﻿ 基于RTT实现的量子超代数的坐标超代数

# 基于RTT实现的量子超代数的坐标超代数Coordinate Superalgebras of Quantum Superalgebras Based on the RTT Relation

Abstract: A quantum superalgebra can be given by an R-matrix and the corresponding RTT relation. For a quantum superalgebra U(R) defined by the R-matrix  , we verified that the matrix generators of its coordinate superalgebra A(R) also satisfy the RTT relation in this paper. And we further illustrate the above results by taking quantum superalgebras Uq(glm|n) and Uq(qn) as examples.

1. 引言

2. 量子超代数

$\mathbb{K}$ 是一个特征为零的域。考虑域 $\mathbb{K}$ 上的超线性空间(即 $ℤ/2ℤ$ -分次的线性空间)， $V={\mathbb{K}}^{m|n}$。V有标准基 $\left\{{v}_{1},\cdots ,{v}_{m+n}\right\}$，其中 ${v}_{i}$$ℤ/2ℤ$ -分次中的次数为

$|{v}_{i}|=|i|=\left\{\begin{array}{ll}\stackrel{¯}{0},\hfill & 如果\text{ }\text{ }1\le i\le m,\hfill \\ \stackrel{¯}{1},\hfill & 如果\text{ }\text{ }m+1\le i\le m+n.\hfill \end{array}$

$|{e}_{ij}|=|i|+|j|,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i,j=1,\cdots ,m+n,$

$\mathcal{R}=\underset{i,j,k,l}{\sum }{R}_{kl}^{ij}{e}_{ij}\otimes {e}_{kl}\in \text{End}V\otimes \text{End}V.$

$a\otimes b↦a\otimes b\otimes 1,\text{ }a\otimes b↦a\otimes 1\otimes b,\text{ }a\otimes b↦1\otimes a\otimes b.$

${\mathcal{R}}_{12}{\mathcal{R}}_{13}{\mathcal{R}}_{23}={\mathcal{R}}_{23}{\mathcal{R}}_{13}{\mathcal{R}}_{12}$ (2.1)

${\sum }_{p,r,s}{\left(-1\right)}^{\left(|p|+|i|\right)\left(|b|+|j|\right)}{R}_{br}^{ap}{R}_{cs}^{pi}{R}_{sk}^{rj}={\sum }_{p,r,s}{\left(-1\right)}^{\left(|p|+|i|\right)\left(|b|+|r|\right)}{R}_{cs}^{br}{R}_{sk}^{ap}{R}_{rj}^{pi}$ (2.2)

$\mathcal{R}{T}_{1}{T}_{2}={T}_{2}{T}_{1}\mathcal{R}$ (2.3)

$T=\underset{i,j=1}{\overset{m+n}{\sum }}{e}_{ij}\otimes {t}_{ij}$

${T}_{1},{T}_{2}$ 分别为T在从 $\text{End}V\otimes U\left(\mathcal{R}\right)$$\text{End}V\otimes \text{End}V\otimes U\left(\mathcal{R}\right)$ 的两个典范同态

$a\otimes u↦a\otimes 1\otimes u,\text{ }a\otimes u↦1\otimes a\otimes u$

$\underset{p,r}{\sum }{\left(-1\right)}^{\left(|p|+|i|\right)\left(|b|+|j|\right)}{R}_{br}^{ap}{t}_{pi}{t}_{rj}=\underset{p,r}{\sum }{\left(-1\right)}^{\left(|p|+|i|\right)\left(|b|+|r|\right)}{t}_{br}{t}_{ap}{R}_{rj}^{pi}$

$\Delta \left({t}_{ij}\right)=\underset{k=1}{\overset{m+n}{\sum }}{\left(-1\right)}^{\left(|i|+|k|\right)\left(|k|+|j|\right)}{t}_{ik}\otimes {t}_{kj}$

$\mathcal{R}:={\sum }_{i}{q}^{{\left(-1\right)}^{|i|}}{e}_{ii}\otimes {e}_{ii}+{\sum }_{i\ne j}{e}_{ii}\otimes {e}_{jj}+\left(q-{q}^{-1}\right){\sum }_{i (2.4)

${t}_{ij}^{+}={t}_{ji}^{-}=0,i>j$${t}_{ii}^{+}{t}_{ii}^{-}=1={t}_{ii}^{-}{t}_{ii}^{+},i=1,\cdots ,m+n$

$\mathcal{R}{T}_{1}^{\left(±\right)}{T}_{2}^{\left(±\right)}={T}_{2}^{\left(±\right)}{T}_{1}^{\left(±\right)}\mathcal{R}$

$\mathcal{R}{T}_{1}^{\left(+\right)}{T}_{2}^{\left(-\right)}={T}_{2}^{\left(-\right)}{T}_{1}^{\left(+\right)}\mathcal{R}$

