Maximal stable orders in the Lie algebra sp(2n) (Q922631)

Let R be a principal ideal ring of characteristic 0, K be a quotient field of R, A be a semisimple Lie algebra splitting over K. A lattice \(\Lambda\) in A is called an order, if [u,v]\(\in \Lambda\) for any u,v\(\in \Lambda\). An order \(\Lambda\) is called stable, if in any finite- dimensional A-module V there exists a \(\Lambda\)-invariant lattice. The description of stable orders in the Lie algebra \(A=sp(2n,K)\) is given. Let \(V=K^{2n}=<e_ 1,...,e_{2n}>\) be a standard A-module, \(L=<e_ 1,...,e_ n,d_ 1e_{n+1},...,d_ ne_{2n}>_ R\) a lattice in V. \(\Lambda (d_ 1,...,d_ n)=\{X\in A |\) XL\(\subset L\}\). It is proved that any maximal stable order in A is isomorphic to an order of the kind \(\Lambda (1,...,1,a_ 1,a_ 1a_ 2,...,a_ 1a_ 2...a_ m)\) where \(m=[n/2]\), \(a_ i\in R\), \(a_ i\) are square-free, \(a_ 1,...,a_ m\) are determined uniquely up to associated elements.