×

zbMATH — the first resource for mathematics

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

MSC:
17B37 Quantum groups (quantized enveloping algebras) and related deformations
PDF BibTeX XML Cite
Full Text: DOI