Jacob Greenstein: Double canonical bases

We introduce a new class of bases for quantized universal enveloping algebras and other doubles attached to semisimple and Kac-Moody Lie algebras. These bases contain dual canonical bases of upper and lower halves of the quantum group and are invariant under many symmetries including all Lusztig's symmetries if the corresponding Lie algebra is semisimple. It also turns out that a part of a double canonical basis spans the center of the quantum group.