Double product integrals and Enriquez quantisation of Lie bialgebras II: The quantum Yang-Baxter equation

Higher genus Kashiwara-Vergne problems and the Goldman-Turaev Lie bialgebra

We define a family ${\rm KV}^{(g,n)}$ of Kashiwara-Vergne problems associated with compact connected oriented 2-manifolds of genus $g$ with $n+1$ boundary components. The problem ${\rm KV}^{(0,3)}$ is the classical Kashiwara-Vergne problem from Lie theory. We show the existence of solutions of ${\rm KV}^{(g,n)}$ for arbitrary $g$ and $n$. The key point is the solution of ${\rm KV}^{(1,1)}$ based on the results by B. Enriquez on elliptic associators. Our construction is motivated by applications to the formality problem for the Goldman-Turaev Lie bialgebra $\mathfrak{g}^{(g, n+1)}$. In more det…

The global quantum duality principle: theory, examples, applications

The "quantum duality principle" states that a quantisation of a Lie bialgebra provides also a quantisation of the dual formal Poisson group and, conversely, a quantisation of a formal Poisson group yields a quantisation of the dual Lie bialgebra as well. We extend this to a much more general result: namely, given any principal ideal domain R, for each prime h in R we establish sort of an "inner" Galois' correspondence on the category HA of torsionless Hopf algebras over R, via the definition of two functors (from HA to itself) such that the image of the first, resp. of the second, is the full …

A 2-categorical extension of Etingof-Kazhdan quantisation

Let k be a field of characteristic zero. Etingof and Kazhdan constructed a quantisation U_h(b) of any Lie bialgebra b over k, which depends on the choice of an associator Phi. They prove moreover that this quantisation is functorial in b. Remarkably, the quantum group U_h(b) is endowed with a Tannakian equivalence F_b from the braided tensor category of Drinfeld-Yetter modules over b, with deformed associativity constraints given by Phi, to that of Drinfeld-Yetter modules over U_h(b). In this paper, we prove that the equivalence F_b is functorial in b.

Noncommutative Differentials on Poisson-Lie groups and pre-Lie algebras

We show that the quantisation of a connected simply-connected Poisson-Lie group admits a left-covariant noncommutative differential structure at lowest deformation order if and only if the dual of its Lie algebra admits a pre-Lie algebra structure. As an example, we find a pre-Lie algebra structure underlying the standard 3D differential structure on $\C_q[SU_2]$. At the noncommutative geometry level we show that the enveloping algebra $U(\cm)$ of a Lie algebra $\cm$, viewed as quantisation of $\cm^*$, admits a connected differential exterior algebra of classical dimension if and only if $\cm$…

高亏格下的Goldman-Turaev Lie双代数与Kashiwara-Vergne问题

The Goldman-Turaev Lie bialgebra and the Kashiwara-Vergne problem in higher genera

本文研究高亏格曲面上的Goldman-Turaev形式化问题，通过定义并求解新的高亏格Kashiwara-Vergne问题，建立了GT形式化映射，并揭示了其与Grothendieck-Teichmüller群的联系。

Uniqueness of Coxeter structures on Kac-Moody algebras

Let g be a symmetrisable Kac-Moody algebra, and U_h(g) the corresponding quantum group. We showed in arXiv:1610.09744 and arXiv:1610.09741 that the braided quasi-Coxeter structure on integrable, category O representations of U_h(g) which underlies the R-matrix actions arising from the Levi subalgebras of U_h(g) and the quantum Weyl group action of the generalised braid group B_g can be transferred to integrable, category O representations of g. We prove in this paper that, up to unique equivalence, there is a unique such structure on the latter category with prescribed restriction functors, R-…