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Difference between revisions of "Drinfel'd-Turaev quantization"

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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]]).
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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|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
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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) <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
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i) $\phi(p(q)a) = p(1)\phi(a)$ for all $a\in A$ and all $p(q) \in R[q^{\pm1}]$; 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" />.
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ii) $ab-ba \in (q-1)\phi^{-1}([\phi(a),\phi(b)])$ for all $a,b \in P$.
  
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.
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If $A$ is not required to be free as an $R[z]$-module, one obtains a so-called weak Drinfel'd–Turaev quantization.
  
 
====References====
 
====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>
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<table>
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<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>
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<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>
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<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>
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</table>
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Latest revision as of 20:52, 5 March 2018

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
How to Cite This Entry:
Drinfel'd-Turaev quantization. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Drinfel%27d-Turaev_quantization&oldid=22359
This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article