# Drinfel'd-Turaev quantization

From Encyclopedia of Mathematics

A type of quantization typically encountered in knot theory, for example in Jones–Conway, homotopy or Kauffman bracket skein modules of three-dimensional manifolds ([a3], [a1], [a2], cf. also Skein module).

Fix a commutative ring with identity, . Let be a Poisson algebra over and let be an algebra over which is free as an -module (cf. also Free module). An -module epimorphism is called a Drinfel'd–Turaev quantization of if

i) for all and all ; and

ii) for all .

If is not required to be free as an -module, one obtains a so-called weak Drinfel'd–Turaev quantization.

#### References

[a1] | J. Hoste, J.H. Przytycki, "Homotopy skein modules of oriented -manifolds" Math. Proc. Cambridge Philos. Soc. , 108 (1990) pp. 475–488 |

[a2] | J.H. Przytycki, "Homotopy and -homotopy skein modules of -manifolds: An example in Algebra Situs" , Proc. Conf. in Low-Dimensional Topology in Honor of Joan Birman's 70th Birthday (Columbia Univ./Barnard College, March, 14-15, 1998) , Internat. Press (2000) |

[a3] | V.G. Turaev, "Skein quantization of Poisson algebras of loops on surfaces" Ann. Sci. École Norm. Sup. , 4 : 24 (1991) pp. 635–704 |

**How to Cite This Entry:**

Drinfel'd-Turaev quantization.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Drinfel%27d-Turaev_quantization&oldid=17078

This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article