Quantized universal enveloping algebra

개요

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

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 \(q \to 1\)

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

counit
\(\epsilon(q^{h}) =1\)
\(\epsilon(e_i)=\epsilon(f_i)=0\)
antipode
\(S(q^h) = q^{-h}\) for \(x \in \mathfrak{g}\)
\(S(e_i) =-e_ik_i, S(f_i)=-k_i^{-1}f_i\)