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 Lusztigs 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.
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