Index of Inönü–Wigner Contractions of Semisimple Lie Algebras

We give an explicit formula for the index of a Lie algebra of the shape \( {\mathfrak{g}} _0= {\mathfrak{h}} \oplus( {\mathfrak{g}} / {\mathfrak{h}} )^{ \text{ab} }\), where \( {\mathfrak{g}} \) is a semisimple Lie algebra, \( {\mathfrak{h}} \) is a subalgebra in \( {\mathfrak{g}} \) regarded as a subalgebra in \( {\mathfrak{g}} _0\), and \(( {\mathfrak{g}} / {\mathfrak{h}} )^{ \text{ab} }\) is an \( {\mathfrak{h}} \)-module \( {\mathfrak{g}} / {\mathfrak{h}} \) regarded as an Abelian ideal of \( {\mathfrak{g}} _0\). This formula has applications to Poisson commutative subalgebras in the symmetric algebra \( \operatorname{S} ( {\mathfrak{g}} )\) and to completely integrable systems.

DOI 10.1134/S1061920825600199