Uncommon notation for division algebra

Question

I have found the following notion for a division algebra in a paper. $K=\mathbb{R}(x_1, \dots, x_n)$ is the field of rational functions in $n$ variables over $\mathbb{R}$ and $F=K((t))$ be the field of Laurent series in one variable over $K$. Now it is defined a division algebra $D$ by $D= (\frac{a,t}{F})$, where $a= x_1^2+ \dots +x_n^2$. It is further stated, that this means that "$D$ is the quaterion algebra defined by $i^2=a$, $j^2=t$, etc.".

I do not understand, how elements in $D$ look like and am thankful for every explanation of this notation and for a more concrete description of $D$.

Answer 1

This is explained in the wiki page on quaternion algebras.

The notation in the above cited page is $(a, b)_F$, which should correspond to your $(\frac{a,t}{F})$.

Concretely, $D=(\frac{a,t}{F})$ is a four-dimensional central algebra over $F$, with basis $\{1, i, j, k\}$, subject to the multiplication rules:

• $i^2 = a$;
• $j^2 = t$;
• $ij = k = - ji$.

See the cited page and the references there for details.