Higher structures, quantum groups and genus zero modular operad

@article{Manin2019HigherSQ,
  title={Higher structures, quantum groups and genus zero modular operad},
  author={Yuri I. Manin},
  journal={Journal of the London Mathematical Society},
  year={2019},
  volume={100}
}
  • Y. Manin
  • Published 12 February 2018
  • Mathematics
  • Journal of the London Mathematical Society
In my Montreal lecture notes of 1988, it was suggested that the theory of linear quantum groups can be presented in the framework of the category of quadratic algebras (imagined as algebras of functions on ‘quantum linear spaces), and quadratic algebras of their inner (co)homomorphisms. 
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References

SHOWING 1-10 OF 52 REFERENCES
Quantum Groups
Here is an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and Drinfeld's recent fundamental contributions. It presents the quantum groups
Quadratic Algebras Related to the Bi-Hamiltonian Operad
We prove the conjectures on dimensions and characters of some quadratic algebras stated by B.L.Feigin. It turns out that these algebras are naturally isomorphic to the duals of the components of the
Towards Motivic Quantum Cohomology of M̄0,S
Abstract We explicitly calculate some Gromov–Witten correspondences determined by maps of labelled curves of genus 0 to the moduli spaces of labelled curves of genus 0. We consider these calculations
Gromov-Witten classes, quantum cohomology, and enumerative geometry
The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic
Towards motivic quantum cohomology of $\bar{M}_{0,S}$
We explicitly calculate some Gromov--Witten correspondences determined by maps of labeled curves of genus zero to the moduli spaces of labeled curves of genus zero. We consider these calculations as
Some remarks on Koszul algebras and quantum groups
La categorie des algebres quadratiques est munie d'une structure tensorielle. Ceci permet de construire des algebres de Hopf du type «(semi)-groupes quantiques»
Grothendieck-Verdier duality patterns in quantum algebra
After a brief survey of the basic definitions of the Grothendieck--Verdier categories and dualities, I consider in this context introduced earlier dualities in the categories of quadratic algebras
Frobenius manifolds, quantum cohomology, and moduli spaces
Introduction: What is quantum cohomology? Introduction to Frobenius manifolds Frobenius manifolds and isomonodromic deformations Frobenius manifolds and moduli spaces of curves Operads, graphs, and
Quantization of Lie Groups and Lie Algebras
Monoidal structures on the categories of quadratic data
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