Related papers: On Low-Dimensional Locally Compact Quantum Groups

In this note, we revisit the quantization of Lie bialgebras described by the second author, placing it in the more general framework of the quantization of moduli spaces developed in our previous work. In particular, we show that embeddings…

We describe a new method of quantization of Lie bialgebras, based on a construction of Hopf algebras out of a cocommutative coalgebra and a braided comonoidal functor.

It is known that Hochschild cohomology groups are represented by crossed extensions of associative algebras. In this paper, we introduce crossed $n$-fold extensions of a Lie algebra $\mathfrak{g}$ by a module $M$, for $n \geq 2$. The…

We establish two conditions equivalent to coamenability for type I locally compact quantum groups. The first condition is concerned with the spectra of certain convolution operators on the space…

For a perfect Lie algebra $\mathfrak{h}$ we classify all Lie algebras containing $\mathfrak{h}$ as a subalgebra of codimension $1$. The automorphism groups of such Lie algebras are fully determined as subgroups of the semidirect product…

Starting with a finite-dimensional complex Lie algebra, we extend scalars using suitable commutative topological algebras. We study Birkhoff decompositions for the corresponding loop groups. Some results remain valid for loop groups with…

A locally conformally product (LCP) structure on a compact conformal manifold is a closed non-exact Weyl connection (i.e.~a linear connection which is locally but not globally the Levi-Civita connection of Riemannian metrics in the…

We introduce the categories of infinitesimal Hopf modules and bimodules over an infinitesimal bialgebra. We show that they correspond to modules and bimodules over the infinitesimal version of the double. We show that there is a natural,…

A notion of L^2-homology for compact quantum groups is introduced, generalizing the classical notion for countable, discrete groups. If the compact quantum group in question has tracial Haar state, it is possible to define its L^2-Betti…

Quantum duality principle is applied to study classical limits of quantum algebras and groups. For a certain type of Hopf algebras the explicit procedure to construct both classical limits is presented. The canonical forms of quantized…

Star products on the classical double group of a simple Lie group and on corresponding symplectic grupoids are given so that the quantum double and the "quantized tangent bundle" are obtained in the deformation description. "Complex"…

We give an interpretation of the Brauer group of a purely inseparable extension of exponent 1, in terms of restricted Lie-Rinehart cohomology. In particular, we define and study the category $p$-$\rm{LR}(A)$ of restricted Lie-Rinehart…

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…

The notion of a matched pair of Lie algebras was introduced in the study of Lie bialgebras and Poisson-Lie groups. In this paper, we introduce representations and cohomology of a matched pair of Lie algebras. We show that there is a…

By using cocycle deformation, we construct a certain class of Hopf algebras, containing the quantized enveloping algebras and their analogues, from what we call pre-Nichols algebras. Our construction generalizes in some sense the known…

We propose a unified framework in which the different constructions of cohomology groups for topological and Lie groups can all be treated on equal footings. In particular, we show that the cohomology of "locally continuous" cochains…

In this paper we use invariant theory to develop the notion of cohomological detection for Type I classical Lie superalgebras. In particular we show that the cohomology with coefficients in an arbitrary module can be detected on smaller…

In the finite-dimensional setting, every Hermitian-symmetric space of compact type is a coadjoint orbit of a finite-dimensional Lie group. It is natural to ask whether every infinite-dimensional Hermitian-symmetric space of compact type,…

We study left invariant locally conformally product structures on simply connected Lie groups and give their complete description in the solvable unimodular case. Based on previous classification results, we then obtain the complete list of…

Let $\G$ be a locally compact group satisfying some technical requirements and $\wG$ its unitary dual. Using the theory of twisted crossed product $C^*$-algebras, we develop a twisted global quantization for symbols defined on $\G\times\wG$…