Figure 45

Figure 45: Level decomposition of the adjoint representation ℛad = 8 of 𝔰𝔩(3,ℝ) into representations of the subalgebra 𝔰𝔩(2,ℝ ). The labels 1 and 2 indicate the simple roots α1 and α 2. Level zero corresponds to the horizontal axis where we find the adjoint representation (0) ℛ ad = 30 of 𝔰𝔩(2,ℝ ) (red nodes) and the singlet representation (0) ℛs = 10 (green circle about the origin). At level one we find the two-dimensional representation (1) ℛ = 21 (green nodes). The black arrow denotes the negative level root − α2 and so gives rise to the level ℓ = − 1 representation ℛ (−1) = 2 (− 1). The blue arrows represent the fundamental weights Λ1 and Λ2.