(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

What is the centre of the ring of the quaternions defined by:

[tex]\mathbf{H}=\{ \begin{pmatrix}

a & b \\

-\bar{b} & \bar{a} \end{pmatrix} | a,b \in \mathbf{C} \}[/tex]?

2. Relevant equations

The definition of the centre of a ring:

The centre Z of a ring R is defined by [tex]Z(R)=\{A | AX=XA, \forall X \in R\}[/tex]

3. The attempt at a solution

I figured that multiples of the 2x2 identity matrix must be in the centre.

Also if we denote an element of H by:

[tex]\begin{pmatrix} x & y \\

-\bar{y} & \bar{x} \end{pmatrix}[/tex]

where [tex]x=x_1 + ix_2[/tex] and similarly for a,b and y that:

1. [tex]b\bar{y}=\bar{b}y[/tex]

2. [tex]y(a-\bar{a})=b(x-\bar{x})[/tex]

3. [tex]\bar{b}(x-\bar{x})=\bar{y}(a-\bar{a})[/tex]

Then for instance we get from the first equation that:

[tex]b_2x_1=a_1y_2[/tex]

But I am not sure whether this approach really is any useful at all. Some hints would be greatly appreciated.

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# Centre of the ring of quaternions

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