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��II�� No.11

��Υơ��: $\fbox{ �� ${\mathbb {C}}[G]$ �δĤȤ��Ƥι�¤ }$

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�� 11.1 ${\mathbb{C}}[G]$ �� $M_k({\mathbb{C}})$ �δ��Ĥ��ľ�ѤȴĤȤ��Ʊ��Ǥ��롣

$\displaystyle {\mathbb{C}}[G]\cong \prod_{i=1}^\ell M_{k_i}({\mathbb{C}})$

${\mathbb{C}}[G]$ ��濴��ˤ��ͭʬ�򤹤뤳�Ȥˤ�äơ��ϼ�� ̿��˵��夵��롣

̿�� 11.2 ${\mathbb{C}}$ ��ͭ�¼��

��

$Z(R)={\mathbb{C}}$
��Ƥκ� -�÷��ϴ��Ǥ��롣

�Ȥ��С�

�Ϥ��륵�� $M_k({\mathbb{C}})$ ��Ʊ��ˤʤ롣

�¤ϡ�̿��11.2��(2)�β��Τ�ȤǤϡ� �濴��ˤ��ͭʬ��ϡ�ξ¦��ǥ��ˤ��ʬ��Ǥ⤢�롣 ��줬��Ǥ��롣

�� 11.1

��̿��11.2��(2)��Ȥ��

��Ǥ�դ�(ξ¦)��ǥ��

��Ф��ơ�

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��¸�ߤ��ơ� $R =I\oplus J$ ��ʤꤿ�ġ� �Ȥ��ˡ�

��̿��11.2�ξ��(1),(2)��Ȥ��С�

�ˤ�� (ξ¦)��ǥ��뤬¸�ߤ��ʤ��

̿��11.2�ξ��ˤϤ��¾�˰ʲ��Ѥ��롣 �ݥ��Ȥϡ��Ū�ʾ�Ｐ(1),(2)��餤��ˤ��ƴ��ܹ�� $E_{ij}$ �� Τ򸫤Ĥ��Ф��Ƥ��뤫�Ȥ��Ȥ��ˤ��롣

�� 11.2 ��

�κ��ǥ��

��äơ�

��(

-��÷��Ȥ��) $J_1\oplus J_2$ ��ʤ�С�

�θ�

�ǡ� ��ξ��Τ�ͣ��¸�ߤ��롣

$% latex2html id marker 1128 $ e_1 \in J_1,\quad e_2 \in J_2.$$
$% latex2html id marker 1132 $ J_1=R e_1,\quad J_2=R e_2.$$

$R=M_2({\mathbb{C}})$ ��Ф��ơ� $J_1=R E_{11}, J_2=R E_{22}$ �Ȥ��Τ� ��ɽŪ��Ǥ��롣

�ʲ��˺٤��ܹ��Ф��Ȥˤʤ롣 ��ξ��Ϥɤ��򤤤Τ�� ٤��äˤʤ뤷��ܺ٤��Τꤿ��ͤϤष��Ϥǹͤ��ɤ��Τǡ� ��٤Ʊ齬��ˤޤ魯��Ȥˤ��롣

�ޤ�� $E_{11},E_{22},E_{33},\dots,E_{kk}$ �ˤ��Τ��Ф��

�� 11.3 ̿��11.2�β��Τ�Ȥǡ�

$R=R a_1\oplus R a_2 \oplus R a_3 \oplus R a_4 \oplus \dots \oplus R a_k$ �� ϴ�� -�÷��Ǥ��褦�� $a_1,\dots,a_k$ ��¸�ߤ��롣
��ˡ� $a_1,a_2,\dots,a_k$ ��

$\displaystyle a_1+a_2+\dots+a_k=1$

��褦��٤롣
$a_1,a_2,a_3,\dots, a_k$ ��Τ褦��Ǥ��ȡ�

$% latex2html id marker 1155 $\displaystyle a_i a_j =\delta_{ij} a_i \qquad i,j\in \{1,2,\dots,k\} $$
��Ω�ġ�(�� $\delta_{ij}$ �ϥ��ͥå��Υǥ륿��)

