View source for Drinfel'd-Turaev quantization

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

Fix a [[Commutative ring|commutative ring]] with identity, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130200/d1302001.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130200/d1302002.png" /> be a [[Poisson algebra|Poisson algebra]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130200/d1302003.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130200/d1302004.png" /> be an algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130200/d1302005.png" /> which is free as an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130200/d1302006.png" />-module (cf. also [[Free module|Free module]]). An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130200/d1302007.png" />-module [[Epimorphism|epimorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130200/d1302008.png" /> is called a Drinfel'd–Turaev quantization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130200/d1302009.png" /> if

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130200/d13020010.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130200/d13020011.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130200/d13020012.png" />; and

ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130200/d13020013.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130200/d13020014.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130200/d13020015.png" /> is not required to be free as an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130200/d13020016.png" />-module, one obtains a so-called weak Drinfel'd–Turaev quantization.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Hoste,   J.H. Przytycki,   "Homotopy skein modules of oriented <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130200/d13020017.png" />-manifolds"  ''Math. Proc. Cambridge Philos. Soc.'' , '''108'''  (1990)  pp. 475–488</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.H. Przytycki,   "Homotopy and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130200/d13020018.png" />-homotopy skein modules of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130200/d13020019.png" />-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)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  V.G. Turaev,   "Skein quantization of Poisson algebras of loops on surfaces"  ''Ann. Sci. École Norm. Sup.'' , '''4''' :  24  (1991)  pp. 635–704</TD></TR></table>