Related papers: PBW bases for modified quantum groups
Let $G$ be a connected simply-connected simple complex algebraic group and $\mathfrak{g}$ the corresponding simple Lie algebra. In the first half of the present paper, we study the relation between the positive part $U_q(\mathfrak{n^+})$ of…
We characterise the symplectic Weyl group elements such that the FFLV basis is compatible with the PBW filtration on symplectic Demazure modules, extending type A results by the second author. Surprisingly, the number of such elements…
For any quantum group of finite ADE type, we prove a new formula for the standard bilinear form evaluated at monomials. Combining this with ideas from the Lusztig-Shoji algorithm, we obtain a new algorithm that computes the canonical basis.…
We study the PBW filtration on irreducible finite--dimensional representations for the Lie algebra of type $\tt B_n$. We prove in several cases, including all multiples of the adjoint representation and all irreducible finite--dimensional…
We introduce the notion of a favourable module for a complex unipotent algebraic group, whose properties are governed by the combinatorics of an associated polytope. We describe two filtrations of the module, one given by the total degree…
We construct the quantized enveloping algebra of any simple Lie algebra of type ADE as the quotient of a Grothendieck ring arising from certain cyclic quiver varieties. In particular, the dual canonical basis of a one-half quantum group…
Many rings and algebras arising in quantum mechanics can be interpreted as skew PBW (Poincar\'e-Birkhoff-Witt) extensions. Indeed, Weyl algebras, enveloping algebras of finite-dimensional Lie algebras (and its quantization), Artamonov…
We classify finite-dimensional irreducible highest weight modules of generalized quantum groups whose positive part is infinite dimensional and has a Kharchenko's PBW basis with an irreducible finite positive root system.
We construct a crystal basis for the negative half of the quantum group U associated to the standard super Cartan datum of gl(m|1), which is compatible with known crystals on Kac modules and simple modules. We show that these crystals admit…
A theory of canonical basis for a two-parameter quantum algebra is developed in parallel with the one in one-parameter case. A geometric construction of the negative part of a two-parameter quantum algebra is given by using mixed perverse…
The $\imath$quantum groups admit two realizations: one via the $\imath$Hall algebras and the other via the quantum Grothendieck rings of quiver varieties, as developed by the first author and Wang. Based on these two realizations, we…
Based on maximally entangled states, we explore the constructions of mutually unbiased bases in bipartite quantum systems. We present a new way to construct mutually unbiased bases by difference matrices in the theory of combinatorial…
In this part one of a series of papers, we introduce a new version of quantum covering and super groups with no isotropic odd simple root, which is suitable for the studies of integrable modules, integral forms and bar-involution. A quantum…
Symmetry adapted bases in quantum chemistry and bases adapted to quantum information share a common characteristics: both of them are constructed from subspaces of the representation space of the group SO(3) or its double group (i.e.,…
In order to construct quantum trigonometric B\'ezier curves with shape parameter, one parameter family of trigonometric Bernstein basis functions are introduced. We study the total positivity of the basis functions to analyze the shape…
We provide a classification of compact quantum groups, which can be obtained by the Woronowicz construction, when the arrays used in the twisted determinant condition are extensions of functions on permutations. General properties of such…
We prove in a very general framework several versions of the classical Poincar\'e-Birkhoff-Witt Theorem, which extend results from [BeGi, BrGa, CS, HvOZ, WW]. Applications and examples are discussed in the last part of the paper.
Let U be the quantum group associated to a Lie algebra g of rank n. The negative part U^- of U has a canonical basis B with favourable properties, introduced by Kashiwara and Lusztig. The approaches of Kashiwara and Lusztig lead to a set of…
We follow the approach developed by Beilinson-Lusztig-MacPherson and modified by Fu and the first author to investigate a new realization for the i-quantum groups U^j(n) of type B, building on the multiplication formulas discovered in…
A geometric categorification is given for arbitrary-large-finite-dimensional quotients of quantum osp(1|2) and the tensor product of its simple modules. The modified quantum osp(1|2) of Clark-Wang, a new version in this paper and the…