## Sunday, January 24, 2016

### Map of Conjugation of Quaternion algebra

Quaternion algebra has following maps of conjugation
\begin{split}
x=E\circ(x_0+x_1i+x_2j+x_3k)&=x_0+x_1i+x_2j+x_3k\\
x^{*_1}=I\circ(x_0+x_1i+x_2j+x_3k)&=x_0-x_1i+x_2j+x_3k\\
x^{*_2}=J\circ(x_0+x_1i+x_2j+x_3k)&=x_0+x_1i-x_2j+x_3k\\
x^{*_3}=K\circ(x_0+x_1i+x_2j+x_3k)&=x_0+x_1i+x_2j-x_3k
\end{split}

A linear map of quaternion algebra
$f:H\rightarrow H\ \ \ y^i=f^i_jx^j$
has form
$f=a_0\circ E+a_1\circ I+a_2\circ J+a_3\circ K$
\begin{split}
f\circ x&=a_0x+a_1\circ I\circ x+a_2\circ J\circ x+a_3\circ K\circ x\\
&=a_0x+a_1x^{*_1}+a_2x^{*_2}+a_3x^{*_3}
\end{split}
where quaternions $$a_0$$, $$a_1$$, $$a_2$$, $$a_3$$ are defined by coordinates $$f^i_j$$ of linear map.

The set $$\mathcal L(R;H;H)$$ of linear maps of quaternion algebra is left $$H$$-vector space and has the basis $$\overline{\overline{e}} =(E,I,J,K)$$.

The set $HE=\{a\circ E:a\in H\}$ is $$R$$-algebra isomorphic to quaternion algebra.