Simple modules of the Witten-Woronowicz algebra.
Let \(k\) be an algebraically closed field, \(D=k[H,C]\) a polynomial algebra, \(\xi\in k^*\) not a root of 1 and \(a=a(H,C)\in D\). A generalized Weyl algebra in the sense of V. V. Bavula is the associative \(k\)-algebra \(A\) generated by the elements \(X,Y,H,C\) subject to the defining relations \[ XH=\xi HX,\;YH=\xi^{-1}HY,\;XC=\xi^p CX,\;YC=\xi^{-p}CY,\;YX=a(H,C),\;XY=a(\xi H,\xi^pC), \] where \(p\) is a positive integer. It is easy to check that the elements \(H,C,H^p-\lambda\) are normal in \(A\) for any \(\lambda\in k\). The main result of the paper shows that a simple \(A\)-module is in fact a module over one of the algebras \[ A/(H),\quad A/(C),\quad A/(H^p-\lambda),\qquad\lambda\in k^*. \] It is mentioned that Witten's second deformation and Woronowicz's deformation are isomorphic and they belong to the class of generalized Weyl algebras.
scientific article; zbMATH DE number 2037113