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