International Congress of Mathematicians

Representations of Yangians 6492. Twisted Yangian of the orthogonal Lie algebra2.1. The associative algebra Y(gl N ,a) is a deformation of the universal envelopingalgebra of the twisted polynomial current Lie algebra{A(x) £ gl N [x] : a(A(xj) = A(-x)} .The deformation Y(gl N ,a) is not a Hopf algebra, but a coideai subalgebra in theHopf algebra Y(gljy). The definition of the twisted Yangian Y(gl N , a) was motivatedby the works of Cherednik [2] and Sklyanin [19] on quantum integrable systems withboundary conditions. This definition was given by Olshanski in [17].As in Subsection 1.1, let the indices i and j range over the set {1, ..., N}. Bydefinition,Y(gl N , a) is the subalgebra in Y(gl JV ) generated by the coefficients of allformal power seriesin x _1 . Due to (1.3), the subalgebra Y(gl N ,a)NJ2Tki(-x)Tkj(x) (2.1)k=iA(Y(gl JV ,a))cY(gl JV ,a)®Y(gl JV ).in Y(gl JV ) is a right coideal:To give the defining relations for the generators of Y(gl N ,a), introduce theextended twisted Yangian X(gl N ,a). The unital associative algebra X(gl N ,a) hasa family of generators Sy where a = 1,2,... . and i, j = 1, ..., N. PutSij(x) = öij • 1 + S^x- 1 + S^x- 2 + ... £ X(gl N ,a) [[x' 1 ]]. (2.2)Defining relations for the generators Sy of the algebra X (gljy, a) can be written as(x 2 - y 2 ) • [Sij(x),S M (y)] = (x + y)- (S kj (x)S a (y) - S kj (y)S a (x))- (x - y) • (Sik(x)Sji(y) - S ki (y)Sij(x)) + S ki (x)Sji(y) - S ki (y)Sji(x) -All these relations can be written as a single reflection equation, see [7]. One candefine a homomorphism TTN '• X(gl N ,a) —¥ Y(gl N ,a) by mapping the series Sij(x)to (2.1). The homomorpism TTN is surjective. As a two-sided ideal of X (gl N , a), thekernel of the homomorphism TTN is generated by the coefficients of all seriesSij(x) + (2x - l)Sij(-x) - 2xSji(x) (2.3)in x^1.This ideal is also generated by certain central elements of X (gl N , a), see [7].The algebra X(gl N ,a) admits an analogue of the automorphism £JV of Y(gl N ).Determine a formal power series S'y (x) in x^1with the coefficients in X (gl N , a) andthe leading term o"y, by the system of equationsJVY^ sik (x) S k j (x) = öij where i, j = 1, ..., N.k=i