# Twisted canonical commutation relations

I am dealing with universal C*-algebra generated by $$x,y$$ with the following relations: $$xy = qyx$$, $$x^{*}y = qyx^{*}$$, $$y^{*}x = qxy^{*}$$, $$x^{*}x = q^2xx^{*} - (1-q^2)yy^{*}$$, $$y^{*}y = q^2yy^{*} - (1-q^2)$$, where q $$\in (0,1)$$. I need to find (the most simple) faithful representation of it but I couldn’t do it myself and the only reference I found is Pusz, Woronovicz “Twisted second quantization”. In this article I couldn’t understand a thing because it is written kind of with a physics notation.

Does someone by any chance know where else it is written? Or may be give me a hint how to construct it?

• Thank you, I figured out how does Fock representation of that work. But haven't you encountered to a faithful representations of TCCR such that generators don't tend to zero as $q$ tends to 1? I am not sure they are even exist... Aug 17, 2019 at 12:57