近年来, Heisenberg李(超)代数的结构和表示一直是非常重要的研究课题, 许多学者对此有着广泛研究.例如, 文[1]研究了特征0代数闭域上$2m+n+1$维Heisenberg李超代数的表示; 文[2]研究了复数域上无限维Heisenberg代数的全形和全形的导子代数, 证明了其全形的导子代数是一个完备李代数; 文[3]研究了特征$2$域上$2n+1$维Heisenberg李代数的同调; 文[4]研究了向量超空间上有限维Heisenberg李超代数不变的超对称和超正交双线性型; 文[5]研究了特征0代数闭域上两种类型Heisenberg李超代数的极小忠实表示.

本文约定在交换环上讨论Heisenberg李代数的自同构群, 在特征0代数闭域上讨论Heisenberg李超代数的自同构群.仿照文[6]中交换环上严格上三角矩阵李代数的自同构和文[7]中复向量空间上Heisenberg李代数的自同构的刻画, 参照文[3, 7]中Heisenberg李代数的定义, 利用文[7]中Heisenberg李代数与线性李代数之间的同构, 本文研究了交换环上Heisenberg李代数的自同构, 包括内自同构、中心自同构、对合自同构, 进而得到其自同构群的子群, 包括内自同构群、中心自同构群、对合自同构群.利用文[5]中有限维Heisenberg李超代数的定义, 本文建立了Heisenberg李超代数与线性李超代数之间的同构, 从而研究了特征0代数闭域上Heisenberg李超代数的自同构, 包括内自同构、中心自同构、对合自同构, 进而得到其自同构群的子群, 包括内自同构群、中心自同构群、对合自同构群.

令$R$是具有单位元的交换环并且$M_{n}(R)$是$R$上所有$n\times n$矩阵构成的集合, 其中$n$是正整数.令$e_{ij}$表示第$i$行第$j$列元素为$1$, 而其余元素为0的矩阵, 其中$i, j$是正整数.

定义2.1^[7] 令$H=\Bigg\{\begin{pmatrix} 0&x&z \\ 0&0&y \\ 0&0&0 \end{pmatrix} \Bigg| x \in R^{1\times n} , z\in R, y\in R^{n\times1}\Bigg\}$, 则$H$关于李运算

$ \begin{eqnarray*} [h_{1}, h_{2}]=h_{1}h_{2}-h_{2}h_{1}, \quad \forall h_{1}, h_{2}\in H \end{eqnarray*} $

作成一个李代数, 称$H$为Heisenberg李代数.

令$\mathbb{F}$是特征0代数闭域并且$M_{n}(\mathbb{F})$是$\mathbb{F}$上所有$n\times n$矩阵构成的集合, 其中$n$是正整数.

定义2.2^[5] $\mathbb{F}$上具有一维中心的二步幂零李超代数称为Heisenberg李超代数, 并且Heisenberg李超代数分为以下两种类型.

(1) 令$\mathfrak{H}_{m, n}=(\mathfrak{H}_{m, n})_{\overline{0}}\oplus(\mathfrak{H}_{m, n})_{\overline{1}}$是具有偶中心的Heisenberg李超代数, 设

$ \{u_{1}, \cdots, u_{m}, v_{1}, \cdots, v_{m}, z\mid w_{1}, \cdots, w_{n}\} $

为它的一个基, 并且李超运算由以下给出$ [u_{i}, v_{i}]=-[v_{i}, u_{i}]=z=[w_{j}, w_{j}], $ $\forall i=1, \cdots, m, $ $j=1, \cdots, n, $其余基元素之间的李超运算均为0.

(2) 令$\mathfrak{H}_{n}=(\mathfrak{H}_{n})_{\overline{0}}\oplus(\mathfrak{H}_{n})_{\overline{1}}$是具有奇中心的Heisenberg李超代数, 设

$ \{v_{1}, \cdots, v_{n}\mid z, w_{1}, \cdots, w_{n}\} $

为它的一个基, 并且李超运算由以下给出$ [v_{i}, w_{i}]=z=-[w_{i}, v_{i}], ~ \forall i=1, \cdots, n, $其余基元素之间的李超运算均为0.

根据定义2.2证得以下两个引理.

引理2.3  令$\mathbb{F}$上$\mathscr{A}=\mathscr{A}_{\overline{0}}\oplus\mathscr{A}_{\overline{1}}$是一个线性李超代数, 其中

$ \begin{eqnarray*}&&\mathscr{A}_{\overline{0}}={\rm{span_{\mathbb{F}}}}\{e_{12}, \cdots, e_{1, m+1}, e_{2, m+2}, \cdots, e_{m+1, m+2}, e_{1, m+2}\}, \\ &&\mathscr{A}_{\overline{1}}={\rm{span_{\mathbb{F}}}}\{\frac{1}{2}e_{1, m+3}+e_{m+3, m+2}, \cdots, \frac{1}{2}e_{1, m+n+2}+e_{m+n+2, m+2}\}.\end{eqnarray*} $

设线性映射$f$

$ \begin{aligned} f: \mathfrak{H}_{m, n}&\rightarrow \mathscr{A}, \\ h&\mapsto f(h), \quad \forall h\in \mathfrak{H}_{m, n}, \end{aligned} $

其中

$ \begin{aligned} &f(u_{i})=e_{1, i+1}\in \mathscr{A}_{\overline{0}}, ~f(v_{i})=e_{i+1, m+2}\in \mathscr{A}_{\overline{0}}, ~f(z)=e_{1, m+2}\in \mathscr{A}_{\overline{0}}, \\ &f(w_{j})=\frac{1}{2}e_{1, m+2+j}+e_{m+2+j, m+2}\in \mathscr{A}_{\overline{1}}, \quad\forall i=1, \cdots, m, j=1, \cdots, n, \end{aligned} $

则$f$是一个李超代数同构.

