What vector space does the Kauffman bracket skein algebra of FxI act on? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T19:21:59Z http://mathoverflow.net/feeds/question/85850 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/85850/what-vector-space-does-the-kauffman-bracket-skein-algebra-of-fxi-act-on What vector space does the Kauffman bracket skein algebra of FxI act on? unknown (google) 2012-01-16T21:54:25Z 2012-01-17T00:14:22Z <p>The Kauffman bracket skein module $K_t(F\times I)$ (where $t$ is an indeterminant and $F$ is a closed surface) is an associative algebra (the operation being "stacking" links in the $I$ direction). It is a quantum deformation of the coordinate ring of the character variety <code>$\operatorname{Hom}(\pi_1(F),\operatorname{SL}(2))//\operatorname{SL}(2)$</code>. This skein algebra naturally acts on the vector space $\mathcal H_F$, the vector space associated to $F$ by the WRT TQFT.</p> <blockquote> <p>Is there a direct construction of this vector space (e.g. in terms of generators and relations) which makes the action of $K_t(F\times I)$ manifest?</p> </blockquote> <p>I know only two constructions of the vector space $\mathcal H_F$, and both are unsatisfactory for the purposes of this question. First, we could pick a complex structure on $F$, and let $\mathcal H_F$ be the global holomorphic sections of a particular line bundle $\mathcal L$ over the $\operatorname{SU}(2)$ character variety of $\pi_1(F)$. In this construction, I don't know of any explicit way to realize the action of $K_t(F\times I)$. Second, we could say the vector space is generated by all $3$-manfolds with boundary $F$, and quotient by the subspace which has zero WFT invariant when paired with all three-manifolds with boundary $-F$. This is unsatisfactory because the relations aren't "local", and they require understanding the WRT invariants for all three-manifolds.</p> <blockquote> <p>I am looking for a definition of the vector space in the spirit of the definition of $K_t(F\times I)$, i.e. some generators modulo some <em>local</em> "picture" relations.</p> </blockquote> http://mathoverflow.net/questions/85850/what-vector-space-does-the-kauffman-bracket-skein-algebra-of-fxi-act-on/85864#85864 Answer by Kevin Walker for What vector space does the Kauffman bracket skein algebra of FxI act on? Kevin Walker 2012-01-17T00:14:22Z 2012-01-17T00:14:22Z <p>The answer is $K_t(H)$, where $H$ is a handlebody with boundary $F$. If $t$ is a root of 1 and we are taking the usual semisimple quotient, then $K_t(F\times I)$ is isomorphic to a matrix algebra and $K_t(H)$ isomorphic to the standard representation. Also, in this case we can let $H$ be <em>any</em> 3-manifold with boundary $F$ -- it doesn't even need to be connected.</p>