International Congress of Mathematicians

Representations of Yangians 6451.2. Let v = (vi,v 2 , ...) be any partition. As usual, the parts of v are arrangedin the non-increasing order: vi ^ v 2 ^ ... ^ 0. Let v' = (v[,v 2 , •••) be thepartition conjugate to v. In particular, v[ is the number of non-zero parts of thepartition v. An irreducible module over the Lie algebra gl^r is called polynomial,if it is equivalent to a submodule in the tensor product of n copies of the defininggl N -module C^, for some integer n ^ 0. The irreducible polynomial gl^r -modulesare parametrized by partitions v such that v[ ^ N. Here n = vi + v 2 + ... . Let V vbe the irreducible module corresponding to v. This gl^r -module is of highest weight(vi, ..., VM). Here we choose the Borei subalgebra in gl^r consisting of the uppertriangular matrices, and fix the basis of the diagonal matrix units En , ..., ENN inthe corresponding Cartan subalgebra of gl N .Take any non-negative integer M. Yet the indices i and j range over the set{1, ..., N + M}. Fix the basis of the matrix units E t j in the Lie algebra gijv+M •We suppose that the subalgebras gl N and gt M in gljv+M are spanned by elementsEij where respectively i,j = 1, ... ,N and i,j = N + 1, ...,N + M. Yet X and pbe two partitions, such that A{ ^ N + M and p[ ^ M. Consider the irreduciblemodules Y\ and V ß over the Lie algebras gijv+M an d B'M- The vector spaceHom 0[ M (^>V\) ( L5 )comes with a natural action of the Lie algebra gl^r • This action of gl N may bereducible. The vector space (1.5) is non-zero, if and only if X k ^ Pk and X' k —p' k ^ Nfor each k = 1,2,...; see for instance [8].Denote by Ajv(Af) the centralizer of the subalgebra U(gl M ) C U(gljv+M)- Thecentralizer Ajv(Af ) c U(gljv +Af ) contains U(gl JV ) as a subalgebra, and acts naturallyin the vector space (1.5). This action is irreducible. For every AT, Olshanski [16]defined a homomorphism of associative algebras Y(gl JV ) —t Ajv(Af). Along withthe centre of the algebra U(gljv+M)) the image of this homomorphism generates thealgebra Ajv(Af). We use a version of this homomorphism, it is denoted by UMM-The subalgebra in Y(gl JV+M ) generated by Ty where i,j = 1, ...,N, bydefinitioncoincides with the Yangian Y(gl N ). Denote by 1, Xi > j > Pi} .