representation theory of 𝔰⁢𝔩2⁢ℂ

The special linear Lie algebra of 2×2 matricies, denoted by 𝔰⁢𝔩2⁢ℂ, is defined to be the span (over ℂ) of the matricies


with Lie bracket given by the commutatorPlanetmathPlanetmath of matricies: [X,Y]:=X⋅Y-Y⋅X. The matricies E,F,H satisfy the commutation relations: [E,F]=H,[H,E]=2⁢E,[H,F]=-2⁢F.

The representation theory of 𝔰⁢𝔩2⁢ℂ is a very important tool for understanding the structure theory and representation theory of other Lie algebrasMathworldPlanetmath (semi-simplePlanetmathPlanetmath finite dimensional Lie algebras, as well as infinite dimensional Kac-Moody Lie algebras).

The finite dimensional, irreducible, representations of 𝔰⁢𝔩2⁢ℂ are in bijection with the non-negative integers ℤ≥0 as follows. Let k∈ℤ≥0, V be a ℂ-vector spaceMathworldPlanetmath spanned by vectors v0,…,vk. The following action of E,H,F on V define the unique (up to isomorphismPlanetmathPlanetmathPlanetmathPlanetmath) irreducible representation of 𝔰⁢𝔩2⁢ℂ of dimensionPlanetmathPlanetmath k+1 (or of highest weight k):⁢(k-i+1)⁢vi-1 ∀ 1≤i≤⁢i)⁢vi ∀ 1≤i≤ ∀ 0≤i<kF.vk=0

The main points are that the one dimensional spaces ℂ⋅vi are eigenspacesMathworldPlanetmath for H with eigenvalueMathworldPlanetmathPlanetmathPlanetmathPlanetmath k-2⁢i, the operator corresponding to E kills v0 and otherwise sends ℂ⋅vi→ℂ⋅vi-1, while F kills vk and otherwise sends ℂ⋅vi→ℂ⋅vi+1. The operator corresponding to E is often called a raising operator since it raises the eigenvalue for H, and that of F is called a lowering operator since it lowers the eigenvalue for H.

𝔰⁢𝔩2⁢ℂ is a simple Lie algebra, thus by Weyl’s Theorem all finite dimensional representations for 𝔰⁢𝔩2⁢ℂ are completely reducible. So any finite dimensional representation of 𝔰⁢𝔩2⁢ℂ splits into a direct sumPlanetmathPlanetmathPlanetmath of irreducible representations for various non-negative integers as described above.

Title representation theory of 𝔰⁢𝔩2⁢ℂ
Canonical name RepresentationTheoryOfmathfraksl2mathbbC
Date of creation 2013-03-22 15:30:29
Last modified on 2013-03-22 15:30:29
Owner benjaminfjones (879)
Last modified by benjaminfjones (879)
Numerical id 7
Author benjaminfjones (879)
Entry type Definition
Classification msc 22E60
Classification msc 22E47
Defines sl_2
Defines special linear Lie algebra of 2x2 matricies