양자 다이로그 항등식 (quantum dilogarithm identities)

개요

• 함수 다이로그 항등식(functional dilogarithm identity)의 양자화

바일 대수(Weyl algebra)와 양자 다이로그 항등식

• 양자 바일 대수와 양자평면 참조
• \(\mathbb{C}[q,q^{-1}]\) 위에서 u,v 로 생성되는 대수, \(uv=qvu\) 를 만족시킴
  • q-이항계수 (가우스 다항식) 에서 양자평면이라는 이름으로 사용됨
• Schützenberger 항등식

\[(u;q)_{\infty}(v;q)_{\infty}=(u+v;q)_{\infty}\]

• Faddeev-Volkov 항등식

\[(v;q)_{\infty}(u;q)_{\infty}=(u+v-vu;q)_{\infty}\]

• Faddeev-Kashaev 양자 5항 관계식 (5-term relation)

\[(v;q)_{\infty}(u;q)_{\infty}=(u;q)_{\infty}(-vu;q)_{\infty}(v;q)_{\infty}\]

리뷰논문, 에세이, 강의노트

• Quiver mutations and quantum dilogarithm identities, presentation, Isle of Skye, June 27, 2011
• Quantum dilogarithm identities from quiver mutations, video of a talk given at Banff, September 9, 2010.

관련논문

• Ip, Ivan Chi-Ho, and Masahito Yamazaki. ‘Quantum Dilogarithm Identities at Root of Unity’. arXiv:1412.5777 [hep-Th], 18 December 2014. http://arxiv.org/abs/1412.5777.
• Fu, Changjian, and Liangang Peng. 2013. “Quantum Dilogarithm Identities and Cyclic Quivers.” arXiv:1305.5395 (May 23). http://arxiv.org/abs/1305.5395.
• Bytsko, Andrei, and Alexander Volkov. 2013. “Tetrahedron Equation and Cyclic Quantum Dilogarithm Identities.” arXiv:1304.1641 (April 5). http://arxiv.org/abs/1304.1641.
• Faddeev, L. D. 2012. “Volkov’s Pentagon for the Modular Quantum Dilogarithm.” arXiv:1201.6464 (January 31). http://arxiv.org/abs/1201.6464.
• Kashaev, Rinat M., and Tomoki Nakanishi. 2011. “Classical and Quantum Dilogarithm Identities.” arXiv:1104.4630 (April 24). doi:10.3842/SIGMA.2011.102. http://arxiv.org/abs/1104.4630.
• Keller, Bernhard. 2011. “On Cluster Theory and Quantum Dilogarithm Identities.” arXiv:1102.4148 (February 21). http://arxiv.org/abs/1102.4148.
• Quantum Dilogarithm L.D.Fadeev and R.M.Kashaev, Mod. Phys. Lett. A. 9 (1994) p.427–434 MR1264393(95i:11150)