"Quantized coordinate ring"의 두 판 사이의 차이
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imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) |
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==introduction== | ==introduction== | ||
| − | * | + | * <math>\mathfrak{g}</math> : simple Lie algebra over <math>\mathbb{C}</math> |
| − | * | + | * <math>G</math> : connected, simply-connected simple algebraic group with Lie algebra <math>\mathfrak{g}</math> |
==dual of quantized enveloping algebras== | ==dual of quantized enveloping algebras== | ||
===QEA=== | ===QEA=== | ||
| − | * | + | * <math>q\in \mathbb{C}^{\times}</math> not a root of 1 |
| − | * | + | * <math>U_q:=U_q(\mathfrak{g})=\langle k_i^{\pm},e_i,f_i :i\in I \rangle</math> : quantum enveloping algebra |
===quantized coordinate algebra=== | ===quantized coordinate algebra=== | ||
| − | * | + | * <math>U_q^{*}=\operatorname{Hom}_{\mathbb{Q}(q)}(U_q,\mathbb{Q}(q))</math> |
| − | * | + | * <math>A_q(\mathfrak{g}):=\{\varphi \in U_q^{*} | \dim U_q \varphi, \dim \varphi U_q <\infty \}</math> |
| − | ** also denoted by | + | ** also denoted by <math>\mathbb{C}_q[G]</math> |
| − | * We call | + | * We call <math>A_q(\mathfrak{g})</math> the quantized coordinate ring |
==comodules and modules== | ==comodules and modules== | ||
| − | * | + | * <math>\mathbb{C}_q[G]</math>-comodules = locally finite <math>U_q(\mathfrak{g})</math>-modules of type 1 |
;thm (Soibelman) | ;thm (Soibelman) | ||
| 23번째 줄: | 23번째 줄: | ||
==result of Kuniba, Okado and Yamada== | ==result of Kuniba, Okado and Yamada== | ||
;thm | ;thm | ||
| − | The transition matrix of PBW-type bases of the positive-half of a quantized universal enveloping algebra | + | The transition matrix of PBW-type bases of the positive-half of a quantized universal enveloping algebra <math>U_q(\mathfrak{g})</math> coincides with a matrix coefficients of the intertwiner between certain irreducible modules over the corresponding quantized coordinate ring <math>A_q(\mathfrak{g})</math>, introduced by Soibelman. |
==cluster theory== | ==cluster theory== | ||
* [[Monoidal categorifications of cluster algebras]] | * [[Monoidal categorifications of cluster algebras]] | ||
| − | * | + | * <math>\mathbb{C}[N]</math> is Hopf dual to <math>U(\mathfrak{n})</math> where <math>\mathfrak{n}=Lie(N)</math> |
| − | * Ringel, Lusztig : Geometric realization of | + | * Ringel, Lusztig : Geometric realization of <math>U_q(\mathfrak{n})</math> via constructible sheaves on varieties of <math>\mathbb{C}Q</math>-modules |
| − | * Lusztig : Geometric realization of | + | * Lusztig : Geometric realization of <math>U(n)</math> via constructible functions on varieties of <math>\Lambda</math>-modules |
* Geiss-Leclerc-S : Dualizing Lusztig's construction, get a cluster character | * Geiss-Leclerc-S : Dualizing Lusztig's construction, get a cluster character | ||
2020년 11월 13일 (금) 17:57 기준 최신판
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.