引理2.4  令$\mathbb{F}$上$\mathscr{B}=\mathscr{B}_{\overline{0}}\oplus\mathscr{B}_{\overline{1}}$是一个线性李超代数, 其中

$ \mathscr{B}_{\overline{0}}={\rm{span_{\mathbb{F}}}}\{e_{12}, \cdots, e_{1, n+1}\}, ~~ \mathscr{B}_{\overline{1}}={\rm{span_{\mathbb{F}}}}\{e_{1, n+2}, e_{2, n+2}, \cdots, e_{n+1, n+2}\}. $

设线性映射$g$

$ \begin{aligned} g: \mathfrak{H}_{n}&\rightarrow \mathscr{B}, \\ h&\mapsto g(h), \quad \forall h\in \mathfrak{H}_{n}, \end{aligned} $

其中$\begin{aligned} g(v_{i})=e_{1, i+1}\in \mathscr{B}_{\overline{0}}, ~ g(z)=e_{1, n+2}\in \mathscr{B}_{\overline{1}}, ~ g(w_{i})=e_{i+1, n+2}\in \mathscr{B}_{\overline{1}}, ~~ \forall i=1, \cdots, n, \end{aligned}$则$g$是一个李超代数同构.

记${\rm{Aut}}(H)$为Heisenberg李代数$H$的自同构群.

定理3.1  设$d\in M_{n+2}(R)$为可逆的对角矩阵, $x\in H$.令$\alpha=d+x$, 则$\alpha$可逆.设映射

$ \begin{aligned} \sigma_{\alpha}: H&\rightarrow H, \\ h&\mapsto\sigma_{\alpha}(h), \quad\forall h\in H, \end{aligned} $

其中$\begin{aligned} \sigma_{\alpha}(h)=\alpha h\alpha^{-1}, \end{aligned}$则$\sigma_{\alpha}$是$H$的一个自同构, 称为$H$的内自同构.令$G$是$H$的所有内自同构构成的集合, 则$G$是${\rm{Aut}} (H)$的子群, 称为$H$的内自同构群.

证  易知$\sigma_{\alpha}$是双射且是线性变换.由已知得$\alpha^{-1}\alpha=e$, 其中$e$是单位矩阵. $\forall h_{1}, h_{2}\in H$, 可得

$ \begin{aligned} &\sigma_{\alpha}([h_{1}, h_{2}])=\sigma_{\alpha}(h_{1}h_{2}-h_{2}h_{1})=\alpha(h_{1}h_{2})\alpha^{-1}-\alpha(h_{2}h_{1})\alpha^{-1}, \\ &[\sigma_{\alpha}(h_{1}), \sigma_{\alpha}(h_{2})]=\sigma_{\alpha}(h_{1})\sigma_{\alpha}(h_{2})-\sigma_{\alpha}(h_{2})\sigma_{\alpha}(h_{1})=\alpha(h_{1}h_{2})\alpha^{-1}-\alpha(h_{2}h_{1})\alpha^{-1}, \end{aligned} $

因此$\sigma_{\alpha}([h_{1}, h_{2}])=[\sigma_{\alpha}(h_{1}), \sigma_{\alpha}(h_{2})].$故$\sigma_{\alpha}$是$H$的一个自同构. $\forall \sigma_{\alpha}, \sigma_{\beta}\in G$, $h\in H$, 可得

$ \begin{aligned} \sigma_{\alpha}\sigma_{\beta}(h)=(\alpha\beta)h(\alpha\beta)^{-1}=\sigma_{\alpha\beta}(h), \end{aligned} $

因此$\begin{aligned} \sigma_{\alpha}\sigma_{\beta}=\sigma_{\alpha\beta}\in G. \end{aligned}$ $\forall\sigma_{\alpha}\in G$, $h\in H$, 可得

$ \begin{aligned} \sigma_{\alpha}\sigma_{\alpha^{-1}}=\alpha\alpha^{-1}h(\alpha^{-1})^{-1}\alpha^{-1}=h, \end{aligned} $

因此$\begin{aligned} \sigma_{\alpha^{-1}}=\sigma_{\alpha}^{-1}\in G. \end{aligned}$故$G$是${\rm{Aut}}(H)$的子群.

