Spectra of tensor products of finite dimensional representations of Yangians. (English) Zbl 0868.17009

Knight, Harold

J. Algebra 174, No. 1, 187-196 (1995).

The author generalizes the notion of character of representations in the case of representations of Yangians. Let \(V\) be a finite-dimensional representation of the Yangian \(Y(g)\), \(\{h_{ik}\}\) canonical generators generating the commutative subalgebra \(H\) of \(Y(g)\), \(V=\bigoplus_\beta V_\beta\), where \(V_\beta\) is a generalized eigenspace for \(h_{ik}\) \((V_\beta= \text{Ker} (h_{ik}-\beta I)^r\) for some \(r\)); \(d^\beta_{ik}\) generalized eigenvalues such that \(h_{ik}- d^\beta_{ik} I\) acts nilpotently on \(V_\beta\). For each sequence \(\beta= (\beta_1,\dots, \beta_n)\) define \(\beta(u_1,\dots, u_n)= \prod^n_{i=1} \beta_i(u_i)\), where \(\beta_i(u_i)= 1+\sum^\infty_{k=1} d^\beta_{ik} u_i^{-k-1}\). Consider the group \(L_n= \{f(u_1,\dots,u_n)= \prod^n_{i=1} f_i(u_i)\}\) over all possible Laurent series \(f_i\) with the usual multiplication of Laurent series. The group algebra \(\mathbb{C}[L_n]\) has as a basis the set of formal exponentials \(\{e(f(u_1,\dots, u_n))\}\) with multiplication \[ e(f(u_1,\dots, u_n)) e(g(u_1,\dots, u_n))= e(f(u_1,\dots, u_n)g(u_1,\dots, u_n)). \] If \(V\) is a finite- dimensional representation of \(Y(g)\) then the character \(\text{ch}(V)\) is an element of the group \(\mathbb{C}[L_n]\) given by \(\text{ch}(V)= \sum_\beta \dim(V_\beta) e(\beta(u_1,\dots, u_n))\). The author proves the following results:
(1) Let \(A\) be a finite-dimensional representation of \(Y(g)\), \(B\) a subrepresentation, \(C=A/B\). Then \(\text{ch}(A)= \text{ch}(B)+ \text{ch}(C)\).
(2) \(\text{ch}(V\otimes W)=\text{ch}(V) \text{ch}(W)\) for finite-dimensional representations \(V\), \(W\).
Finally, the author computes the character of the \((m+1)\)-dimensional representation \(W_m(c)\) of \(Y({\mathfrak sl}_2)\) in terms of Drinfeld polynomials.

Reviewer: V.Stukopin (Rostov-na-Donu)

MSC:

17B37

Quantum groups (quantized enveloping algebras) and related deformations

Keywords:

character formula; Drinfeld polynomials; representations of Yangians