geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
Let be a finite-dimensional Euclidean vector space, i.e. a real vector space equipped with a positive-definite inner product . For convenience, given any vectors with , we define the notation . Notice that this is precisely twice the coefficient of when we take the orthogonal projection of onto the line generated by . We denote by the reflection in the hyperplane , which is given by the formula
A root system in is a finite set of nonzero vectors satisfying the following axioms:
spans .
If , then the reflection permutes .
(Integrality) If then .
If in addition for every we have then we say that the root system is reduced. Equivalently, a root system is reduced if each determines a unique reflection. Many authors require that root systems are reduced, but we will specify here when root systems are reduced or not.
The Weyl group of a root system in is the finite subgroup of the orthogonal group generate by the reflections , . Thus, the Weyl group can also be seen as a subgroup of the permutation group of .
Given root systems in and in , we define a morphism of root systems from to to be a linear map such that and for all .
Howard Georgi, §6 in: Lie Algebras In Particle Physics, Westview Press (1999), CRC Press (2019) [doi:10.1201/9780429499210]
with an eye towards application to (the standard model of) particle physics
Peter Woit, Topics in representation theory: Roots and weights (pdf)
The Unapologetic Mathematician, The category of root systems (blog)
Last revised on September 7, 2023 at 12:31:11. See the history of this page for a list of all contributions to it.