定理3.2  令$F=\{f\in {\rm{Hom}}_{R}(H, R)\mid f(y)=0, \forall y\in \delta^{[1]}(H)\}$, 其中$\delta^{[1]}(H)=[H, H]$. $\forall f\in F$, 设映射

$ \begin{aligned} \psi_{f}: H&\rightarrow H, \\ h&\mapsto\psi_{f}(h), \quad\forall h\in H, \end{aligned} $

其中$\begin{aligned} \psi_{f}(h)=h+f(h)e_{1, n+2}, \end{aligned}$则$\psi_{f}$是$H$的一个自同构, 称为$H$的中心自同构.令$S$是$H$的所有中心自同构构成的集合, 则$S$是${\rm{Aut}}(H)$的子群, 称为$H$的中心自同构群.

证  易知$\psi_{f}$是双射且是线性变换. $\forall h_{1}, h_{2}\in H$, 可得

$ \begin{aligned} \psi_{f}([h_{1}, h_{2}])=&[h_{1}, h_{2}]+f([h_{1}, h_{2}])e_{1, n+2}=[h_{1}, h_{2}], \\ [\psi_{f}(h_{1}), \psi_{f}(h_{2})]=&[h_{1}, h_{2}]+[h_{1}, f(h_{2})e_{1, n+2}]+[f(h_{1})e_{1, n+2}, h_{2}]\\ &+[f(h_{1})e_{1, n+2}, f(h_{2})e_{1, n+2}] =[h_{1}, h_{2}], \end{aligned} $

因此$\begin{aligned} \psi_{f}([h_{1}, h_{2}])=[\psi_{f}(h_{1}), \psi_{f}(h_{2})], \end{aligned}$故$\psi_{f}$是$H$的一个自同构. $\forall \psi_{f}, \psi_{g}\in S$, $h\in H$, 可得

$ \begin{aligned} \psi_{f}\psi_{g}(h)=h+(g(h)+f(h))e_{1, n+2}=\psi_{f+g}(h), \end{aligned} $

因此$\begin{aligned} \psi_{f}\psi_{g}=\psi_{f+g}\in S. \end{aligned}$ $\forall \psi_{f}\in S$, $h\in H$, 可得$\begin{aligned} \psi_{f}\psi_{-f}(h)=(h-f(h)e_{1, n+2})+f(h)e_{1, n+2}=h, \end{aligned}$因此$\begin{aligned} \psi_{-f}=\psi_{f}^{-1}\in S. \end{aligned}$故$S$是${\rm{Aut}}(H)$的子群.

定理3.3  令$\gamma=e_{1, n+2}+e_{2, n+1}+\cdots+e_{n+2, 1}$.设映射

$ \begin{aligned} w_{0}: H&\rightarrow H, \\ h&\mapsto w_{0}(h), \quad\forall h\in H, \end{aligned} $

其中$\begin{aligned} w_{0}(h)=-\gamma h^{{\rm{T}}}\gamma, \end{aligned}$则$w_{0}$是$H$的一个自同构, 称为$H$的对合自同构.令$W=\{\iota, w_{0}\}$, 其中$\iota$是恒等变换, 则$W$是${\rm{Aut}} (H)$的子群, 称为$H$的对合自同构群.

证  易知$w_{0}$是双射且是线性变换.由已知得$\gamma^{2}=e$, $\gamma^{{\rm{T}}}=\gamma$, 其中$e$是单位矩阵. $\forall h_{1}, h_{2}\in H$, 可得

$ \begin{aligned} &w_{0}([h_{1}, h_{2}])=w_{0}(h_{1}h_{2}-h_{2}h_{1})=-\gamma h_{2}^{{\rm{T}}}h_{1}^{{\rm{T}}}\gamma-(-\gamma h_{1}^{{\rm{T}}}h_{2}^{{\rm{T}}}\gamma), \\ &[w_{0}(h_{1}), w_{0}(h_{2})]=[-\gamma h_{1}^{{\rm{T}}}\gamma, -\gamma h_{2}^{{\rm{T}}}\gamma]=\gamma h_{1}^{{\rm{T}}}h_{2}^{{\rm{T}}}\gamma-\gamma h_{2}^{{\rm{T}}}h_{1}^{{\rm{T}}}\gamma, \end{aligned} $

因此$\begin{aligned} w_{0}([h_{1}, h_{2}])=[w_{0}(h_{1}), w_{0}(h_{2})]. \end{aligned}$故$w_{0}$是$H$的一个自同构. $\forall w_{0}\in W$, $h\in H$, 可得$\begin{aligned} w_{0}w_{0}(h)=-\gamma(-\gamma h^{{\rm{T}}}\gamma)^{{\rm{T}}}\gamma=h, \end{aligned}$因此$\begin{aligned} w_{0}^{2}=\iota\in W. \end{aligned}$故$W$是${\rm{Aut}}(H)$的子群.

记${\rm{Aut}}(\mathfrak{H}_{m, n})$, ${\rm{Aut}}(\mathfrak{H}_{n})$分别为Heisenberg李超代数$\mathfrak{H}_{m, n}$, $\mathfrak{H}_{n}$的自同构群.

