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). 1. Definition of crystal bases The goal is to find a basis of a representation V of Uq(g) such that the Kashiwara operators Ẽi, F̃i act by partial permutations. This will allow us to “draw ” V (λ) as a colored directed graph. However, simple calculations show that this is impossible, even for the adjoint representation of sl3. But, in a sense that will be made precise today, this will work “at q = 0”. Note: we have switched conventions from last week, so that our crystals are at q = 0 rather then q = ∞. The first part of the theory works just as well at either 0 or ∞, but once we start discussing tensor products our choice of coproduct will determine this choice. So, the reason we have changed this convention is that we are changing our convention for coproduct. I did this to match Kashiwara’s conventions in [K1, K2], which are the main references for the next few lectures. Perhaps this is a good time to note that there are four choices of coproduct that work equally well in this theory: (i) ∆(Ei) = Ei ⊗K−1i + 1 ⊗ Ei; ∆(Fi) = Fi ⊗ 1 +Ki ⊗ Fi (ii) ∆(Ei) = Ei ⊗Ki + 1 ⊗ Ei; ∆(Fi) = Fi ⊗ 1 +K−1i ⊗ Fi (iii) ∆(Ei) = Ei ⊗ 1 +Ki ⊗ Ei; ∆(Fi) = Fi ⊗K−1i + 1 ⊗ Fi (iv) ∆(Ei) = Ei ⊗ 1 +K−1i ⊗ Ei; ∆(Fi) = Fi ⊗Ki + 1 ⊗ Fi. One must make a choice, and all four choices have been used in the literature. We will be using the first one from now on, whereas last week we used the second one. Let us also recall the definition of Kashiwara operators on a representation V of Uq(g). For each i, 〈Ei, Fi,K±1i 〉 is a copy of Uq(sl2) inside Uq(g). The irreducible representations of Uq(sl2) are root strings