Abstract: Let $$ \mathfrak{a} $$ be a finite-dimensional Lie algebra and $$ Y\left( \mathfrak{a} \right) $$ the $$ \mathfrak{a} $$ invariant subalgebra of its symmetric algebra $$ S\left( \mathfrak{a} \right) $$ under adjoint action. Recently there has been considerable interest in studying situations when $$ Y\left( \mathfrak{a} \right) $$ may be polynomial on index $$ \mathfrak{a} $$ generators, for example if $$ \mathfrak{a} $$ is a biparabolic or a centralizer $$ {\mathfrak{g}^x} $$ in a semisimple Lie algebra $$ \mathfrak{g} $$. Here a sum rule of Ooms and Van den Bergh on the degrees d i of the generators is extended to the case when $$ \mathfrak{a} $$ is unimodular and the fundamental semi-invariant $$ {p_\mathfrak{a}} $$ of $$ \mathfrak{a} $$ is an invariant. It is noted that these last two conditions always hold for a centralizer $$ {\mathfrak{g}^x} $$ and moreover that one has GKdim $$ Y\left( {{\mathfrak{g}^x}} \right) $$ = index $$ {\mathfrak{g}^x} $$, even though $$ S\left( {{\mathfrak{g}^x}} \right) $$ may admit proper semi-invariants. Applications are given to the notion of an adapted pair (h, η) for $$ \mathfrak{a} $$. Under the above conditions it is shown that the set of eigenvalues {m i} of -ad h on $$ {\mathfrak{a}^\eta } $$ must equal {(d i - 1)}. This gives rise to a condition under which $$ Y\left( \mathfrak{a} \right) $$ cannot be polynomial on index $$ \mathfrak{a} $$ generators. Finally the theorem of Bolsinov is extended to the case when $$ {p_\mathfrak{a}} $$ is nonscalar.

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{{Explor lien
   |wiki=    Wicri/Mathematiques
   |area=    BourbakiV1
   |flux=    Istex
   |étape=   Corpus
   |type=    RBID
   |clé=     ISTEX:1B8DBEFFE1EAC7A2EDF4CEDAE8DC11BCECDB299D
   |texte=   Polynomiality of invariants, unimodularity and adapted pairs
}}