定理3.4  设$d=\sum\limits_{k=1}^{m+n+2}a_{kk}e_{kk}\in M_{m+n+2}(\mathbb{F})$为可逆的矩阵, 并且满足

$ a_{11}a_{m+2, m+2}=(a_{m+2+j, m+2+j})^{2}, \forall j=1, \cdots, n, x\in (\mathfrak{H}_{m, n})_{\overline{0}}. $

令$\alpha=d+x$, 则$\alpha$可逆.设映射

$ \begin{aligned} \sigma_{\alpha}: \mathfrak{H}_{m, n}&\rightarrow \mathfrak{H}_{m, n}, \\ h&\mapsto\sigma_{\alpha}(h), \quad\forall h\in \mathfrak{H}_{m, n}, \end{aligned} $

其中$\begin{aligned} \sigma_{\alpha}(h)=\alpha h\alpha^{-1}, \end{aligned}$则$\sigma_{\alpha}$是$\mathfrak{H}_{m, n}$的一个自同构, 称为$\mathfrak{H}_{m, n}$的内自同构.令$G_{1}$是$\mathfrak{H}_{m, n}$所有内自同构构成的集合, 则$G_{1}$是${\rm{Aut}}(\mathfrak{H}_{m, n})$的子群, 称为$\mathfrak{H}_{m, n}$的内自同构群.

证  易知$\sigma_{\alpha}$是双射且是线性变换.由已知设

$ \begin{aligned} \alpha=&\sum\limits_{k=1}^{m+n+2}a_{kk}e_{kk}+\sum\limits_{i=1}^{m}(x_{i}e_{1, i+1}+y_{i}e_{i+1, m+2})+pe_{1, m+2}, \quad a_{kk}\neq0, x_{i}, y_{i}, p\in\mathbb{F}, \end{aligned} $

则有

$ \begin{aligned} \alpha^{-1}=&\sum\limits_{k=1}^{m+n+2}a_{kk}^{-1}e_{kk}-a_{11}^{-1}\sum\limits_{i=1}^{m}a_{i+1, i+1}^{-1}x_{i}e_{1, i+1}-a_{m+2, m+2}^{-1}\sum\limits_{i=1}^{m}a_{i+1, i+1}^{-1}y_{i}e_{i+1, m+2}\\ &-a_{11}^{-1}a_{m+2, m+2}^{-1}\bigg(p-\sum\limits_{i=1}^{m}a_{i+1, i+1}^{-1}x_{i}y_{i}\bigg)e_{1, m+2}, \quad a_{kk}^{-1}\neq0, x_{i}, y_{i}, p\in\mathbb{F}, \end{aligned} $

其中$a_{kk}a_{kk}^{-1}=1$.由引理2.3, 设

$ \begin{align} h_{\overline{0}}=\sum\limits_{i=1}^{m}(x_{i}e_{1, i+1}+y_{i}e_{i+1, m+2})+pe_{1, m+2}\in(\mathfrak{H}_{m, n})_{\overline{0}}, \quad x_{i}, y_{i}, p\in\mathbb{F}, \end{align} $

(3.1)

$ \begin{align} h_{\overline{1}}=\sum\limits_{j=1}^{n}\gamma_{j}\Big(\frac{1}{2}e_{1, m+2+j}+e_{m+2+j, m+2}\Big)\in(\mathfrak{H}_{m, n})_{\overline{1}}, \quad \gamma_{j}\in\mathbb{F}, \end{align} $

(3.2)

则有

$ \begin{aligned} \sigma_{\alpha}(h_{\overline{0}})=&a_{11}\sum\limits_{i=1}^{m}a_{i+1, i+1}^{-1}x_{i}e_{1, i+1}+a_{m+2, m+2}^{-1}\sum\limits_{i=1}^{m}a_{i+1, i+1}y_{i}e_{i+1, m+2}\\ &+a_{m+2, m+2}^{-1}\bigg(a_{11}p-a_{11}\sum\limits_{i=1}^{m}a_{i+1, i+1}^{-1}x_{i}y_{i}+\sum\limits_{i=1}^{m}x_{i}y_{i}\bigg)e_{1, m+2}\in(\mathfrak{H}_{m, n})_{\overline{0}}, \\ \sigma_{\alpha}(h_{\overline{1}})=&\sum\limits_{j=1}^{n}\gamma_{j}\Big(\frac{1}{2}a_{11}a_{m+2+j, m+2+j}^{-1}e_{1, m+2+j}+a_{m+2, m+2}^{-1}a_{m+2+j, m+2+j}e_{m+2+j, m+2}\Big)\\ &\in(\mathfrak{H}_{m, n})_{\overline{1}}, \end{aligned} $

故$\sigma_{\alpha}$是偶的线性变换.由引理2.3, 设

$ \begin{aligned} &h_{1}=\sum\limits_{i=1}^{m}(x'_{i}e_{1, i+1}+y'_{i}e_{i+1, m+2})+\sum\limits_{j=1}^{n}\gamma'_{j}\Big(\frac{1}{2}e_{1, m+2+j}+e_{m+2+j, m+2}\Big)+ce_{1, m+2}, \\ &h_{2}=\sum\limits_{i=1}^{m}(x''_{i}e_{1, i+1}+y''_{i}e_{i+1, m+2})+\sum\limits_{j=1}^{n}\gamma''_{j}\Big(\frac{1}{2}e_{1, m+2+j}+e_{m+2+j, m+2}\Big)+qe_{1, m+2}, \\ & x'_{i}, x''_{i}, y'_{i}, y''_{i}, \gamma'_{j}, \gamma''_{j}, c, q\in\mathbb{F}, \nonumber \end{aligned} $