�� 11.4 �嵭��β��θ��ǡ� $a_1,a_2,\dots,a_k$ ��(1) ��(2)��Ω�Ĥ褦�� Ǥ��ȡ�

�ϴĤˤʤꡢ�� ǥ��(

��

-��ʬ�÷�)��⤿�ʤ��

�� 11.5 ${\mathbb{C}}$ ��ͭ�¼��δ�

��ʺ��ǥ��⤿�ʤ��ʤ�С� $S={\mathbb{C}}$ �Ǥ��롣�äˡ��11.4��

�� ${\mathbb{C}}$ �˰��פ��롣

$E_{ij}$ �ˤ��Τ��Ф��Τ��11.7�Ǥ��롣

��ơ��11.1�ˤ�ꡢ ��ɽ��Τ��ʤ��ʬ�� ɽ��Ĵ�٤뤳�Ȥ˵��夵��롣�¤��Ĥ�ɽ��ϸ¤�줿��Τ��ʤ��

�� 11.3 ${\mathbb{C}}^k$ �ϼ��ʤ�� $M_k({\mathbb{C}})$ -�÷��ȸ��뤳�Ȥ��Ǥ��롣 ��򤳤��Ǥ�

�Ƚ񤯤ȡ� $M_k({\mathbb{C}})$ ��Ǥ�դ�ͭ�¼��ɽ��

��ͭ�¸Ĥ�ľ�¤�( $M_k({\mathbb{C}})$ �÷��Ȥ��) Ʊ��Ǥ��롣 �Ȥ��ˡ� $M_k({\mathbb{C}})$ ��ͭ�¼��ɽ��ɬ��

��Ʊ��Ǥ��롣

�� 11.1 $R=M_3({\mathbb{C}})$ �ΤȤ��11.3 �ξ��(1),(2),(3)��

�� Ĥ��Ф��

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�� 11.2

�κ�

-��ʬ�÷�(�Ĥޤꡢ

�κ��ǥ��) $% latex2html id marker 1220 $ J(\neq 0)$$ ��

-�÷��Ȥ��ƴ��ʤ�С� �� $a\in J$ ��äơ�

��Ω�Ĥ��Ȥ򼨤�� (�ҥ��:

��Ǥ�դθ�

��Ф��ơ�

��

�κ��ǥ��Ǥ��롣 )

�� 11.3 ��11.3 ��(1),(2) ��Ω�Ĥ褦�� $a_1,a_2,\dots,a_k$ ��Ф��ơ� Ʊ��(3)��Ω�Ĥ��Ȥ򼨤�� (�ҥ��: $a_1=a_1(a_1+a_2+\dots+a_k)$ �α��դ�Ÿ��ξ�դ�ʬ��(1)�� ؤ��ʬ��Ӥ��衣

�� 11.4 ��11.3�β��Τ�Ȥ� �ɤ� $x,y\in R$ �ˤ��Ƥ⡢ $% latex2html id marker 1244 $ x R y\neq 0 $$ �Ǥ��뤳�ȡ� ��ʤ�� $w\in R$ ��ä� $% latex2html id marker 1248 $ xwy \neq 0$$ �Ǥ��뤳�Ȥ򼨤��ʤ�� (�ҥ��:

��ĸ��ꤹ��ˡ� $% latex2html id marker 1252 $ \{x\in R; xRy\neq 0\}$$ ��

��ξ¦ ��ǥ��ˤʤ롣)

�� 11.5 ��

��ͭ�¼��δ�

��ʺ��ǥ��⤿�ʤ��Ȳ��ꤹ�롣 ��ΤȤ��

$\displaystyle (a,b\in S$ �� $\displaystyle ab=0) \implies \ (a=0$ �ޤ�� $\displaystyle b=0)$

��Ω�Ĥ��Ȥ򼨤��ʤ�� (�ҥ��:

�� 0 �Ǥʤ��Ȥ��ȡ�

��

�κ��ǥ��ǡ�0 �ǤϤʤ��顢

�ˤʤ餶��ʤ��

�ʤ� $x\in S$ ��¸�ߤ�Ƴ��)

�� 11.6 ��11.5��衣(�ҥ��:Ǥ�դ� $z\in S$ �ˤ��ơ� $1,z,z^2,z^3,\dots,z^{\dim_{\mathbb{C}}S}$ �ϰ켡��Ω��ʤ��ᡢ

��

�夢��ߤ��Τ褦��Τ�� Ǿ��Τ��(

�κǾ�¿�༰) ��

�Ȥ��ȡ��ؤδ��ˤ��

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�ΰ켡��Ѥ�ʬ�򤹤롣�Ĥޤꡢ

$\displaystyle (z-c_1)(z-c_2)(z-c_3)\dots(z-c_l)=0$

�ʤ� $c_1,c_2,c_3,\dots, c_l$ ��¸�ߤ��롣 ��Ȥ��롣)

�� 11.7 ��11.3�β��Τ�Ȥǡ� $\dim_{{\mathbb{C}}}(a_i R a_j)=1$ �Ǥ��뤳�Ȥ򼨤��ʤ�� (�ҥ��: ��11.4��Ѥ��ơ�0 �Ȱۤʤ�褦��

,

��դ��Ƥ��ơ�� 11.5 ��Ѥ��롣)

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2003/7/5