Does some version of U_q(gl(1|1)) have a basis like Lusztig's basis for \dot{U(sl_2)}? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T04:16:08Z http://mathoverflow.net/feeds/question/4764 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/4764/does-some-version-of-u-qgl11-have-a-basis-like-lusztigs-basis-for-dotusl Does some version of U_q(gl(1|1)) have a basis like Lusztig's basis for \dot{U(sl_2)}? Sammy Black 2009-11-09T20:07:52Z 2011-01-08T12:22:14Z <p>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.</p> <p>Lusztig's basis $\mathbb{\dot{B}}$ for $\dot{U}$ has nice positivity properties: structure coefficients are in $\mathbb{Z}[q,q^{-1}]$.</p> <p>Has anyone tried to write down a similar type of basis for the algebra associated to $U_q (gl_{1|1})$?</p> http://mathoverflow.net/questions/4764/does-some-version-of-u-qgl11-have-a-basis-like-lusztigs-basis-for-dotusl/37696#37696 Answer by David Hill for Does some version of U_q(gl(1|1)) have a basis like Lusztig's basis for \dot{U(sl_2)}? David Hill 2010-09-04T06:01:07Z 2010-09-04T06:01:07Z <p><a href="http://front.math.ucdavis.edu/1007.4105" rel="nofollow">Kashiwara</a> has developed some crystal theoretic methods for the Lie superalgebra $\mathfrak{q}(n)$. However, I think you should look at <a href="http://front.math.ucdavis.edu/1007.3517" rel="nofollow">Khovanov</a>, to get an idea of what it should look like.</p>