Rank 2 Affine Mv Polytopes

We give a realization of the crystal B(−∞) for ̂ sl2 using decorated polygons. The construction and proof are combinatorial, making use of Kashiwara and Saito’s characterization of B(−∞), in terms of the ∗ involution. The polygons we use have combinatorial properties suggesting they are the ̂ sl2 analogues of the Mirković-Vilonen polytopes defined by Anderson and the third author in finite type. Using Kashiwara’s similarity of crystals we also give MV polytopes for A (2) 2 , the other rank 2 affine Kac-Moody algebra.