International Congress of Mathematicians

644 Alaxim NazarovThe unital associative algebra Y(gl JV ) over C has a family of generators Tywhere a = 1,2,... and i, j = 1, ..., N. The defining relations for these generatorscan be written in terms of the formal power seriesTij(x) = öij • 1 + T^x- 1 + T^x- 2 + ... £ Y(gl N ) [[x' 1 ]]. (1.1)Here x is the formal parameter. Let y be another formal parameter, then thedefining relations in the associative algebra Y(gl JV ) can be written as(x- y )-[Tij(x),Tk l (y)]= T kj (x)T a (y)-T kj (y)T a (x), (1.2)where i,j,k,l = 1, ...,N. The square brackets in (1.2) denote usual commutator.In terms of the formal series (1.1), the coproduct A : Y(gl JV ) —t Y(gt N ) ® Y(gt N )is defined byA(Tij(x)) = J2 T i*(x) ® Tkj(x) ; (1.3)k=ithe tensor product on the right hand side of the equality (1.3) is taken over thesubalgebra C[[a: _1 ]] C Y(gl JV ) [[i -1 ]]- The counit homomorphism e : Y(gl JV ) —t Cis determined by the assignment e : Tj(u) >-¥ % • 1.For each i and j one can determine a formal power series Tj(x) in x^1withthe coefficients in Y(gl JV ) and the leading term o"y, by the system of equationsJVY^ Tik (x) T k j (x) = öij where i, j = 1, ..., N.k=iThe antipode S on Y(gl JV ) is the anti-automorphism of the algebra Y(gl N ), definedby the assignment S : Ty (x) >-¥ Tj (x). We also use the involutive automorphism£iv of the algebra Y(gt N ), defined by the assignment £JV : Tij(x) >-¥ Tij(-x).Take any formal power series f(x) £ C[[a: _1 ]] with the leading term 1. TheassignmentTj(x)^ f(x)-Tj(x) (1.4)defines an automorphism of the algebra Y(gl N ), this follows from (1.1) and (1.2).The Yangian Y(sljv) is the subalgebra in Y(gl JV ) consisting of all elements, whichare invariant under every automorphism (1.4).It also follows from (1.1) and (1.2) that for any z £ C, the assignmentT Z : Tij(x) H> Tj(xdefines an automorphism T Z of the algebra Y(gl N ). Here the formal power seriesin (x — z)^1should be re-expanded in x _1 . Regard the matrix units E t j £ gl N asgenerators of the universal enveloping algebra U(gl JV ). The assignment-z)«iv : Tij(x) >-¥ öij • 1 — Eji x^1defines a homomorphism ctN '• Y(gl N ) —¥ U(gl JV ). By definition, the homomorphism«iv is surjective. For more details and references on the definition of the YangianY(gl N ), see [7].