop quantum gravity (LQG) is a theory of quantum gravity, which aims to reconcile quantum mechanics and general relativity, incorporating matter of the Standard Model into the framework established for the intrinsic quantum gravity case. It is an attempt to develop a quantum theory of gravity based directly on Einstein's geometric formulation rather than the treatment of gravity as a mysterious mechanism (force). As a theory LQG postulates that the structure of space and time is composed of finite loops woven into an extremely fine fabric or network.

Cartan decomposition

In mathematics, the Cartan decomposition is a decomposition of a semisimple Lie group or Lie algebra, which plays an important role in their structure theory and representation theory. It generalizes the polar decomposition or singular value decomposition of matrices. Its history can be traced to the 1880s work of Élie Cartan and Wilhelm Killing. Let be a real semisimple Lie algebra and let be its Killing form. An involution on is a Lie algebra automorphism of whose square is equal to the identity.

Quantum gravity

Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics. It deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vicinity of black holes or similar compact astrophysical objects, such as neutron stars as well as in the early stages of the universe moments after the Big Bang Three of the four fundamental forces of nature are described within the framework of quantum mechanics and quantum field theory: the electromagnetic interaction, the strong force, and the weak force; this leaves gravity as the only interaction that has not been fully accommodated.