论文标题
Chebyshev-frobenius同构代表了三个manifolds的绞线模块
The Chebyshev-Frobenius homomorphism for stated skein modules of 3-manifolds
论文作者
论文摘要
我们研究了标记的3个manifolds的规定绞线模块。我们将表面的绞线代数的分裂同构概括为3个manifolds所述的绞线模块的分裂同态。我们表明，存在3个manifolds陈述的绞线模块的Chebyshev-frobenius同构，这扩展了最初由Bonahon和Wong最初建造的未标记表面的Skein代数的Chebyshev同态性。此外，我们证明了Chebyshev-Frobenius地图以分裂的同构为通勤。然后用来表明，在表面的符号代数的情况下，Chebyshev-Frobenius地图是$ \ Mathcal {o} _ {o} _ {q^2}（sl（2）（sl（2））$ triangular triangular decomptostion triand decomptostion的双重frobenius地图（在lusztig的意义上）的独特扩展。特别是，这给出了Lusztig的Frobenius同构的HopF双重型的绞kin理论构建。给出了第二个概念框架，该框架表明，表面陈述的链球代数的Chebyshev-frobenius同构是通过量子痕迹图对量子Tori的Frobenius同构的独特限制。
We study the stated skein modules of marked 3-manifolds. We generalize the splitting homomorphism for stated skein algebras of surfaces to a splitting homomorphism for stated skein modules of 3-manifolds. We show that there exists a Chebyshev-Frobenius homomorphism for the stated skein modules of 3-manifolds which extends the Chebyshev homomorphism of the skein algebras of unmarked surfaces originally constructed by Bonahon and Wong. Additionally, we show that the Chebyshev-Frobenius map commutes with the splitting homomorphism. This is then used to show that in the case of the stated skein algebra of a surface, the Chebyshev-Frobenius map is the unique extension of the dual Frobenius map (in the sense of Lusztig) of $\mathcal{O}_{q^2}(SL(2))$ through the triangular decomposition afforded by an ideal triangulation of the surface. In particular, this gives a skein theoretic construction of the Hopf dual of Lusztig's Frobenius homomorphism. A second conceptual framework is given, which shows that the Chebyshev-Frobenius homomorphism for the stated skein algebra of a surface is the unique restriction of the Frobenius homomorphism of quantum tori through the quantum trace map.
