TITLE: Does some version of U_q(gl(1|1)) have a basis like Lusztig's basis for \dot{U(sl_2)}?
QUESTION [7 upvotes]: There's a non-unital algebra $\dot{U}$ formed from $U_q (sl_2)$ by including a system of mutually orthogonal idempotents $1_n$, indexed by the weight lattice.  You can think of this as a category with objects $\mathbb{Z}$ if you prefer.
Lusztig's basis $\mathbb{\dot{B}}$ for $\dot{U}$ has nice positivity properties:  structure coefficients are in $\mathbb{Z}[q,q^{-1}]$.
Has anyone tried to write down a similar type of basis for the algebra associated to $U_q (gl_{1|1})$?

REPLY [3 votes]: Kashiwara has developed some crystal theoretic methods for the Lie superalgebra $\mathfrak{q}(n)$. However, I think you should look at Khovanov, to get an idea of what it should look like.