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Skein quantization of Poisson algebras of loops on surfaces. (English) Zbl 0758.57011

Let \(K\) be a commutative ring with 1, \(F\) a surface, denote by \(\hat\pi\) the set of free homotopy classes of maps \(S^ 1\to F\), and let \(Z\) be the free \(K\)-module on \(\hat \pi\). Using composition and intersection of loops, W. Goldman [Invent. Math. 85, 263-302 (1986; Zbl 0619.58021)] introduced a structure of a Lie algebra on \(Z\). By general principles, the symmetric algebra \(S=S(Z)\) then inherits a structure of a Poisson algebra. In the paper under review a link between the skein invariants of 3-manifolds generalizing the Jones polynomial and the Poisson algebras of loops on surfaces is obtained in various ways by means of quantization, involving notions such as Lie bialgebras and Poisson Lie groups introduced by V. G. Drinfel’d [Dokl. Akad. Nauk SSSR 268, 285-287 (1983); Engl. transl.: Sov. Math., Dokl. 27, 68-71 (1983; Zbl 0526.58017)]. While the quantum nature of knots and links has already recently shown up at various other places, cf. e.g. E. Witten [Commun. Math. Phys. 121, 351-399 (1989; Zbl 0667.57005)], N. Reshetikhin and the author [Invent. Math. 103, 547-597 (1991; Zbl 0725.57007)], the present work shows in particular that also the algebraic notions of Lie bialgebras and Poisson Lie groups arise in the purely topological study of loops on surfaces.
The specific results of the paper under review fall into four main parts. Firstly a structure of non-commutative algebra \(A\) on certain skein modules defined over the product \(F\times I\) of \(F\) with an interval is obtained furnishing a quantization of the Poisson algebra \(S\). The notion of quantization coming into play here is closely related to but somewhat weaker than that of usual deformation quantization [cf., e.g., F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, Ann. Phys. 111, 61-110, 111-151 (1978; Zbl 0377.53024, Zbl 0377.53025)]; a quantization in the present sense consists of an algebra \(A\) containing an element \(h\) serving as deformation parameter together with a map \(p: A\to S\) having suitable properties but the algebra \(A\) is not required to be a topologically free \(K[[h]]\)-module. In the paper, for an arbitrary oriented 3-manifold \(M\), with \(\hat \pi\) momentarily denoting the set of free homotopy classes of loops in \(M\), a map from the skein module \(A(M)\) to the symmetric algebra \(S(M)\) of the free \(K\)-module on \(\hat\pi\) is obtained by sending an oriented link to the element in \(\hat \pi\) represented by it. For \(M=F\times I\), with reference to the obvious orientation of \(I\), given two links, we can put one on top of the other; this operation induces an in general non- commutative product on the skein module \(A(M)\). The quantum nature of the construction is reflected by the fact that the product of two elements represented by two links depends on the order in which one is put on top of the other. In a sense, the construction yields a rigorous quantization of what are called Wilson loop observables in \(M=F\times I\). Also a version of the construction is given involving non-oriented geometric data.
The results in the remaining three parts are more technical and we content ourselves with a few remarks: The Lie structure on \(Z\) is extended to that of a Lie bialgebra in the paper, and a suitable bialgebra (or Hopf algebra) quantizes this structure. This is the second quantization result. The third one refers to a certain “ infinitesimal Poisson Lie group” derived from a suitable Lie subalgebra \(Z_ 0\) of \(Z\). The fourth result involves bi-Poisson bialgebras and biquantization; these notions came up in a search for adequate treatment of geometric situations.

MSC:

57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
53D50 Geometric quantization
81Q99 General mathematical topics and methods in quantum theory
81T99 Quantum field theory; related classical field theories
57N10 Topology of general \(3\)-manifolds (MSC2010)
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References:

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