Perfect crystals of quantum affine Lie algebras
The perfect crystals play a crucial role in realizing the crystal bases of integrable irreducible representations of quantum affine Lie algebras and in computing the one-point functions of vertex models. In this paper, the authors give explicit forms of perfect crystals of an arbitrary level for quantum affine Lie algebras \(U_ q({\mathfrak g})\) of type \(A_ n^{(1)}\), \(B_ n^{(1)}\), \(C_ n^{(1)}\), \(D_ n^{(1)}\), \(A_{2n}^{(2)}\), \(A^{(2)}_{2n-1}\), or \(D^{(2)}_{n+1}\). The main idea of the construction is as follows. For a given level \(\ell\), they construct a finite-dimensional irreducible representation \(V_ \ell\) of \(U_ q'({\mathfrak g})\) with a crystal pseudobase such that its associated crystal is perfect of level \(\ell\). They start with a suitable crystal \(B\) for \(U_ q({\mathfrak g}_{I\setminus\{0\}})\), and define the 0-arrows in such a way that \(B\) becomes a virtual crystal for \(U_ q'({\mathfrak g})\) and prove in a purely combinatorial way that \(B\) is a perfect crystal of level \(\ell\). On the other hand, let \(V\) be a certain finite-dimensional module over \(U_ q({\mathfrak g}_{I\setminus\{0\}})\) with a crystal base \(B_ 1\) which is characterized by a polarization on \(V\). \(V\) can also be regarded as an irreducible \(U_ q'({\mathfrak g})\)-module. The polarization on \(V\) for \(U_ q({\mathfrak g}_{I\setminus \{0\}})\) is also a polarization for \(U_ q'({\mathfrak g})\). Then \(B_ 1\) can be extended to a crystal for \(U_ q'({\mathfrak g})\) and \(B_ 1\) is a perfect crystal of level 1. After getting a certain finite-dimensional submodule \(V_ \ell\) of \(V^ \ell\) for \(U_ q'({\mathfrak g})\), they show that the polarization on \(V\) for \(U_ q'({\mathfrak g})\) induces a polarization on \(V_ \ell\), and it has a crystal pseudobase and its associated crystal is isomorphic to \(B\) as crystals for \(U_ q({\mathfrak g}_ J)\) for some subset \(J\) of \(I\). Finally, they prove that this isomorphism can be uniquely extended to that of crystals for \(U_ q'({\mathfrak g})\) and thus conclude that \(V_ \ell\) has a crystal pseudobase whose associated crystal is perfect of level \(\ell\). As an example, the authors construct the perfect crystals of level 2 for \(U_ q(C_ 2^{(1)})\).
scientific article; zbMATH DE number 179206
scientific article