# Quantized universal enveloping algebra

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## 개요

• $$U_{q}(\mathfrak{g})$$ : Kac-Moody 대수의 UEA $$U(\mathfrak{g})$$ 의 deformation
• 호프 대수(Hopf algebra)의 구조를 가짐
• 양자군 (quantum group)의 예

## Cartan datum

• Cartan datum $$(A,P^{\vee},P,\Pi^{\vee},\Pi)$$
• $$A=(a_{ij})_{i,j\in I}$$ symmetrizable GCM
• $$D=\operatorname{diag}(s_i\in\mathbb{Z}_{\geq 0})_{i \in I}$$ diagonal matrix s.t. DA is symmetric
• $$P^{\vee}=(\bigoplus_{i\in I}\mathbb{Z}h_{i})\bigoplus(\bigoplus_{j=1}^{\operatorname{corank}(A)}\mathbb{Z}d_{j})$$ : dual weight lattice
• $$\mathfrak{h}=\mathbb{Q}\otimes_{\mathbb{Z}} P^{\vee}$$ : Cartan subalgebra
• $$P=\{\lambda\in\mathfrak{h}^{*}|\lambda(P^{\vee})\subset \mathbb{Z}\}$$ : weight lattice
• $$\Pi^{\vee}=\{h_{i}|i\in I\}$$ : simple coroots
• $$\Pi=\{\alpha_{i}\in\mathfrak{h}^{*}|i\in I, \alpha_{i}(h_j)=a_ {ji}\}$$ : simple roots
• $$(\cdot|\cdot)$$ symmetric bilinear form on $$\mathfrak{g}^{*}$$
• $$s_{i}=\frac{(\alpha_{i}|\alpha_{i})}{2}\in \mathbb{Z}_{>0}$$
• q: indeterminate
• $$q_i=q^{s_{i}}$$
• q-초기하급수(q-hypergeometric series)와 양자미적분학(q-calculus)

## 정수의 q-analogue

• 정수 n에 대하여 다음과 같이 정의$[n]_{q_i} =\frac{q_ {i}^{n}-q_ {i}^{-n}}{q_i-q_i^{-1}}$$[0]_{q_i} =1$$[n]_{q_i}!=[n]_ {q_i}[n]_ {q_i}\cdots [n]_{q_i}$${m \choose n}_{q_{i}}=\frac{[m]_{q!}}{[n]_{q_{i}!}[m-n]_{q_i}!}$
• 극한 $$q \to 1$$

## quantized universal enveloping algebra $$U_{q}(\mathfrak{g})$$

• 생성원 $$e_i,f_i , (i\in I)$$, $$q^{h} (h\in P^{\vee})$$
• 관계식
• $$q^0=1$$
• $$q^{h}q^{h'}=q^{h+h'}$$
• $$e_if _j-f_je _i=\delta_{i,j}\frac{k_i-k_i^{-1}}{q_i-q_i^{-1}}$$ 여기서 $$k_{i}=q^{h_is _i}$$
• $$q^he_jq^{-h}=q^{\alpha_j(h)}e_j$$
• $$q^hf_jq^{-h}=q^{-\alpha_j(h)}f_j$$
• $$\sum_{k=0}^{1-a_{i,j}}(-1)^k \binom{1-a_{i,j}}{k}_{q_{i}}e_ {i}^{1-a_{i,j}-k}e_{j}e_ {i}^k=0$$ ($$i \neq j$$)
• $$\sum_{k=0}^{1-a_{i,j}}(-1)^k \binom{1-a_{i,j}}{k}_{q_{i}}f_ {i}^{1-a_{i,j}-k}f_{j}f_ {i}^k=0$$ ($$i \neq j$$)

### 호프 대수 구조

• comultiplication

$\Delta : U_{q}(\mathfrak{g}) \to U_{q}(\mathfrak{g}) \otimes U_{q}(\mathfrak{g})$ $\Delta(q^{h}) =q^{h}\otimes q^{h}$ $\Delta(e_i)=e_i\otimes k_i+1\otimes e_i$ $\Delta(f_i)=f_i\otimes 1+ k_i^{-1}\otimes f_i$

• counit

$\epsilon(q^{h}) =1$ $\epsilon(e_i)=\epsilon(f_i)=0$

• antipode

$S(q^h) = q^{-h}$ for $$x \in \mathfrak{g}$$ $S(e_i) =-e_ik_i^{-1}, S(f_i)=-k_if_i$