Segal-Sugawara vectors for the Lie algebra of type $G_2$

Abstract

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 $U(\mathfrak{g})$. We also calculate the eigenvalues of the Hamiltonians on the Bethe vectors in the Gaudin model associated with $\mathfrak{g}$.

26 pages

References in corpus (6)

• Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence
• On higher order Sugawara operators
• Quantization of the shift of argument subalgebras in type A
• Classical W-algebras in types A, B, C, D and G
• Eigenvalues of Bethe vectors in the Gaudin model
• A new formula for Pfaffian-type Segal-Sugawara vector

Cited by in corpus (6)

• Center of the quantum affine vertex algebra in type A
• On Segal--Sugawara vectors and Casimir elements for classical Lie algebras
• Casimir elements and Sugawara operators for Takiff algebras
• On the Gaudin model of type G${}_2$
• Segal-Sugawara vectors for orthosymplectic Lie superalgebras
• Introduction to finite $W$-algebras