Lecture 2: Crystal Bases

Today I’ll define crystal bases, and discuss their basic properties. This will include the tensor product rule and the relationship between the crystals B(λ) of highest weight modules and the infinity crystal B(∞). I’ll then discuss an intrinsic characterization of crystal lattices and crystal bases in terms of a natural bilinear form (sometimes called “polarization”). Unless otherwise stated, the results here are due to Kashiwara, and proofs can be found in [K1]. For today, g is a symmetrizable Kac–Moody algebra, Uq(g) is its quantized universal enveloping algebra, that V is a representation which is a (not necessarily finite) direct sum of integrable highest weight modules V (λ) (i.e., an object in Oint).