Knowlpedia

Construction

Root sl2\mathfrak{sl}_2-subalgebra

The canonical rank-one subalgebra generated by the root spaces for a root and its negative.

Core idea

Let g\mathfrak g be a complex with h\mathfrak h, and let α\alpha be a . Choose nonzero root vectors eα∈gαe_\alpha\in\mathfrak g_\alpha and fα∈g−αf_\alpha\in\mathfrak g_{-\alpha}, normalized so that

hα=[eα,fα],α(hα)=2.h_\alpha=[e_\alpha,f_\alpha],\qquad \alpha(h_\alpha)=2.

Then

sα=Ceα⊕Chα⊕Cfα≅sl2(C)\mathfrak s_\alpha =\mathbb C e_\alpha\oplus\mathbb C h_\alpha\oplus\mathbb C f_\alpha \cong\mathfrak{sl}_2(\mathbb C)

is the root sl2\mathfrak{sl}_2-subalgebra associated with α\alpha.

The sl2\mathfrak{sl}_2 relations

The normalization gives

[hα,eα]=2eα,[hα,fα]=−2fα,[eα,fα]=hα.[h_\alpha,e_\alpha]=2e_\alpha, \qquad [h_\alpha,f_\alpha]=-2f_\alpha, \qquad [e_\alpha,f_\alpha]=h_\alpha.

Thus (eα,hα,fα)(e_\alpha,h_\alpha,f_\alpha) is an sl2\mathfrak{sl}_2-triple. Replacing eαe_\alpha by a nonzero scalar multiple and fαf_\alpha by its inverse multiple changes the chosen basis but not the resulting subalgebra.

Action on the ambient Lie algebra

Restricting the of g\mathfrak g to sα\mathfrak s_\alpha organizes roots into finite α\alpha-strings. On a gβ\mathfrak g_\beta, the element hαh_\alpha acts with integral weight

β(hα)=⟨β,α∨⟩.\beta(h_\alpha)=\langle\beta,\alpha^\vee\rangle.

The raising and lowering operators eαe_\alpha and fαf_\alpha move between the root spaces indexed by β+α\beta+\alpha and β−α\beta-\alpha whenever those are roots. This rank-one representation theory is the source of the integrality and root-string restrictions in a crystallographic .

Scope

A root sl2\mathfrak{sl}_2-subalgebra depends on a Cartan subalgebra and a root. It is a special case of an sl2\mathfrak{sl}_2-triple, but not every sl2\mathfrak{sl}_2-subalgebra is presented as a root subalgebra for a fixed Cartan. The Jacobson–Morozov theorem concerns the more general construction of an sl2\mathfrak{sl}_2-triple from a nilpotent element.

References
  1. James E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, 1972, §§9–10 and 25. Publisher record.
  2. Jean-Pierre Serre, Complex Semisimple Lie Algebras, Springer, 1987, Chapter VI. Publisher record.