Bethe Subalgebras in Twisted Yangians

We study analogues of the Yangian of the Lie algebra glN for the other classical Lie algebras soN and spN . We call them twisted Yangians. They are coideal subalgebras in the Yangian of glN and admit homomorphisms onto the universal enveloping algebras U(soN ) and U(spN ) 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(soN ) and U(spN ) of these subalgebras are also maximal commutative. Introduction In this article we study the Yangian of the Lie algebra glN and its analogues for the other classical Lie algebras soN and spN . The Yangian Y(glN ) is a deformation of the universal enveloping algebra U(glN [t ]) in the class of Hopf algebras [D1]. Moreover, it contains the universal enveloping algebra U(glN ) as a subalgebra and admits a homomorphism π : Y(glN ) → U(glN ) identical on U(glN ) . Let aN be one of the Lie algebras soN and spN . In [D1] the Yangian Y(aN ) was defined as a deformation of the Hopf algebra U(aN [t ]) . It contains U(aN ) as a subalgebra but does not admit a homomorphism Y(aN ) → U(aN ) identical on U(aN ) . In the present article we consider another analogue of the Yangian Y(glN ) for the classical Lie algebra aN . It has been introduced in [O2] and called twisted Yangian; see also [MNO]. The definition in [O2] was motivated by [O1] and [C2 , S]. Algebras closely related to this analogue of Y(glN ) were recently studied in [NS]. Consider aN as a fixed point subalgebra in the Lie algebra glN with respect to an involutive automorphism σ . The twisted Yangian Y(glN , σ) is a subalgebra in Y(glN ) . Moreover, it is a left coideal in the Hopf algebra Y(glN ) . It also contains U(aN ) as a subalgebra and does admit a homomorphism ρ : Y(glN , σ) → U(aN ) identical on U(aN ) ; see Section 3. The algebra Y(glN , σ) is a deformation of the universal enveloping algebra for the twisted current Lie algebra { F (t) ∈ glN [t ] | σ ( F (t) ) = F (−t) } . There is a remarkable family of maximal commutative subalgebras in Y(glN ) . They are parametrized by the regular semisimple elements of glN . As well as the Yangian Y(glN ) itself, these subalgebras were studied in the works by mathematical physicists from St. Petersburg on Bethe Ansatz; see for instance [KBI] and [KR ,KS]. These subalgebras were also studied in [C1]. We will call them Bethe subalgebras. In Section 1 of the present article we recall their definition. Their images in U(glN ) with respect to the homomophism π are also maximal commutative; see Section 2. The main aim of this article is to construct analogues of the Bethe subalgebras in Y(glN ) for the twisted Yangian Y(glN , σ) . In Section 3 for any element Z ∈ aN we construct certain commutative subalgebra in Y(glN , σ) . This construction is a generalization of one result from [S]. If the element Z ∈ aN is regular semisimple then the corresponding commutative subalgebra in Y(glN , σ) is maximal. Moreover, the image of this subalgebra in U(aN ) with respect to the homomorphism ρ is also maximal commutative; see Section 4. This image in U(aN ) is a quantization of a maximal involutive subalgebra in the Poisson algebra S(aN ) obtained by the so called shift of argument method; see [K2] and [MF]. For further details on the involutive subalgebras in S(aN ) obtained by this method see for instance [RS]. Some results on the quantization of these subalgebras can be found in [V]. We are indebted to M.Räıs who explained to us that the methods of [K1] can be applied to the current Lie algebras; see [RT]. Together with [M] and [MNO] the present article is a part of a project on representation theory of Yangians initiated by [O1 ,O2]. It is our joint project with A.Molev, and we are grateful to him for collaboration. 1. Bethe subalgebras in Yangians We will start this section with recalling several known facts from [D1] and [KR ,KS] about the Yangian of the Lie algebra glN ; see also [MNO , Sections 1–2]. This is a complex associative unital algebra Y(glN ) with the countable set of generators T (r) ij where r = 1, 2, . . . and i, j = 1, . . . , N . The defining relations in the algebra Y(glN ) are (1.1) [T (p+1) ij , T (q) kl ]− [T (p) ij , T (q+1) kl ] = T (p) kj T (q) il − T (q) kj T (p) il ; p, q = 0, 1, 2, . . . where T (0) ij = δij · 1 . The collection (1.1) is equivalent to the collection of relations (1.2) [T (p) ij , T (q) kl ] = min(p,q) ∑ r=1 ( T (r−1) kj T (p+q−r) il −T (p+q−r) kj T (r−1) il ) ; p, q = 1, 2, . . . . Let Eij ∈ End(C N ) be the standard matrix units. We will also use the following matrix form of the relations (1.1). Introduce a formal variable u and consider the Yang R -matrix