“Generalized” algebraic Bethe ansatz, Gaudin-type models and Z p -graded classical r-matrices

摘要

We consider quantum integrable systems associated with reductive Lie algebra g l ( n ) and Cartan-invariant non-skew-symmetric classical r -matrices. We show that under certain restrictions on the form of classical r -matrices “nested” or “hierarchical” Bethe ansatz usually based on a chain of subalgebras g l ( n ) ? g l ( n ? 1 ) ? . . . ? g l ( 1 ) is generalized onto the other chains or “hierarchies” of subalgebras. We show that among the r -matrices satisfying such the restrictions there are “twisted” or Z p -graded non-skew-symmetric classical r -matrices. We consider in detail example of the generalized Gaudin models with and without external magnetic field associated with Z p -graded non-skew-symmetric classical r -matrices and find the spectrum of the corresponding Gaudin-type hamiltonians using nested Bethe ansatz scheme and a chain of subalgebras g l ( n ) ? g l ( n ? n 1 ) ? g l ( n ? n 1 ? n 2 ) ? g l ( n ? ( n 1 + . . . + n p ? 1 ) ) , where n 1 + n 2 + . . . + n p = n .