Branching rule

A branching rule decomposes a representation into irreducible representations after restriction from a Lie group or Lie algebra to a subgroup or subalgebra.

Table of contents
- Adjoint branching to root sl2 subalgebras in rank two Branching rule

Adjoint branching to root sl2 subalgebras in rank two ( $g ↓_{sl_{2} (α)}$ )

A root string of length

Λ + 1

gives an irreducible sl2 Lie algebra module

R (Λ)

when restricting the Adjoint representation to the sl2 subalgebra associated with a root. Include also the root's own

R (2)

and one commuting Cartan direction

R (0)

. For

A_{2}

either simple root gives

R (2) \oplus 2 R (1) \oplus R (0)

. For

B_{2}

, the long root gives

R (2) \oplus 2 R (1) \oplus 3 R (0)

and the short root gives

3 R (2) \oplus R (0)

. For

G_{2}

, the long root gives

R (2) \oplus 4 R (1) \oplus 3 R (0)

and the short root gives

R (2) \oplus 2 R (3) \oplus 3 R (0)

.

 Ancestors (9)

 Incoming links (2)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version