The quantum dilogarithm and unitary representations of the cluster mapping class groups

@misc{citeulike:1340740, abstract = {Cluster modular groups are discrete groups acting by automorphisms of cluster varieties. They include the classical modular groups of punctured surfaces. also known as the mapping class groups. Our main result is construction of infinite dimensional unitary projective representations of cluster modular groups, defined explicitly via the quantum dilogarithm. It can be viewed as an analog of the Weil representation. In both cases representations are given by integral operators. The kernels of these intertwiners are exponentials of quadratic forms for the latter, and quantum dilogarithms and exponentials of quadratic forms in our case. One of applications of our construction is quantization of higher Teichm\"uller spaces. We show how the symplectic double appears in higher Teichmuller theory. This is new even in the classical situation.}, author = {Fock, V. V. and Goncharov, A. B. }, citeulike-article-id = {1340740}, eprint = {math.QA/0702397}, keywords = {claster, fock}, month = {Feb}, posted-at = {2007-05-29 12:29:37}, priority = {2}, title = {The quantum dilogarithm and unitary representations of the cluster mapping class groups}, url = {http://arxiv.org/abs/math.QA/0702397}, year = {2007} }

TY - ELEC ID - citeulike:1340740 L3 - citeulike-article-id:1340740 TI - The quantum dilogarithm and unitary representations of the cluster mapping class groups N2 - Cluster modular groups are discrete groups acting by automorphisms of cluster varieties. They include the classical modular groups of punctured surfaces. also known as the mapping class groups. Our main result is construction of infinite dimensional unitary projective representations of cluster modular groups, defined explicitly via the quantum dilogarithm. It can be viewed as an analog of the Weil representation. In both cases representations are given by integral operators. The kernels of these intertwiners are exponentials of quadratic forms for the latter, and quantum dilogarithms and exponentials of quadratic forms in our case. One of applications of our construction is quantization of higher Teichm\"uller spaces. We show how the symplectic double appears in higher Teichmuller theory. This is new even in the classical situation. KW - claster KW - fock AU - Fock, VV AU - Goncharov, AB PY - 2007/Feb/19/ UR - http://arxiv.org/abs/math.QA/0702397 ER -