Primitive and Poisson spectra of single-eigenvalue twists of polynomial algebras.
Suppose that \(S=\bigoplus_{d\geqslant 0}S_d\) is a graded associative algebra generated by \(S_1\) with an automorphism \(\sigma\) of \(S_1\) as a linear space which can be extended to an automorphism \(\sigma\) of \(S\). Then there is a twisted algebra \(S^\sigma\) with multiplication defined as \(a*b=a\cdot\sigma(b)\) for homogeneous elements \(a,b\in S\). The author considers the case when \(S\) is the complex polynomial algebra on \(Y_1,\dots,Y_n\) and \(\sigma\) is defined as \(\sigma(Y_1)=Y_1\) and \(\sigma(Y_{i+1})=Y_{i+1}+Y_i\) for \(i>0\). There is given a classification of the primitive ideals in \(S^\sigma\) which do not contain \(Y_1\). In particular, every primitive ideal in \(S^\sigma\) is generated by a regular sequence of homogeneous elements \(g_1,\dots,g_t\) where each \(g_i\) is irreducible and \(\sigma\)-invariant modulo the ideal generated by the preceding elements. So there exists a one to one correspondence between primitive ideals of \(S^\sigma\) and symplectic leaves of the Poisson structure induced by \(\sigma\). This result can be generalized to the case when an automorphism of \(S\) has a single eigenvalue. In this case the leaves are algebraic and realizable by orbits of an algebraic group.
scientific article; zbMATH DE number 5125404