Ring $R_{ 196 }$

Description:

et $V = \Bbb Q\langle\{x_{ij},y_{ij}: 1\le i,j \le 2\}\rangle/(XY - I_2, YX - I_2)$, where $X = (x_{ij})$, $Y = (y_{ij})$, $I_2$ is the $2 \times 2$ identity matrix, and the relations are understood componentwise. It is known that the free product $V \sqcup V$, the $\Bbb Q$-algebra generated by 16 elements $x_{ij}, y_{ij}, x'_{ij}, y'_{ij}$ with defining relations $XY = YX = X'Y' = Y'X' = I_2$, embeds into a skew field $D$, and the subalgebra $F = \Bbb Q \langle \{ x_{ij}, x'_{ij}\}\rangle \subset D$ is free on its generators. Then $R$ is the trivial extension of the $F$-bimodule $D/(V\sqcup V)$ by $F$.

Keywords free algebra subring trivial extension