On quantization of quasi-Lie bialgebras (Q2346369)

The quantization of Lie bialgebras over a field \(\mathbb K\) was first obtained by \textit{P. Etingof} and \textit{D. Kazhdan} [Sel. Math., New Ser. 2, No. 1, 1--41 (1996; Zbl 0863.17008)]. Their method is explicit, depending on the choice of a Drinfeld associator. A quasi-Lie bialgebra \(\mathfrak g\) differs from a Lie bialgebra by the use of an element from the triple exterior product of \(\mathfrak g\). The quantization problem is to give a quasi-bialgebra structure (see [\textit{V. G. Drinfel'd}, Leningr. Math. J. 1, No. 6, 1419--1457 (1990; Zbl 0718.16033); translation from Algebra Anal. 1, No. 6, 114--148 (1989)]) on \(\mathcal U_\hbar\mathfrak g\) deforming the bialgebra \(\mathcal U\mathfrak g\). The authors, adjusting the methods of Etingof and Kazhdan, do this in two steps: (Notation to be explained later) (1) Construct a monoidal category \(\mathcal A^\Phi\) which is a deformation of the category of \(\mathcal U\mathfrak g\)-modules; (2) Construct an object \(C\) of \(\mathcal A^\Phi\) together with morphisms \(C\to C\otimes C\) tensor \(C\) and \(C\to 1\) making \(\Hom(C, - )\) into a quasi-monoidal functor. This makes \(\Hom(C,C)\) into a quasi-bialgebra. Then find an explicit isomorphism of vector spaces of \(\Hom(C,C)\) and \(\mathcal U\mathfrak g\) and set \(\mathcal U_\hbar\mathfrak g =\Hom(C,C)\). We summarize the steps in this quantization. (A) If \(F: \mathcal C\to \mathrm{Vec}\) is a representable, strong quasi-monoidal functor, then \(\mathrm{End}(F)\) is a quasi-bialgebra. (B) The notion of a Karoubi envelope \(\mathrm{Split}(\mathcal C)\) of a monoidal category \(\mathcal C\) is recalled, and show that a quasi-coalgebra in \(\mathrm{Split}(\mathcal C)\) defines a quasi-monoidal represented functor from \(\mathcal C\) to \(\mathrm{Vec}\). (C) \(\mathfrak g\) is a Lie subalgebra of the Drinfeld double \(\mathfrak p\). In the category \(\mathcal P\) of of \(\mathfrak p\)-modules, find a commutative subalgebra \(A\) and construct the category \(\mathcal F'\) of free \(A\)-modules. (D) Show \(\mathcal F'\) is isomorphic to the category \(\mathcal P'\) whose objects are objects in \(\mathcal P\) but whose morphisms are taken from \(\mathcal G\), the category of \(\mathfrak g\)-modules. This isomorphism is used to view \(\mathcal U\mathfrak g\) as as isomorphic to \(\mathrm{End}(h: \mathcal F\to \mathcal V^{(cpl)}\), where \(h\) is the functor represented by a certain object in \(\mathrm{Split}(\mathcal F')\). (E) A category \((\mathcal F')_h\) is introduced, whose objects are those of \(\mathcal F'\), but the hom-sets are the hom-sets in \(\mathcal F'\) tensored with \(K[[h]]\). This makes \(\mathcal U\mathfrak g(h)\) isomorphic to \(\mathrm{End}(h: (F_h)'\to (V^{cpl}))_{(K[[h]])}\). (F) An associator \(\mathcal P_h\) is used to change the associativity and braiding isomorphisms in \(\mathcal P\) to get a new braided monoidal category \((\mathcal P_h)^\Phi\). \(A\) stays commutative there, and the category \(((\mathcal F')_h)^\Phi\) of free \(A\)-modules in \((\mathcal P_h)^\Phi\) is formed. (G) Construct a map from \(\mathrm{Split}(\mathcal F')_h )\) to \(\mathrm{Split}((\mathcal F')_h)^\Phi)\). A certain element of the image is identified, and the corresponding represented functor is \[ h^\Phi: ((F')_h)^\Phi\to (V^{cpl})_{(K[[h]])}. \] (H) A quasi-coalgebra in \(\mathrm{Split}(((\mathcal F')_h)^\Phi\) is identified, and thus \(h^\Phi\) is a is a quasi-monoidal functor. From (A) it follows that \(\mathrm{End}(h^\Phi)\) is a quasi-bialgebra. (I) Get an isomorphism of \(K[[h]]\)-modules from \(\mathcal U\mathfrak g[[h]]\) to \(\mathrm{End}(h^\Phi: ((\mathcal F')_h)^\Phi \to (V^{(cpl)})_{(K[[h]])}\). The quasi-bialgebra structure on \(\mathrm{End}(h^\Phi)\) yields a new quasi-bialgebra structure on \(\mathcal U\mathfrak g[[h]]\). (J) As categories, \(\mathcal P_h\) and \((\mathcal P_h)^\Phi\) are isomorphic. Show that the quasi-monoidal functors \[ (\mathcal P_h)^\Phi=\mathcal P_h \to (\mathcal F')_h\to (V^{(h-cpl)}))_{(K[[h]])} \] (here the first arrow is taking the completed tensor product with \(A\), and the second arrow is \(h\)) and \[ (\mathcal P_h)^\Phi\to ((\mathcal F')_h)^\Phi\to(V^{(cpl)})_{K[[h]]} \] (here the first arrow is taking the completed tensor product with \(A\), and the second arrow is \(h^\Phi)\) are naturally isomorphic. This isomorphism is not quasi-monoidal and gives rise to a twist whose whose classical limit is computed. (K) (J) is used to show that (I) gives a quantization of the quasi-Lie bialgebra \(\mathfrak g\). The quantization of quasi-Lie bialgebras was addressed by \textit{B. Enriquez} and \textit{G. Halbout} [J. Am. Math. Soc. 23, No. 3, 611--653 (2010; Zbl 1328.17016)] using different methods. They compare the cohomologies of the prop of quasi-Lie bialgebras with those of the prop of Lie bialgebras in order to reduce their problem to the one solved by Etingof and Kazhdan. The main difference with the paper under review is that the paper under review replaces a certain free module with a projective module and uses that projections can be explicitly deformed.