Related papers: PBW bases for $\imath$quantum groups
Easy quantum groups are compact matrix quantum groups, whose intertwiner spaces are given by the combinatorics of categories of partitions. This class contains the symmetric group and the orthogonal group as well as Wang's quantum…
In this paper, the singular Ringel-Hall algebra for a tame quiver is introduced and shown to be isomorphic to the positive part of the quantum extended Kac-Moody algebra. A PBW basis is constructed and a new class of perverse sheaves is…
We clarify some aspects of quantum group gauge theory and its recent generalisations (by T. Brzezinski and the author) to braided group gauge theory and coalgebra gauge theory. We outline the diagrammatic version of the braided case. We…
The well known incompatibility between inhomogeneous quantum groups and the standard q-deformation is shown to disappear (at least in certain cases) when admitting the quantum group to be braided. Braided quantum ISO(p,N-p) containing…
We construct finite-dimensional pointed Hopf algebras \mathfrak u_{r,s}(G_2) (i.e. restricted 2-parameter quantum groups) from the 2-parameter quantum group U_{r,s}(G_2) defined in \cite{HS}, which turn out to be of Drinfel'd doubles, where…
Let $(\bf U, \bf U^\imath)$ be a quantum symmetric pair of Kac-Moody type. The $\imath$quantum groups $\bf U^\imath$ and the universal $\imath$quantum groups $\widetilde{\bf U}^\imath$ can be viewed as a generalization of quantum groups and…
We investigate the differential smoothness of bi-quadratic algebras with PBW basis.
We define invariants of braids rather than invariants of conjugacy classes of braids. For any pure three-braid we give effective upper and lower bounds for these invariants. This is done in terms of a natural syllable decomposition of the…
We generalize categories of spatial partitions in the sense of C\'ebron-Weber by introducing new base partitions. This allows us to construct additional examples of free orthogonal quantum groups but yields the same class of spatial…
This paper consists of three parts: (I) To develop general theory of a (large) class of central simple finite dimensional algebras and answering some natural questions about them (that in general situation it is not even clear how to…
In this article we construct a large family of $R$-matrices for various extensions of small quantum groups by grouplike elements. The extensions are in correspondence to lattices between root and weight lattice and admit $R$-matrices in…
We prove a conjecture proposed in [DSKV16] describing the Lax type operator L(z) for the quantum finite W-algebras of gl_N in terms of a PBW generating system for the W-algebra. In doing so, we extend this result to an arbitrary good…
We initiate the study of several distinguished bases for the positive half of a quantum supergroup $U_q$ associated to a general super Cartan datum $(\mathrm{I}, (\cdot,\cdot))$ of basic type inside a quantum shuffle superalgebra. The…
A deformation $U$, of a graded $K$-algebra $A$ is said to be of PBW type if $grU$ is $A$. It has been shown for Koszul and $N$-Koszul algebras that the deformation is PBW if and only if the relations of $U$ satisfy a Jacobi type condition.…
We provide a classification of finite-dimensional graded pointed Majid algebras generated by finite abelian groups as group-like elements and a set of quasi-commutative skew-primitive elements. This amounts to a classification of finite…
We study and give examples of braided groupoids, and, a fortiori, non-degenerate solutions of the quiver-theoretical braid equation.
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…
The present survey results from the will to reconcile two approaches to quantum probabilities: one rather physical and coming directly from quantum mechanics, the other more algebraic. The second leading idea is to provide a unified picture…
We investigate the rational cohomology of the quotient of (generalized) braid groups by the commutator subgroup of the pure braid groups. We provide a combinatorial description of it using isomorphism classes of certain families of graphs.…
This is a survey of some recent progress on quantum symmetric pairs and applications. The topics include quasi K-matrices, $\imath$Schur duality, canonical bases, super Kazhdan-Lusztig theory, $\imath$Hall algebras, current presentations…