On Irreducibility of Tensor Products of Yangian Modules

of generators T (s) ij with s = 1, 2 , . . . and i , j = 1 , . . . , N is given by (1.1), (1.2). Comultiplication ∆ : Y(glN ) → Y(glN )⊗Y(glN ) is defined by (1.1), (1.16). The algebra Y(glN ) admits an alternative definition in terms of the ascending chain U(gl1) ⊂ U(gl2) ⊂ . . . of classical universal enveloping algebras [O]. For any non-negative integer M consider the commutant in U(glM+N ) of the subalgebra U(glM ) . This commutant is generated by the centre of the subalgebra U(glM ) and a homomorphic image of the Yangian Y(glN ) , see Proposition 1.1. The intersection of the kernels in Y(glN ) of all these homomorphisms for M = 0, 1, 2 , . . . is zero. For any dominant integral weights λ and μ of the Lie algebras glM+N and glM consider the subspace Vλ,μ in the irreducible glM+N -module Vλ of highest weight λ formed by all singular vectors with respect to glM of weight μ . The algebra Y(glN ) acts in Vλ,μ irreducibly through the above homomorphism. Further, for any complex number h there is an automorphism τh of the algebra Y(glN ) defined in terms of the generating series (1.1) by the assignment Tij(u) 7→ Tij(u+ h) . By pulling back the Y(glN )-module Vλ,μ through this automorphism we obtain an irreducible Y(glN )-module, which we denote by Vλ,μ(h) and call elementary. Any formal series f(u) ∈ 1 + uC [[u]] also defines an automorphism ωf of the algebra Y(glN ) by Tij(u) 7→ f(u) · Tij(u) . Further, there is a canonical chain of algebras Y(gl1) ⊂ . . . ⊂ Y(glN ). The elementary modules are distinguished amongst all irreducible finite-dimensional Y(glN )-modules W by the following fact. Take the commutative subalgebra A(glN ) in Y(glN ) generated by the centres of all algebras in the latter chain. Then the action of this subalgebra in W is semi-