Double canonical bases

Abstract

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.

62 pages; some typos corrected, some examples streamlined

References in corpus (7)

• Quantum groups via cyclic quiver varieties I
• Remarks on quantum unipotent subgroup and dual canonical basis
• Modules over quantized coordinate algebras and PBW-bases
• Semisimplicity of certain representation categories
• Canonical bases of quantum Schubert cells and their symmetries
• Generalized Joseph's decompositions
• Quantum seaweed algebras and quantization of affine Cremmer-Gervais r-matrices

Cited by in corpus (9)

• The Berenstein-Zelevinsky quantum cluster algebra conjecture
• Remarks on quantum unipotent subgroup and dual canonical basis
• Modules over quantized coordinate algebras and PBW-bases
• Quantum Grothendieck rings as quantum cluster algebras
• Duals of semisimple Poisson-Lie groups and cluster theory of moduli spaces of G-local systems
• Integral quantum cluster structures
• Root of unity quantum cluster algebras and discriminants
• Canonical bases of quantum Schubert cells and their symmetries
• Quantized nilradicals of parabolic subalgebras of $\mathfrak{sl}(n)$ and algebras of coinvariants