Quasi-Lie bialgebras of loops in quasisurfaces. (English) Zbl 1520.57018

Turaev, Vladimir

Kyoto J. Math. 63, No. 1, 87-116 (2023).

W. M. Goldman [Invent. Math. 85, 263–302 (1986; Zbl 0619.58021)] defined a Lie bracket in the module generated by free homotopy classes of loops in an oriented surface. V. G. Turaev [Ann. Sci. Éc. Norm. Supér. (4) 24, No. 6, 635–704 (1991; Zbl 0758.57011)] defined a Lie cobracket such that, together with the Goldman bracket, they form a Lie bialgebra. In the article under review the author considers similar operations on loops in more general topological spaces called quasisurfaces.
A quasisurface \(X\) is a space obtained by gluing a surface \(\Sigma\) to an arbitrary topological space \(Y\) along a mapping of several disjoint subsegments of \(\partial \Sigma\) to \(Y\). The space \(X\) splits as a union of the surface core and the singular part, which meet at the subsegments.
Considering the loops in \(X\) and their intersections in \(\Sigma\), the author defines a bracket in the module \(M\) freely generated by the set of free homotopy classes of loops in \(X\). This bracket is skew-symmetric but may not satisfy the Jacobi identity. Considering self-intersections of loops, a skew-symmetric cobracket in \(M\) is also obtained.
These operations on loops induce a quasi-Lie bialgebra structure in the quotient of the module \(M\) by the homotopy class of contractible loops in \(X\). For loops in the surface core \(\Sigma\) of \(X\), the standard Lie bialgebra of loops in \(\Sigma\) is recovered.

Reviewer: Miguel Paternain (Montevideo)

MSC:

57K20	2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.)
17B62	Lie bialgebras; Lie coalgebras

Keywords:

brackets; cobrackets; quasisurfaces; surfaces

Citations:

Zbl 0619.58021; Zbl 0758.57011

References:

[1]	M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. Lond. Ser. A 308 (1983), no. 1505, 523-615. · Zbl 0509.14014 · doi:10.1098/rsta.1983.0017
[2]	M. Chas, Combinatorial Lie bialgebras of curves on surfaces, Topology 43 (2004), 543-568. · Zbl 1050.57014 · doi:10.1016/j.top.2003.03.003
[3]	V. V. Fock and A. A. Rosly, Poisson structure on moduli of flat connections on Riemann surfaces and the r-matrix, (in Russian) Moscow Seminar in Mathematical Physics, 67-86; English translation in Amer. Math. Soc. Transl. Ser. 2 191, Adv. Math. Sci. 43, Amer. Math. Soc., Providence, 1999. · Zbl 0945.53050 · doi:10.1090/trans2/191/03
[4]	W. M. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. Math. 54 (1984), no. 2, 200-225. · Zbl 0574.32032 · doi:10.1016/0001-8708(84)90040-9
[5]	W. M. Goldman, Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Invent. Math. 85 (1986), no. 2, 263-302. · Zbl 0619.58021 · doi:10.1007/BF01389091
[6]	N. Kawazumi and Y. Kuno, The logarithms of Dehn twists, Quantum Topol. 5 (2014), no. 3, 347-423. · Zbl 1361.57027 · doi:10.4171/QT/54
[7]	N. Kawazumi and Y. Kuno, Intersection of curves on surfaces and their applications to mapping class groups, Ann. Inst. Fourier (Grenoble) 65 (2015), no. 6, 2711-2762. · Zbl 1370.57009
[8]	G. Massuyeau and V. Turaev, Quasi-Poisson structures on representation spaces of surfaces, Int. Math. Res. Not. 2014, no. 1, 1-64 · Zbl 1298.53085 · doi:10.1093/imrn/rns215
[9]	V. Turaev, Skein quantization of Poisson algebras of loops on surfaces, Ann. Sci. Éc. Norm. Supér. (4) 24 (1991), no. 6, 635-704. · Zbl 0758.57011
[10]	V. Turaev, Loops in surfaces and star-fillings; English translation in Progr. Math. 340, 617-643, Springer, Cham, 2021. · Zbl 1497.55009 · doi:10.1007/978-3-030-78148-4_21
[11]	S. Wolpert, On the symplectic geometry of deformations of a hyperbolic surface, Ann. of Math. (2) 117 (1983), 207-234. · Zbl 0518.30040 · doi:10.2307/2007075