By Faddeev L., Moerbeke P.V., Lambert F. (eds.)

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Firstly we construct a Hirota-like operator (Hirota map) acting on representations of finitedimensional Lie algebras almost as an intertwining operator. This allows us to construct highest weight vectors of irreducible representations. Such irreducibles are associated with polynomial functions but in the limit of infinitedimensional representations the Hirota map becomes the Hirota derivative acting on analytic functions. Secondly, we are able to push this procedure through for S L n (C), constructing Hirota maps, analyzing their actions on representations and their infinite-dimensional limits, recovering old and new Hirota derivatives.

K j . ) ⊗ (. . li . . l j + 1 . ) − (. . ki . . k j + 1 . ) ⊗ (. . li + 1 . . l j . ). It is important to note that there are, up to linear dependence, n − 1 such D operators and that they alter the weights of the modules on which they act. Their crucial property is the following. They commute with the ei j except when one of their indices coincides with the first of the indices on ei j : ⎧ 12 ⎪ ⎨ D jl i = k, j = l 12 [Di j , ekl ] = Dli12 j = k, i = l (16) ⎪ ⎩ 0 otherwise. and they commute with most of the Cartan subalgebra: ⎧ 12 Di j i =l = j −1 ⎪ ⎪ ⎪ 12 ⎪ ⎪ ⎨ −Di j i − 1 = l = j Di12j j =l =i −1 [Di12j , h l ] = ⎪ ⎪ 12 ⎪ −Di j j − 1 = l = i ⎪ ⎪ ⎩ 0 otherwise (17) Consequently, if v ∈ N1 ⊗ N2 is a highest weight vector according to some choice of Borel subalgebra, B, there will be a subset, DB , of the Hirota operators Algebraic Hirota Maps 21 defined above which commute with the nilpotent part of the B action, so that D(v) ∈ N1 +1 ⊗ N2 +1 is again highest weight, with respect to B, but with a different weight value (because of the nontrivial relations (17)).

Algebraic Hirota Maps 23 We will choose the Borel subalgebra B = h ⊕ {e12 , e23 , e13 }. The operator D12 23 commutes with the nilpotent part and satisfies the following relations on h: 12 [D12 23 , h 1 ] = −D23 , [D12 23 , h 2 ] = 0. (26) Consequently if v ∈ N ⊗ M is a highest weight vector of weight ( p, q) ∈ Z2 , D12 23 (v) will be a highest weight vector in N +1 ⊗ M+1 of weight ( p + 1, q). The module N has weight (0, N ) but not all sl3 modules are of this type. 1 ⊗ 1 ≡ (0,1) ⊗ (0,1) is a nine-dimensional module.

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