##
**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.

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.

Reviewer: J.Huebschmann (Villeneuve d’Ascq)

### 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) |

### Keywords:

skein invariants of 3-manifolds; Jones polynomial; Poisson algebras of loops on surfaces; quantization; Lie bialgebras; Poisson Lie groups; set of free homotopy classes of loops; skein module; Wilson loop observables; infinitesimal Poisson Lie group; bi-Poisson bialgebras; biquantization### Citations:

Zbl 0619.58021; Zbl 0526.58017; Zbl 0667.57005; Zbl 0725.57007; Zbl 0377.53024; Zbl 0377.53025
PDF
BibTeX
XML
Cite

\textit{V. G. Turaev}, Ann. Sci. Éc. Norm. Supér. (4) 24, No. 6, 635--704 (1991; Zbl 0758.57011)

### References:

[1] | V. G. DRINFELD , Hamiltonian Structures on Lie Groups, Lie Bialgebras and the Meaning of the classical Yang-Baxter Equation (Doklady A.N. S.S.S.R., Vol. 268, 1982 , pp. 285-287 ; English translation : Soviet Math. Dokl., Vol. 27, No. 1, 1983 ). MR 84i:58044 | Zbl 0526.58017 · Zbl 0526.58017 |

[2] | V. G. DRINFELD , Quantum Groups (Proc. Internat. Congress Math., Berkeley, 1986 , Amer. Math. Soc., Providence, R.I., 1987 , pp. 798-820). MR 89f:17017 | Zbl 0667.16003 · Zbl 0667.16003 |

[3] | W. M. GOLDMAN , The Symplectic Nature of Fundamental Groups of Surfaces (Adv. Math., Vol. 54, 1984 , pp. 200-225). MR 86i:32042 | Zbl 0574.32032 · Zbl 0574.32032 |

[4] | W. M. GOLDMAN , Invariant Functions on Lie Groups and Hamiltonian Flows of Surface Group Representations (Invent. Math., Vol. 85, 1986 , pp. 263-302). MR 87j:32069 | Zbl 0619.58021 · Zbl 0619.58021 |

[5] | J. HOSTE and M. K. KIDWELL , Dichromatic Link Invariants , (Trans. Amer. Math. Soc., Vol. 321, 1990 , pp. 197-229). MR 90m:57007 | Zbl 0702.57002 · Zbl 0702.57002 |

[6] | F. JAEGER , Composition Products and Models for the Homfly Polynomial , preprint, Grenoble, 1988 . |

[7] | L. H. KAUFFMAN , New Invariants in the Theory of Knots (Am. Math. Mon., Vol. 95, 1988 , pp. 195-242). MR 89d:57005 | Zbl 0657.57001 · Zbl 0657.57001 |

[8] | M. KONTSEVICH , Rational Conformal Field Theory and Invariants of 3-Manifolds , preprint, 1989 . |

[9] | W. B. R. LICKORISH , Polynomials for Links (Bull. London Math. Soc., Vol. 20, 1988 , pp. 558-588). MR 90d:57004 | Zbl 0685.57001 · Zbl 0685.57001 |

[10] | W. B. R. LICKORISH and K. C. MILLETT , A Polynomial Invariant of Oriented Links (Topology, Vol. 26, 1987 , pp. 107-141). MR 88b:57012 | Zbl 0608.57009 · Zbl 0608.57009 |

[11] | J. H. PRZYTYCKI , Skein Modules of 3-manifolds , preprint, 1987 . · Zbl 0762.57013 |

[12] | N. Y. RESHETIKHIN , Quantized Universal Enveloping Algebras, the Yang-Baxter Equation and Invariants of Links , I and II, L.O.M.I. preprints E-4-87 and E-17-87, Leningrad, 1988 . |

[13] | N. Y. RESHETIKHIN and V. G. TURAEV , Invariants of 3-manifolds via Link Polynomials and Quantum Groups , (Invent. Math., Vol. 103, 1991 , pp. 547-597). MR 92b:57024 | Zbl 0725.57007 · Zbl 0725.57007 |

[14] | M. A. SEMENOV-TIAN-SHANSKY , Dressing Transformations and Poisson Group Actions (Publ. Res. Inst. Math. Sci. Kyoto Univ., Vol. 21, 1985 , pp. 1237-1260). Article | MR 88b:58057 | Zbl 0674.58038 · Zbl 0674.58038 |

[15] | J.-P. SERRE , Lie Algebras and Lie Groups , Benjamin, New York-Amsterdam, 1965 . MR 36 #1582 | Zbl 0132.27803 · Zbl 0132.27803 |

[16] | V. G. TURAEV , Intersections of Loops in Two-Dimensional Manifolds (Matem. Sbornik, Vol. 106, 1978 , pp. 566-588 ; English translation : Math. U.S.S.R. Sbornik, Vol. 35, 1979 , pp. 229-250). MR 80d:57019 | Zbl 0422.57005 · Zbl 0422.57005 |

[17] | V. G. TURAEV , Multivariable Generalizations of the Seifert form of Classical Knot (Matem. Sbornik, Vol. 116, 1981 , pp. 370-397 ; English translation : Math. U.S.S.R. Sbornik, (Vol. 44, 1983 , pp. 335-361). Zbl 0508.57005 · Zbl 0508.57005 |

[18] | V. G. TURAEV , The Yang-Baxter Equation and Invariants of Links (Invent. Math., Vol. 92, 1988 , pp. 527-553). MR 89e:57003 | Zbl 0648.57003 · Zbl 0648.57003 |

[19] | V. G. TURAEV , The Conway and Kauffman Modules of the Solid Torus (Zap. nauchn. sem. L.O.M.I., Vol. 167, 1988 , pp. 79-89 ; English translation : J. Soviet. Math.). MR 90f:57012 | Zbl 0673.57004 · Zbl 0673.57004 |

[20] | V. G. TURAEV , Algebras of Loops on Surfaces, Algebras of Knots and Quantization , preprint L.O.M.I., Leningrad, 1988 . · Zbl 0853.57016 |

[21] | J.-L. VERDIER , Groupes quantiques (d’après V. G. Drinfel’d) (Astérisque, Vol. 152-153, 1987 , pp. 305-319). Numdam | MR 89f:17021 | Zbl 0645.16006 · Zbl 0645.16006 |

[22] | E. WITTEN , Quantum Field Theory and the Jones Polynomial , (Comm. Math. Phys., Vol. 121, 1989 , pp. 351-399). Article | MR 90h:57009 | Zbl 0667.57005 · Zbl 0667.57005 |

[23] | S. WOLPERT , On the Symplectic Geometry of Deformations of Hyperbolic Surface (Ann. Math., Vol. 117, 1983 , pp. 207-234). MR 85e:32028 | Zbl 0518.30040 · Zbl 0518.30040 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.