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 ([a3], [a1], [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

[a1]	J. Hoste, J.H. Przytycki, "Homotopy skein modules of oriented $3$-manifolds" Math. Proc. Cambridge Philos. Soc. , 108 (1990) pp. 475–488
[a2]	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)
[a3]	V.G. Turaev, "Skein quantization of Poisson algebras of loops on surfaces" Ann. Sci. École Norm. Sup. , 4 : 24 (1991) pp. 635–704