$\mathcal{R}:={\sum }_{i,j}{q}^{\phi \left(i,j\right)}{e}_{ii}\otimes {e}_{jj}+\left(q-{q}^{-1}\right){\sum }_{i (2.5)

$\mathcal{R}{T}_{1}{T}_{2}={T}_{2}{T}_{1}\mathcal{R}$

${t}_{ii}{t}_{-i,-i}=1={t}_{-i,-i}{t}_{ii},i=-n,\cdots ,-1,1,\cdots ,n$，其中

$T={\sum }_{i\le j}{t}_{ij}\otimes {e}_{ij}.$

3. 坐标代数

$U{\left(\mathcal{R}\right)}^{\circ }:=\left\{f\in U{\left(\mathcal{R}\right)}^{*}|U\left(\mathcal{R}\right)如果存在余维数有限的理想\text{ }I\text{ }使得\text{ }f\left(I\right)=0\right\}.$

$〈fg,x〉={\sum }_{\left(x\right)}{\left(-1\right)}^{|{x}_{\left(1\right)}||g|}〈f,{x}_{\left(1\right)}〉〈g,{x}_{\left(2\right)}〉,如果\Delta \left(x\right)={\sum }_{\left(x\right)}{x}_{\left(1\right)}\otimes {x}_{\left(2\right)},$

$〈f,xy〉={\sum }_{\left(f\right)}{\left(-1\right)}^{|{f}_{\left(2\right)}||x|}〈{f}_{\left(1\right)},x〉〈{f}_{\left(2\right)},y〉,如果{\Delta }^{\circ }\left(f\right)={\sum }_{\left(f\right)}{f}_{\left(1\right)}\otimes {f}_{\left(2\right)},$

$\rho :U\left(\mathcal{R}\right)\to \text{End}V,\text{ }T↦\mathcal{R}\text{.}$

$\rho \left({t}_{ij}\right)={\sum }_{k,l}{R}_{ij}^{kl}{e}_{kl},\text{ }i,j=1,\cdots ,m+n.$

$\rho \left(x\right)\cdot {v}_{l}={\sum }_{k}〈{\tau }_{kl},x〉{v}_{k},\text{ }x\in U\left(\mathcal{R}\right).$ (3.1)

$〈{\tau }_{kl},{t}_{ij}〉={R}_{ij}^{kl}.$

${\Delta }^{\circ }\left({\tau }_{kl}\right)={\sum }_{p}{\left(-1\right)}^{\left(|k|+|p|\right)\left(|p|+|l|\right)}{t}_{kp}\otimes {t}_{pl}.$ (3.2)

$\rho \left(xy\right)\cdot {v}_{l}={\sum }_{k}〈{\tau }_{kl},xy〉{v}_{k}.$

$\rho \left(xy\right)\cdot {v}_{l}=\rho \left(x\right)\rho \left(y\right)\cdot {v}_{l}={\sum }_{p}〈{\tau }_{pl},y〉\rho \left(x\right)\cdot {v}_{p}={\sum }_{k,p}〈{\tau }_{kp},x〉〈{\tau }_{pl},y〉{v}_{k}.$

$〈{\tau }_{kl},xy〉={\sum }_{p}〈{\tau }_{kp},x〉〈{\tau }_{pl},y〉.$

$\mathcal{R}{X}_{1}{X}_{2}={X}_{2}{X}_{1}\mathcal{R},$

$\Delta \left({t}_{ck}\right)={\sum }_{s}{\left(-1\right)}^{\left(|c|+|s|\right)\left(|s|+|k|\right)}{t}_{cs}\otimes {t}_{sk}.$

$〈{\tau }_{pi}{\tau }_{rj},{t}_{ck}〉={\sum }_{s}{\left(-1\right)}^{\left(|c|+|s|\right)\left(|s|+|k|\right)+\left(|c|+|s|\right)\left(|r|+|j|\right)}〈{\tau }_{pi},{t}_{cs}〉〈{\tau }_{rj},{t}_{sk}〉={\sum }_{s}{R}_{cs}^{pi}{R}_{sk}^{rj},$

$\begin{array}{l}〈{\sum }_{p,r}{\left(-1\right)}^{\left(|p|+|i|\right)\left(|b|+|j|\right)}{R}_{br}^{ap}{\tau }_{pi}{\tau }_{rj},{t}_{ck}〉\\ ={\sum }_{p,r,s}{\left(-1\right)}^{\left(|p|+|i|\right)\left(|b|+|j|\right)}{R}_{br}^{ap}{R}_{cs}^{pi}{R}_{sk}^{rj}\\ ={\sum }_{p,r,s}{\left(-1\right)}^{\left(|p|+|i|\right)\left(|b|+|r|\right)}{R}_{cs}^{br}{R}_{sk}^{ap}{R}_{rj}^{pi}\\ =〈{\sum }_{p,r}{\left(-1\right)}^{\left(|p|+|i|\right)\left(|b|+|r|\right)}{\tau }_{br}{\tau }_{ap}{R}_{rj}^{pi},{t}_{ck}〉.\end{array}$