则有

$ \begin{array}{l} {\sigma _\alpha }([{h_1},{h_2}]) = {\sigma _\alpha }(\sum\limits_{i = 1}^m {({{x'}_i}{{y''}_i} - {{y'}_i}{{x''}_i})} {e_{1,m + 2}} + \sum\limits_{j = 1}^n {{{\gamma '}_j}} {{\gamma ''}_j}{e_{1,m + 2}})\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \alpha (\sum\limits_{i = 1}^m {{{x'}_i}} {{y''}_i}{e_{1,m + 2}}){\alpha ^{ - 1}} - \alpha (\sum\limits_{i = 1}^m {{{y'}_i}} {{x''}_i}{e_{1,m + 2}}){\alpha ^{ - 1}} + \alpha (\sum\limits_{j = 1}^n {{{\gamma '}_j}} {{\gamma ''}_j}{e_{1,m + 2}}){\alpha ^{ - 1}},\\ [{\sigma _\alpha }({h_1}),{\sigma _\alpha }({h_2})] = [\alpha (\sum\limits_{i = 1}^m {{{x'}_i}} {e_{1,i + 1}}){\alpha ^{ - 1}},\alpha (\sum\limits_{i = 1}^m {{{y''}_i}} {e_{i + 1,m + 2}}){\alpha ^{ - 1}}] + [\alpha (\sum\limits_{i = 1}^m {{{y'}_i}} {e_{i + 1,m + 2}}){\alpha ^{ - 1}},\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\alpha (\sum\limits_{i = 1}^m {{{x''}_i}} {e_{1,i + 1}}){\alpha ^{ - 1}}] + [\alpha (\sum\limits_{j = 1}^n {{{\gamma '}_j}} (\frac{1}{2}{e_{1,m + 2 + j}} + {e_{m + 2 + j,m + 2}})){\alpha ^{ - 1}},\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\alpha (\sum\limits_{j = 1}^n {{{\gamma ''}_j}} (\frac{1}{2}{e_{1,m + 2 + j}} + {e_{m + 2 + j,m + 2}})){\alpha ^{ - 1}}]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \alpha (\sum\limits_{i = 1}^m {{{x'}_i}} {{y''}_i}{e_{1,m + 2}}){\alpha ^{ - 1}} - \alpha (\sum\limits_{i = 1}^m {{{y'}_i}} {{x''}_i}{e_{1,m + 2}}){\alpha ^{ - 1}} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\alpha (\sum\limits_{j = 1}^n {{{\gamma '}_j}} {{\gamma ''}_j}{e_{1,m + 2}}){\alpha ^{ - 1}}, \end{array} $

因此$\begin{aligned} \sigma_{\alpha}([h_{1}, h_{2}])=[\sigma_{\alpha}(h_{1}), \sigma_{\alpha}(h_{2})]. \end{aligned}$故$\sigma_{\alpha}$是$\mathfrak{H}_{m, n}$的一个自同构.由定理3.1的类似证明可得$G_{1}$是${\rm{Aut}}(\mathfrak{H}_{m, n})$的子群.

定理3.5  令$a=(a_{1}, b_{1}, \cdots, a_{m}, b_{m})\in \mathbb{F}^{2m}$.设映射

$ \begin{aligned} \psi_{a}: \mathfrak{H}_{m, n}&\rightarrow \mathfrak{H}_{m, n}, \\ h&\mapsto\psi_{a}(h), \quad\forall h\in \mathfrak{H}_{m, n}, \end{aligned} $

其中$\begin{aligned} \psi_{a}(h)=h+\sum_{i=1}^{m}(a_{i}x_{i}+b_{i}y_{i})e_{1, m+2}, \end{aligned}$这里

$ h=\sum\limits_{i=1}^{m}(x_{i}e_{1, i+1}+y_{i}e_{i+1, m+2})+\sum\limits_{j=1}^{n}\gamma_{j}\Big(\frac{1}{2}e_{1, m+2+j}+e_{m+2+j, m+2}\Big)+pe_{1, m+2}, ~~ x_{i}, y_{i}, \gamma_{j}, \\ p\in \mathbb{F}, $

则$\psi_{f}$是$\mathfrak{H}_{m, n}$的一个自同构, 称为$\mathfrak{H}_{m, n}$的中心自同构.令$S_{1}$是$\mathfrak{H}_{m, n}$所有中心自同构构成的集合, 则$S_{1}$是${\rm{Aut}}(\mathfrak{H}_{m, n})$的子群, 称为$\mathfrak{H}_{m, n}$的中心自同构群.

