Sunday, January 24, 2016

Map of Conjugation of Quaternion algebra

Quaternion algebra has following maps of conjugation

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
f\circ x&=a_0x+a_1\circ I\circ x+a_2\circ J\circ x+a_3\circ K\circ x\\
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.

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