A263005 Dimensions of the simple Lie algebras over complex numbers (with repetitions), sorted nondecreasingly.

3, 8, 10, 14, 15, 21, 21, 24, 28, 35, 36, 36, 45, 48, 55, 55, 57, 63, 66, 78, 78, 78, 80, 91, 99, 105, 105, 120, 120, 133, 136, 136, 143, 153, 168, 171, 171, 190, 195, 210, 210, 224, 231, 248, 253, 253, 255, 276, 288, 300, 300

Offset: 1

Author

Wolfdieter Lang, Oct 23 2015

Keywords

• nonn
• easy

Comments

This sequence gives the dimensions of the (compact) simple Lie algebras A_l, l >= 1, B_l, l >= 2, C_l >= 3, D_l, l >= 4, E_6, E_7, E_8, F_4 and G_2 which are l*(l+2), l*(2*l + 1), l*(2*l + 1), l*(2*l - 1), 78, 133, 248, 52 and 14, respectively. These are also the dimensions of the adjoint representations of these Lie algebras. For the l-ranges see the Humphreys reference, p. 58, and for the dimensions, e.g., the Samelson link, Theorem A, p. 74.

The dimension duplications occur for the B_l and C_l series for l >= 3.

References

• E. Cartan, Sur la structure des groupes de transformation finis et continus. Thèse Paris 1894. Oeuvres Complètes, I,1, pp. 137-287, Paris 1952.
• J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer, 1972.

Links

• W. Killing, Die Zusammensetzung der stetigen endlichen Transformationsgruppen, Mathematische Ann. I: 31 (1888) 252-290, II: 33 (1889) 1-48, III: 34 (1889) 57-122, IV: 36 (1890) 161-189: I, II, III, IV.
• Hans Samelson, Notes on Lie Algebras.

Crossrefs

Cf. A104599, A121214, A121732, A121736, A121737, A121738, A121739, A121741.