View source for Drinfel'd-Turaev quantization

A type of quantization typically encountered in [[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]]).

Fix a [[commutative ring]] with identity, $R$. Let $P$ be a [[Poisson algebra]] over $R$ and let $A$ be an algebra over $R[q^{\pm1}]$ which is free as an $R[q^{\pm1}]$-module (cf. also [[Free module]]). An $R$-module [[epimorphism]] $\phi:A \rightarrow P$ is called a Drinfel'd–Turaev quantization of $P$ if

i) $\phi(p(q)a) = p(1)\phi(a)$ for all $a\in A$ and all $p(q) \in R[q^{\pm1}]$; and

ii) $ab-ba \in (q-1)\phi^{-1}([\phi(a),\phi(b)])$ for all $a,b \in P$.

If $A$ is not required to be free as an $R[z]$-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 $3$-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 $q$-homotopy skein modules of $3$-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>

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