${\sum }_{p,r}{\left(-1\right)}^{\left(|p|+|i|\right)\left(|b|+|j|\right)}{R}_{br}^{ap}{\tau }_{pi}{\tau }_{rj}={\sum }_{p,r}{\left(-1\right)}^{\left(|p|+|i|\right)\left(|b|+|r|\right)}{\tau }_{br}{\tau }_{ap}{R}_{rj}^{pi},$

$\mathcal{R}{X}_{1}{X}_{2}={X}_{2}{X}_{1}\mathcal{R}$

$\mathcal{R}{X}_{1}{X}_{2}={X}_{2}{X}_{1}\mathcal{R},$ (3.3)

$\rho :{U}_{q}\left(\mathfrak{g}{\mathfrak{l}}_{m|n}\right)\to \text{End}V,\text{ }{T}^{\left(+\right)}↦\mathcal{R},\text{\hspace{0.17em}}{T}^{\left(-\right)}↦{\mathcal{R}}^{-1},$

${\mathcal{R}}^{-1}={\sum }_{i}{q}^{-{\left(-1\right)}^{|i|}}{e}_{ii}\otimes {e}_{ii}+{\sum }_{i\ne j}{e}_{ii}\otimes {e}_{jj}-\left(q-{q}^{-1}\right){\sum }_{i

$\mathcal{R}$ 的逆矩阵，因此，相应的自然表示的矩阵元 ${\tau }_{kl},k,l=1,\cdots ,m+n$ 满足

$〈{\tau }_{kl},{t}_{ii}^{+}〉={q}^{{\left(-1\right)}^{|i|}{\delta }_{il}},〈{\tau }_{kl},{t}_{ii}^{-}〉={q}^{-{\left(-1\right)}^{|i|}{\delta }_{il}},$

$〈{\tau }_{kl},{t}_{ij}^{+}〉={\delta }_{jk}{\delta }_{il}\left(q-{q}^{-1}\right){\left(-1\right)}^{|j|},〈{\tau }_{kl},{t}_{ji}^{-}〉=-{\delta }_{ik}{\delta }_{jl}\left(q-{q}^{-1}\right){\left(-1\right)}^{|j|},i

${\left({\tau }_{ak}\right)}^{2}=0,|a|+|k|=1,$

${\tau }_{ak}{\tau }_{bk}={\left(-1\right)}^{\left(|a|+|k|\right)\left(|b|+|k|\right)}{q}^{{\left(-1\right)}^{|k|}}{\tau }_{bk}{\tau }_{ak},a>b,$

${\tau }_{ak}{\tau }_{al}={\left(-1\right)}^{\left(|a|+|k|\right)\left(|b|+|l|\right)}{q}^{{\left(-1\right)}^{|a|}}{\tau }_{al}{\tau }_{ak},k>l,$

${\tau }_{ak}{\tau }_{al}={\left(-1\right)}^{\left(|a|+|k|\right)\left(|b|+|l|\right)}{q}^{{\left(-1\right)}^{|a|}}{\tau }_{al}{\tau }_{ak},k>l,$

${\tau }_{ak}{\tau }_{bl}={\left(-1\right)}^{\left(|a|+|k|\right)\left(|b|+|l|\right)}{\tau }_{bl}{\tau }_{ak}+{\left(-1\right)}^{|a|\left(|b|+|l|\right)+|b||l|}\left(q-{q}^{-1}\right){\tau }_{bk}{\tau }_{al},a>b,k>l.$

$\mathcal{R}{X}_{1}{X}_{2}={X}_{2}{X}_{1}\mathcal{R},$ (3.4)

$\rho \left({t}_{ii}\right)={\sum }_{j}{q}^{\phi \left(j,i\right)}{e}_{jj},\text{ }\rho \left({t}_{ij}\right)=\left(q-{q}^{-1}\right){\left(-1\right)}^{|i|}\left({e}_{ji}-{e}_{-j,-i}\right).$

$〈{\tau }_{kl},{t}_{ii}〉={q}^{\phi \left(l,i\right)}{\delta }_{kl},\text{ }〈{\tau }_{kl},{t}_{ij}〉=\left(q-{q}^{-1}\right){\left(-1\right)}^{|i|}\left({\delta }_{il}{\delta }_{jk}+{\delta }_{-i,l}{\delta }_{-j,k}\right),\text{ }i

$\begin{array}{l}{q}^{\phi \left(i,j\right)}{\tau }_{ia}{\tau }_{jb}-{\left(-1\right)}^{\left(|i|+|a|\right)\left(|j|+|b|\right)}{q}^{\phi \left(a,b\right)}{\tau }_{jb}{\tau }_{ia}\\ ={\left(-1\right)}^{|i||j||+|j||b|+|b||i|}\left(q-{q}^{-1}\right)\left(\left({\delta }_{a

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