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.