Segal-Sugawara vectors for the Lie algebra of type \(G_2\). (English) Zbl 1338.17023
Summary: Explicit formulas for Segal-Sugawara vectors associated with the simple Lie algebra \(\mathfrak{g}\) of type \(G_2\) are found by using computer-assisted calculations. This leads to a direct proof of the Feigin-Frenkel theorem describing the center of the corresponding affine vertex algebra at the critical level. As an application, we give an explicit solution of Vinberg’s quantization problem by providing formulas for generators of maximal commutative subalgebras of \(\operatorname{U}(\mathfrak{g})\). We also calculate the eigenvalues of the Hamiltonians on the Bethe vectors in the Gaudin model associated with \(\mathfrak{g}\).
MSC:
| 17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |
| 17B35 | Universal enveloping (super)algebras |
Software:
FORMReferences:
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