Publication date:
20 August 2017
Source:Advances in Mathematics, Volume 316
Author(s): Arkady Berenstein, Jacob Greenstein
We introduce a new class of bases for quantized universal enveloping algebras $U_q(\mathfrak{g})$ and other doubles attached to semisimple and Kac–Moody Lie algebras. These bases contain dual canonical bases of upper and lower halves of $U_q(\mathfrak{g})$ and are invariant under many symmetries including all Lusztig's symmetries if $\mathfrak{g}$ is semisimple. It also turns out that a part of a double canonical basis of $U_q(\mathfrak{g})$ spans its center and consists of higher Casimirs which suggests physical applications.