Level one perfect crystals for \(B_n^{(1)}\), \(C_n^{(1)}\), and \(D_n^{(1)}\)
Let \({\mathfrak g}\) be an affine Lie algebra and \({\mathcal U}_q({\mathfrak g})\) denote the associated quantum affine algebra. The crystal base for the integrable highest weight representations of \({\mathcal U}_q({\mathfrak g})\) can be realized in terms of certain combinatorial objects called ``paths'' which are elements of the semi-infinite tensor product \(\cdots\oplus B\oplus B\oplus B\), where \(B\) is a perfect crystal of corresponding level associated with a finite-dimensional representation of the derived quantum affine algebra \({\mathcal U}_q'({\mathfrak g})\). For \({\mathfrak g}={\mathfrak A}_n^{(1)}\), it has been shown in [\textit{S. J. Kang} et al., Duke Math. J. 68, 499-607 (1992; Zbl 0774.17017)] that \(B^{k,\ell}= B(\ell\Lambda_k)\) is a perfect crystal of level \(\ell\) where \(\Lambda_k\) denotes the \(k\)th fundamental weight of \({\mathcal U}_q(A_n)\). In this paper the author constructs new examples of level one perfect crystals for the quantum affine algebras \({\mathcal U}_q(B_n^{(1)})\), \({\mathcal U}_q(C_n^{(1)})\), \({\mathcal U}_q(D_n^{(1)})\). It is conjectured that any perfect crystal of level one for these quantum affine algebras is isomorphic to one of these crystals or the tensor product of them.
scientific article; zbMATH DE number 1340615
scientific article