2000/06/13 by J. A. de Azcárraga, J. A. de Azcarraga, A. J. Macfarlane +1
Mathematics · Physics and Astronomy · #Adjoint representation #Advanced Topics in Algebra #Affine Lie algebra #Algebra over a field #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #Casimir effect #Casimir element #Cohomology #Combinatorics #Current algebra #Eigenvalues and eigenvectors #Fundamental representation #Lie algebra #Lie group #Mathematics #Physics #Pure mathematics #Quantum mechanics #Rank (graph theory) #Representation (politics) #Simple (philosophy) #hep-th #math-ph #math.GR #math.MP #math.RT
paper · pdf · doi:10.1063/1.1322076
Latex, 16 pages
arxiv created 2000/06/13 · openalex publication_date 2001/01/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
This paper deals with the striking fact that there is an essentially canonical path from the ith Lie algebra cohomology cocycle, i=1,2,…,l, of a simple compact Lie algebra 𝔤 of rank l to the definition of its primitive Casimir operators C(i) of order mi. Thus one obtains a complete set of Racah–Casimir operators C(i) for each 𝔤 and nothing else. The paper then goes on to develop a general formula for the eigenvalue c(i) of each C(i) valid for any representation of 𝔤, and thereby to relate c(i) to a suitably defined generalized Dynkin index. The form of the formula for c(i) for su(n) is known sufficiently explicitly to make clear some interesting and important features. For the purposes of illustration, detailed results are displayed for some classes of representation of su(n), including all the fundamental ones and the adjoint representation.