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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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The goal of this work is to provide an elementary construction of the canonical basis $mathbf B(w)$ in each quantum Schubert cell~$U_q(w)$ and to establish its invariance under modified Lusztigs symmetries. To that effect, we obtain a direct characterization of the upper global basis $mathbf B^{up}$ in terms of a suitable bilinear form and show that $mathbf B(w)$ is contained in $mathbf B^{up}$ and its large part is preserved by modified Lusztigs symmetries.
We generalize a construction in [BW18] (arXiv:1610.09271) by showing that the tensor product of a based $textbf{U}^{imath}$-module and a based $textbf{U}$-module is a based $textbf{U}^{imath}$-module. This is then used to formulate a Kazhdan-Lusztig theory for an arbitrary parabolic BGG category $mathcal{O}$ of the ortho-symplectic Lie superalgebras, extending a main result in [BW13] (arXiv:1310.0103).
We provide a construction of global bases for quantum Borcherds-Bozec algebras and their integrable highest weight representations.
For quantum group of affine type, Lusztig gave an explicit construction of the affine canonical basis by simple perverse sheaves. In this paper, we construct a bar-invariant basis by using a PBW basis arising from representations of the corresponding tame quiver. We prove that this bar-invariant basis coincides with Lusztigs canonical basis and obtain a concrete bijection between the elements in theses two bases. The index set of these bases is listed orderly by modules in preprojective, regular non-homogeneous, preinjective components and irreducible characters of symmetric groups. Our results are based on the work of Lin-Xiao-Zhang and closely related with the work of Beck-Nakajima. A crucial method in our construction is a generalization of that by Deng-Du-Xiao.
We have reviewed some results on quantized shuffling, and in particular, the grading and structure of this algebra. In parallel, we have summarized certain details about classical shuffle algebras, including Lyndon words (primes) and the construction of bases of classical shuffle algebras in terms of Lyndon words. We have explained how to adapt this theory to the construction of bases of quantum group algebras in terms of Lyndon words. This method has a limited application to the specific case of the quantum group parameter being a root of unity, with the requirement that specialization to the root of unity is non-restricted. As an additional, applied part of this work, we have implemented a Wolfram Mathematica package with functions for quantum shuffle multiplication and constructions of bases in terms of Lyndon words.
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