Quantized coordinate ring
Pythagoras0 (토론 | 기여)님의 2020년 11월 13일 (금) 17:57 판
introduction
- \(\mathfrak{g}\) : simple Lie algebra over \(\mathbb{C}\)
- \(G\) : connected, simply-connected simple algebraic group with Lie algebra \(\mathfrak{g}\)
dual of quantized enveloping algebras
QEA
- \(q\in \mathbb{C}^{\times}\) not a root of 1
- \(U_q:=U_q(\mathfrak{g})=\langle k_i^{\pm},e_i,f_i :i\in I \rangle\) : quantum enveloping algebra
quantized coordinate algebra
- \(U_q^{*}=\operatorname{Hom}_{\mathbb{Q}(q)}(U_q,\mathbb{Q}(q))\)
- \(A_q(\mathfrak{g}):=\{\varphi \in U_q^{*} | \dim U_q \varphi, \dim \varphi U_q <\infty \}\)
- also denoted by \(\mathbb{C}_q[G]\)
- We call \(A_q(\mathfrak{g})\) the quantized coordinate ring
comodules and modules
- \(\mathbb{C}_q[G]\)-comodules = locally finite \(U_q(\mathfrak{g})\)-modules of type 1
- thm (Soibelman)
result of Kuniba, Okado and Yamada
- thm
The transition matrix of PBW-type bases of the positive-half of a quantized universal enveloping algebra \(U_q(\mathfrak{g})\) coincides with a matrix coefficients of the intertwiner between certain irreducible modules over the corresponding quantized coordinate ring \(A_q(\mathfrak{g})\), introduced by Soibelman.
cluster theory
- Monoidal categorifications of cluster algebras
- \(\mathbb{C}[N]\) is Hopf dual to \(U(\mathfrak{n})\) where \(\mathfrak{n}=Lie(N)\)
- Ringel, Lusztig : Geometric realization of \(U_q(\mathfrak{n})\) via constructible sheaves on varieties of \(\mathbb{C}Q\)-modules
- Lusztig : Geometric realization of \(U(n)\) via constructible functions on varieties of \(\Lambda\)-modules
- Geiss-Leclerc-S : Dualizing Lusztig's construction, get a cluster character
expositions
- Saito, Yoshihisa (University of Tokyo) Quantized coordinate rings, PBW bases and q-boson algebras
- Tanisaki, Toshiyuki (Osaka City University) Modules over quantized coordinate algebras and PBW-bases
articles
- Geiss, Christof, Bernard Leclerc, and Jan Schröer. “Quivers with Relations for Symmetrizable Cartan Matrices III: Convolution Algebras.” arXiv:1511.06216 [math], November 19, 2015. http://arxiv.org/abs/1511.06216.
- Geiss, Christof, Bernard Leclerc, and Jan Schröer. “Quivers with Relations for Symmetrizable Cartan Matrices II: Change of Symmetrizers.” arXiv:1511.05898 [math], November 18, 2015. http://arxiv.org/abs/1511.05898.
- Oya, Hironori. “Representations of Quantized Function Algebras and the Transition Matrices from Canonical Bases to PBW Bases.” arXiv:1501.01416 [math], January 7, 2015. http://arxiv.org/abs/1501.01416.
- Tanisaki, T. “Modules over Quantized Coordinate Algebras and PBW-Bases.” arXiv:1409.7973 [math], September 28, 2014. http://arxiv.org/abs/1409.7973.
books
- Korogodski, Leonid I., and Yan S. Soibelman. Algebras of Functions on Quantum Groups. American Mathematical Soc., 1998.