Quantized coordinate ring - 편집 역사

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@@ 1번째 줄: / 1번째 줄: @@
 ==introduction==
-* $\mathfrak{g}$ : simple Lie algebra over $\mathbb{C}$
+* <math>\mathfrak{g}</math> : simple Lie algebra over <math>\mathbb{C}</math>
-* $G$ : connected, simply-connected simple algebraic group with Lie algebra $\mathfrak{g}$
+* <math>G</math> : connected, simply-connected simple algebraic group with Lie algebra <math>\mathfrak{g}</math>
 ==dual of quantized enveloping algebras==
 ===QEA===
-* $q\in \mathbb{C}^{\times}$ not a root of 1
+* <math>q\in \mathbb{C}^{\times}</math> 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
+* <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===
-* $U_q^{*}=\operatorname{Hom}_{\mathbb{Q}(q)}(U_q,\mathbb{Q}(q))$
+* <math>U_q^{*}=\operatorname{Hom}_{\mathbb{Q}(q)}(U_q,\mathbb{Q}(q))</math>
-* $A_q(\mathfrak{g}):=\{\varphi \in U_q^{*} | \dim U_q \varphi, \dim \varphi U_q <\infty \}$
+* <math>A_q(\mathfrak{g}):=\{\varphi \in U_q^{*} | \dim U_q \varphi, \dim \varphi U_q <\infty \}</math>
-** also denoted by $\mathbb{C}_q[G]$
+** also denoted by <math>\mathbb{C}_q[G]</math>
-* We call $A_q(\mathfrak{g})$ the quantized coordinate ring
+* We call <math>A_q(\mathfrak{g})</math> the quantized coordinate ring
 ==comodules and modules==
-* $\mathbb{C}_q[G]$-comodules = locally finite $U_q(\mathfrak{g})$-modules of type 1
+* <math>\mathbb{C}_q[G]</math>-comodules = locally finite <math>U_q(\mathfrak{g})</math>-modules of type 1
 ;thm (Soibelman)
@@ 23번째 줄: / 23번째 줄: @@
 ==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.
+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==
 * [[Monoidal categorifications of cluster algebras]]
-* $\mathbb{C}[N]$ is Hopf dual to $U(\mathfrak{n})$ where $\mathfrak{n}=Lie(N)$
+* <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 $U_q(\mathfrak{n})$ via constructible sheaves on varieties of $\mathbb{C}Q$-modules
+* 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 $U(n)$ via constructible functions on varieties of $\Lambda$-modules
+* 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

← 이전 판		2020년 11월 14일 (토) 00:51 판
51번째 줄:		51번째 줄:
	==books==		==books==
	* Korogodski, Leonid I., and Yan S. Soibelman. Algebras of Functions on Quantum Groups. American Mathematical Soc., 1998.		* Korogodski, Leonid I., and Yan S. Soibelman. Algebras of Functions on Quantum Groups. American Mathematical Soc., 1998.
		+	[[분류:migrate]]

@@ 44번째 줄: / 44번째 줄: @@
 ==articles==
 * 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.

@@ 44번째 줄: / 44번째 줄: @@
 ==articles==
 * 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.

← 이전 판		2014년 12월 5일 (금) 00:56 판
20번째 줄:		20번째 줄:
	;thm (Soibelman)		;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.

← 이전 판		2014년 8월 9일 (토) 04:40 판
36번째 줄:		36번째 줄:
	==articles==		==articles==
	*		*
		+
		+
		+	==books==
		+	* Korogodski, Leonid I., and Yan S. Soibelman. Algebras of Functions on Quantum Groups. American Mathematical Soc., 1998.

← 이전 판		2014년 8월 9일 (토) 04:25 판
13번째 줄:		13번째 줄:
	* $A_q(\mathfrak{g}):=\{\varphi \in U_q^{*} \| \dim U_q \varphi, \dim \varphi U_q <\infty \}$		* $A_q(\mathfrak{g}):=\{\varphi \in U_q^{*} \| \dim U_q \varphi, \dim \varphi U_q <\infty \}$
	* We call $A_q(\mathfrak{g})$ the quantized coordinate ring		* We call $A_q(\mathfrak{g})$ the quantized coordinate ring
		+
		+
		+	==related items==
		+	* [[Q-boson algebras]]