证  易知$\psi_{a}$是双射且是线性变换.由(3.1)和(3.2)式可得

$ \begin{aligned} &\psi_{a}(h_{\overline{0}})=h_{\overline{0}}+\sum\limits_{i=1}^{m}(a_{i}x_{i}+b_{i}y_{i})e_{1, m+2}\in(\mathfrak{H}_{m, n})_{\overline{0}}, \\ &\psi_{a}(h_{\overline{1}})=h_{\overline{1}}\in(\mathfrak{H}_{m, n})_{\overline{1}}, \end{aligned} $

故$\psi_{a}$是偶的线性变换.由定义2.2和引理2.3可得$\begin{aligned} \psi_{a}(e_{1, m+2})=e_{1, m+2}. \end{aligned}$ $\forall h_{1}, h_{2}\in\mathfrak{H}_{m, n}$, 可得$\psi_{a}([h_{1}, h_{2}])=[h_{1}, h_{2}], $

$ \begin{align*} [\psi_{a}(h_{1}), \psi_{a}(h_{2})]=&[h_{1}, h_{2}]+[h_{1}, \sum\limits_{i=1}^{m}(a_{i}x''_{i}+b_{i}y''_{i})e_{1, m+2}]+[\sum\limits_{i=1}^{m}(a_{i}x'_{i}+b_{i}y'_{i})e_{1, m+2}, h_{2}]\\ &+[\sum\limits_{i=1}^{m}(a_{i}x'_{i}+b_{i}y'_{i})e_{1, m+2}, \sum\limits_{i=1}^{m}(a_{i}x''_{i}+b_{i}y''_{i})e_{1, m+2}] =[h_{1}, h_{2}], \end{align*} $

因此$\begin{aligned} \psi_{a}([h_{1}, h_{2}])=[\psi_{a}(h_{1}), \psi_{a}(h_{2})]. \end{aligned}$故$\psi_{a}$是$\mathfrak{H}_{m, n}$的一个自同构.由定理3.2的类似证明可得$S_{1}$是${\rm{Aut}}(\mathfrak{H}_{m, n})$的子群.

定理3.6  令$\gamma=e_{11}-\sum\limits_{i=1}^{m}e_{i+1, i+1}+\sum\limits_{j=1}^{n}e_{m+2+j, m+2+j}+e_{m+2, m+2}$.设映射

$ \begin{aligned} w_{0}: \mathfrak{H}_{m, n}&\rightarrow \mathfrak{H}_{m, n}, \\ h&\mapsto w_{0}(h), \quad\forall h\in \mathfrak{H}_{m, n}, \end{aligned} $

其中$\begin{aligned} w_{0}(h)=\gamma h\gamma, \end{aligned}$则$w_{0}$是$\mathfrak{H}_{m, n}$的一个自同构, 称为$\mathfrak{H}_{m, n}$的对合自同构.令$W_{1}=\{\iota, w_{0}\}$, 其中$\iota$是恒等变换, 则$W_{1}$是${\rm{Aut}}(\mathfrak{H}_{m, n})$的子群, 称为$\mathfrak{H}_{m, n}$的对合自同构群.

证  事实上$w_{0}\in G_{1}$, $\{\iota, w_{0}\}$构成$G_{1}$的子群.

定理3.7  设$d\in M_{n+2}(\mathbb{F})$为可逆的对角矩阵, $x\in (\mathfrak{H}_{n})_{\overline{0}}$.令$\alpha=d+x$, 则$\alpha$可逆.设映射

$ \begin{aligned} \sigma_{\alpha}: \mathfrak{H}_{n}&\rightarrow \mathfrak{H}_{n}, \\ h&\mapsto\sigma_{\alpha}(h), \quad\forall h\in \mathfrak{H}_{n}, \end{aligned} $

其中$\begin{aligned} \sigma_{\alpha}(h)=\alpha h\alpha^{-1}, \end{aligned}$则$\sigma_{\alpha}$是$\mathfrak{H}_{n}$的一个自同构, 称为$\mathfrak{H}_{n}$的内自同构.令$G_{2}$是$\mathfrak{H}_{n}$所有内自同构构成的集合, 则$G_{2}$是${\rm{Aut}}(\mathfrak{H}_{n})$的子群, 称为$\mathfrak{H}_{n}$的内自同构群.

证  易知$\sigma_{\alpha}$是双射且是线性变换.由已知设

$ \begin{aligned} \alpha=&\sum\limits_{k=1}^{n+2}a_{kk}e_{kk}+\sum\limits_{i=1}^{n}x_{i}e_{1, i+1}, \quad a_{kk}\neq0, x_{i}\in\mathbb{F}, \end{aligned} $

则有

$ \begin{aligned} \alpha^{-1}=&\sum\limits_{k=1}^{n+2}a_{kk}^{-1}e_{kk}-a_{11}^{-1}\sum\limits_{i=1}^{n}a_{i+1, i+1}^{-1}x_{i}e_{1, i+1}, \quad a_{kk}^{-1}\neq0, x_{i}\in\mathbb{F}, \end{aligned} $

其中$a_{kk}a_{kk}^{-1}=1$.由引理2.4设

$ \begin{align} h_{\overline{0}} =\sum\limits_{i=1}^{n}x_{i}e_{1, i+1}\in(\mathfrak{H}_{n})_{\overline{0}}, \quad x_{i}\in\mathbb{F}, \end{align} $

