### A brief review of abelian categorifications.

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We show that a Poisson Lie group (G,π) is coboundary if and only if the natural action of G×G on M=G is a Poisson action for an appropriate Poisson structure on M (the structure turns out to be the well known ${\pi}_{+}$). We analyze the same condition in the context of Hopf algebras. A quantum analogue of the ${\pi}_{+}$ structure on SU(N) is described in terms of generators and relations as an example.

Let G be a complex reductive connected algebraic group equipped with the Sklyanin bracket. A classification of Poisson homogeneous G-spaces with connected isotropy subgroups is given. This result is based on Drinfeld's correspondence between Poisson homogeneous G-spaces and Lagrangian subalgebras in the double D𝖌 (here 𝖌 = Lie G). A geometric interpretation of some Poisson homogeneous G-spaces is also proposed.

We describe a cluster algebra algorithm for calculating $q$-characters of Kirillov–Reshetikhin modules for any untwisted quantum affine algebra ${U}_{q}\left(\widehat{\U0001d524}\right)$. This yields a geometric $q$-character formula for tensor products of Kirillov–Reshetikhin modules. When $\U0001d524$ is of type $A,D,E$, this formula extends Nakajima’s formula for $q$-characters of standard modules in terms of homology of graded quiver varieties.

A generalisation of quantum principal bundles in which a quantum structure group is replaced by a coalgebra is proposed.

We obtain a presentation by generators and relations of any Nichols algebra of diagonal type with finite root system. We prove that the defining ideal is finitely generated. The proof is based on Kharchenko’s theory of PBW bases of Lyndon words. We prove that the lexicographic order on Lyndon words is convex for PBW generators and so the PBW basis is orthogonal with respect to the canonical non-degenerate form associated to the Nichols algebra.

We show by explicit calculations in the particular case of the 4-dimensional irreducible representation of ${\mathcal{U}}_{q}\left(\mathfrak{s}\mathfrak{l}\left(2\right)\right)$ that it is not always possible to generalize to the quantum case the notion of symmetric algebra of a Lie algebra representation.

A new Jordanian quantum complex 4-sphere together with an instanton-type idempotent is obtained as a suspension of the Jordanian quantum group $S{L}_{h}\left(2\right)$.