Bethe subalgebras in twisted Yangians

Abstract

We study analogues of the Yangian of the Lie algebra$\mathfrak{g}\mathfrak{l}_N$ for the other classical Lie algebras$\mathfrak{s}\mathfrak{o}_N$ and$\mathfrak{s}\mathfrak{p}_N$. We call them twisted Yangians. They are coideal subalgebras in the Yangian of$\mathfrak{g}\mathfrak{l}_N$ and admit homomorphisms onto the universal enveloping algebras U($\mathfrak{s}\mathfrak{o}_N$) and U($\mathfrak{s}\mathfrak{p}_N$) respectively. In every twisted Yangian we construct a family of maximal commutative subalgebras parametrized by the regular semisimple elements of the corresponding classical Lie algebra. The images in U($\mathfrak{s}\mathfrak{o}_N$) and U($\mathfrak{s}\mathfrak{p}_N$) of these subalgebras are also maximal commutative.