(3.3)

$ \begin{align} h_{\overline{1}} =\sum\limits_{i=1}^{n}y_{i}e_{i+1, n+2}+pe_{1, n+2}\in(\mathfrak{H}_{n})_{\overline{1}}, \quad y_{i}, p\in\mathbb{F}, \end{align} $

(3.4)

则有

$ \begin{aligned} \sigma_{\alpha}(h_{\overline{0}})&=a_{11}\sum\limits_{i=1}^{n}a_{i+1, i+1}^{-1}x_{i}e_{1, i+1}\in(\mathfrak{H}_{n})_{\overline{0}}, \\ \sigma_{\alpha}(h_{\overline{1}})&=a_{n+2, n+2}^{-1}\bigg(\sum\limits_{i=1}^{n}a_{i+1, i+1}y_{i}e_{i+1, n+2}+\sum\limits_{i=1}^{n}x_{i}y_{i}e_{1, n+2}+a_{11}pe_{1, n+2}\bigg)\in(\mathfrak{H}_{n})_{\overline{1}}, \end{aligned} $

故$\sigma_{\alpha}$是偶的线性变换.由引理2.4设

$ \begin{align} h_{1}=\sum\limits_{i=1}^{n}(x'_{i}e_{1, i+1}+y'_{i}e_{i+1, n+2})+ce_{1, n+2}, \end{align} $

(3.5)

$ \begin{align} h_{2}=\sum\limits_{i=1}^{n}(x''_{i}e_{1, i+1}+y''_{i}e_{i+1, n+2})+qe_{1, n+2}, \quad x'_{i}, x''_{i}, y'_{i}, y''_{i}, c, q\in\mathbb{F}, \end{align} $

(3.6)

则有

$ \begin{aligned} \sigma_{\alpha}([h_{1}, h_{2}])=&\sigma_{\alpha}\bigg(\sum\limits_{i=1}^{n}(x'_{i}y''_{i}-y'_{i}x''_{i})e_{1, n+2}\bigg)\\ =&\alpha\bigg(\sum\limits_{i=1}^{n}x'_{i}y''_{i}e_{1, n+2}\bigg)\alpha^{-1}-\alpha\bigg(\sum\limits_{i=1}^{n}y'_{i}x''_{i}e_{1, n+2}\bigg)\alpha^{-1}, \\ [\sigma_{\alpha}(h_{1}), \sigma_{\alpha}(h_{2})]=&\bigg[\alpha\bigg(\sum\limits_{i=1}^{n}x'_{i}e_{1, i+1}\bigg)\alpha^{-1}, \alpha\bigg(\sum\limits_{i=1}^{n}y''_{i}e_{i+1, n+2}\bigg)\alpha^{-1}\bigg]+\\ &\bigg[\alpha\bigg(\sum\limits_{i=1}^{n}y'_{i}e_{i+1, n+2}\bigg)\alpha^{-1}, \\ &\alpha\bigg(\sum\limits_{i=1}^{n}x''_{i}e_{1, i+1}\bigg)\alpha^{-1}\bigg]\\ =&\alpha\bigg(\sum\limits_{i=1}^{n}x'_{i}y''_{i}e_{1, n+2}\bigg)\alpha^{-1}-\alpha\bigg(\sum\limits_{i=1}^{n}y'_{i}x''_{i}e_{1, n+2}\bigg)\alpha^{-1}, \end{aligned} $

因此$\begin{aligned} \sigma_{\alpha}([h_{1}, h_{2}])=[\sigma_{\alpha}(h_{1}), \sigma_{\alpha}(h_{2})]. \end{aligned}$故$\sigma_{\alpha}$是$\mathfrak{H}_{n}$的一个自同构.由定理3.1的类似证明可得$G_{2}$是${\rm{Aut}}(\mathfrak{H}_{n})$的子群.

定理3.8  令$b=(b_{1}, \cdots, b_{n})\in\mathbb{F}^{n}$.设映射

$ \begin{aligned} \psi_{b}: \mathfrak{H}_{n}&\rightarrow \mathfrak{H}_{n}, \\ h&\mapsto\psi_{b}(h), \quad\forall h\in \mathfrak{H}_{n}, \end{aligned} $

其中$\begin{aligned} \psi_{b}(h)=h+\sum_{i=1}^{n}b_{i}y_{i}e_{1, n+2}, \end{aligned}$这里$\begin{aligned} h=\sum_{i=1}^{n}(x_{i}e_{1, i+1}+y_{i}e_{i+1, n+2})+pe_{1, n+2}, ~ x_{i}, y_{i}, p\in\mathbb{F}, \end{aligned}$则$\psi_{b}$是$\mathfrak{H}_{n}$的一个自同构, 称为$\mathfrak{H}_{n}$的中心自同构.令$S_{2}$是$\mathfrak{H}_{n}$所有中心自同构构成的集合, 则$S_{2}$是${\rm{Aut}}(\mathfrak{H}_{n})$的子群, 称为$\mathfrak{H}_{n}$的中心自同构群.

