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Quantization of coboundary Lie bialgebras. (English) Zbl 1242.17021

Let \(a\) be a Lie bialgebra over a field of characteristic zero. The problem of quantizing \(a\), as posed by V. G. Drinfeld [Lect. Notes Math. 1510, 1–8 (1992; Zbl 0765.17014)] was solved by P. Etingof and D. Kazhdan [Sel. Math., New Ser. 2, No. 1, 1–41 (1996; Zbl 0863.17008); Sel. Math., New Ser. 4, No. 2, 213–231, 233–269 (1998; Zbl 0915.17009)]. The first paper was for a finite-dimensional, the second for general \(a\). Roughly speaking, it means to give a topological Hopf algebra structure on \(U(a)[[h]]\), so that when the coproduct is skew-symmetrized, then modulo \(h\) one recovers the Lie bialgebra structure on \(a\). The paper under review achieves a quantization of \(a\) when \(a\) has cobracket given by the derivation induced by an element \(r\) in \(a \wedge a\) which satisfies the modified classical Yang-Baxter equation. Here the coalgebra structure on \(U= U(a)[[h]]\) has an invertible \(R\)-matrix \(R\) satisfying five properties: the opposite coproduct on \(U\) is conjugate to the coproduct via \(R\); \(R^{-1}\) is the standard twist of \(R\); the coproduct on \(U\) is coassociative modulo left actions involving \(R\); \(R= 1\) tensor \(1 \pmod{h}\); and a condition involving the counit of \(U\). \((U, R)\) is called a quantization of \((a,r)\) if a is the classical limit of \(U\), and \(R\) skew-symmetrized is congruent to \(2r \pmod{h}\).
This quantization also answers a a question of Drinfeld in his 1992 Selecta paper (see above). The authors’ solution can be viewed as a completion of the second author’s result on twist quantization [Adv. Math. 207, No. 2, 617–633 (2006; Zbl 1163.17303)], since the authors’ quantization is shown to be compatible with Lie bialgebra twists. A twist on \(a\) is an element \(f\) of \(a \wedge a\) satisfying a cobracket property. Modifying the cobracket of \(a\) by \(ad (f)\) yields another Lie bialgebra called a twist of \(a\). If \((a,r)\) is a coboundary Lie bialgebra, then \(-2r\) is a twist of \((a,r)\), and the twisted Lie bialgebra is the opposite coalgebra of \(a\), and the same holds for their quantized enveloping algebras.
The authors’ quantization of \((a,r)\) is formulated in the language of props. It involves three steps:
(1) They show that the Etingof-Kazhdan quantification functor \(Q\) is compatible with twists;
(2) They show that if \(Q\) corresponds to an even associator, then \(Q\) is compatible with with taking the opposite coalgebra of a Lie bialgebra and of a quantum universal enveloping algebra;
(3) Now, at the proper level, one has an isomorphism between \(Q(a, r)\) and its twist involving \(-2r\). They then prove that there is a twist \(R\) of \((a,r)\) with \(R^{-1}\) equal to the standard twist of \(R\), with \(Q(a,r)\) and its opposite coalgebra being conjugate by \(R\). In a final section, they show how their quantization implies that of certain quasi-Poisson homogeneous spaces.

MSC:

17B62 Lie bialgebras; Lie coalgebras
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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