Level 1 perfect crystals and path realizations of basic representations at q = 0

Abstract

We present a uniform construction of level 1 perfect crystals B for all affine Lie algebras. We also introduce the notion of a crystal algebra and give an explicit description of its multiplication. This allows us to determine the energy function on B ⊗ B completely and thereby give a path realization of the basic representations at q = 0 in the homogeneous picture.

Level 0 Monomial Crystals

“…As the crystal graph is connected, we conclude that the crystal base constructed in [3] are isomorphic to B(W ( )).…”

Existence of Crystal Bases for Kirillov–Reshetikhin Modules of Type $D$

“…The definition of X requires the existence of the crystal base of a KR module. Despite many efforts as in [12,11,25,18,9,14,19,23,2], this existence problem is yet to be settled. For type D for instance, the crystal base has been shown to exist for W (k) l where k = 1, n − 1, n; l ∈ Z >0 in [12] and for W (k) 1 for arbitrary k in [18,14].…”

“…If the weight of J is not of the form of Λ k−2m 1 + Λ k−2m 2 , the assertion is clear. Suppose wt J = Λ k−2m 1 + Λ k−2m 2 . Consider a highest weight vector of the form…”

A generalization of adjoint crystals for the quantized affine algebras of type A n (1) , C n (1) and D n+1 (2)

“…We denote by V (λ) the finitedimensional irreducible module with highest weight λ over U q (g 0 ) and B(λ) its crystal base. In [1], Benkart, Frenkel, Kang and Lee study a U q (g)-module V with the following properties.…”

“…(1) n , 2 1 if g is of type C (1) n , i 0 otherwise, where i denotes the i-th fundamental weight of g 0 and i 0 denotes the node of the Dynkin diagram of g which is joined to the special node 0. If g is an untwisted affine algebra, then θ is the highest root of g 0 .…”