证  易知$\psi_{b}$是双射且是线性变换.由(3.3)和(3.4)式可得

$ \begin{aligned} \psi_{b}(h_{\overline{0}})=h_{\overline{0}}\in(\mathfrak{H}_{n})_{\overline{0}}, ~~ \psi_{b}(h_{\overline{1}})=h_{\overline{1}}+\sum\limits_{i=1}^{n}b_{i}y_{i}e_{1, n+2}\in(\mathfrak{H}_{n})_{\overline{1}}, \end{aligned} $

故$\psi_{b}$是偶的线性变换.由定义2.2和引理2.4可得$\begin{aligned} \psi_{b}(e_{1, n+2})=e_{1, n+2}. \end{aligned}$ $\forall h_{1}, h_{2}\in\mathfrak{H}_{n}$, 可得

$ \begin{aligned} \psi_{b}([h_{1}, h_{2}])=&[h_{1}, h_{2}], \\ [\psi_{b}(h_{1}), \psi_{b}(h_{2})]=&[h_{1}, h_{2}]+[h_{1}, \sum\limits_{i=1}^{n}b_{i}y''_{i}e_{1, n+2}]+[\sum\limits_{i=1}^{n}b_{i}y'_{i}e_{1, n+2}, h_{2}]\\ &+[\sum\limits_{i=1}^{n}b_{i}y'_{i}e_{1, n+2}, \sum\limits_{i=1}^{n}b_{i}y''_{i}e_{1, n+2}] =[h_{1}, h_{2}], \end{aligned} $

因此$\begin{aligned} \psi_{b}([h_{1}, h_{2}])=[\psi_{b}(h_{1}), \psi_{b}(h_{2})]. \end{aligned}$故$\psi_{b}$是$\mathfrak{H}_{n}$的一个自同构.由定理3.2的类似证明可得$S_{2}$是${\rm{Aut}}(\mathfrak{H}_{n})$的子群.

定理3.9  令$\gamma=e_{11}-\sum\limits_{k=1}^{n+1}e_{k+1, k+1}+\sum\limits_{i=1}^{n}e_{1, i+1}+e_{1, n+2}$.设映射

$ \begin{aligned} w_{0}: \mathfrak{H}_{n}&\rightarrow \mathfrak{H}_{n}, \\ h&\mapsto w_{0}(h), \quad\forall h\in \mathfrak{H}_{n}, \end{aligned} $

其中$\begin{aligned} w_{0}(h)=\gamma h\gamma, \end{aligned}$则$w_{0}$是$\mathfrak{H}_{n}$的一个自同构, 称为$\mathfrak{H}_{n}$的对合自同构.令$W_{2}=\{\iota, w_{0}\}$, 其中$\iota$是恒等变换, 则$W_{2}$是${\rm{Aut}}(\mathfrak{H}_{n})$的子群, 称为$\mathfrak{H}_{n}$的对合自同构群.

证  易知$w_{0}$是双射且是线性变换.由(3.3)和(3.4)式可得

$ \begin{aligned} &w_{0}(h_{\overline{0}})=-h_{\overline{0}}\in(\mathfrak{H}_{n})_{\overline{0}}, \\ &w_{0}(h_{\overline{1}})=\sum\limits_{i=1}^{n}y_{i}e_{i+1, n+2}-\sum\limits_{i=1}^{n}y_{i}e_{1, n+2}-pe_{1, n+2}\in(\mathfrak{H}_{n})_{\overline{1}}, \end{aligned} $

故$w_{0}$是偶的线性变换.由已知得$\gamma^{2}=e$, 其中$e$是单位矩阵.由(3.5)和(3.6)式可得

$ \begin{aligned} w_{0}([h_{1}, h_{2}])=&w_{0}\bigg(\sum\limits_{i=1}^{n}(x'_{i}y''_{i}-y'_{i}x''_{i})e_{1, n+2}\bigg) =-\sum\limits_{i=1}^{n}(x'_{i}y''_{i}-y'_{i}x''_{i})e_{1, n+2}, \\ [w_{0}(h_{1}), w_{0}(h_{2})]=&\bigg[\gamma\bigg(\sum\limits_{i=1}^{n}x'_{i}e_{1, i+1}\bigg)\gamma, \gamma\bigg(\sum\limits_{i=1}^{n}y''_{i}e_{i+1, n+2}\bigg)\gamma\bigg]+\bigg[\gamma\bigg(\sum\limits_{i=1}^{n}y'_{i}e_{1, i+1}\bigg)\gamma, \\ &\gamma\bigg(\sum\limits_{i=1}^{n}x''_{i}e_{i+1, n+2}\bigg)\gamma\bigg] =-\sum\limits_{i=1}^{n}(x'_{i}y''_{i}-y'_{i}x''_{i})e_{1, n+2}, \end{aligned} $

因此$\begin{aligned} w_{0}([h_{1}, h_{2}])=[w_{0}(h_{1}), w_{0}(h_{2})]. \end{aligned}$故$w_{0}$是$\mathfrak{H}_{n}$的一个自同构.由定理3.3的类似证明可得$W_{2}$是${\rm{Aut}}(\mathfrak{H}_{n})$的子群.