Hopf structures on ambiskew polynomial rings. (Q2464523)

Let \(R\) be an affine commutative algebra over an algebraically closed field \(k\) of characteristic zero with an automorphism \(\sigma\) and fixed elements \(h\in R\), \(\xi\in k\). An ambiskew polynomial algebra \(A=A(R,\sigma,h,\xi)\) is generated by \(R\) and elements \(X_+,X_-\) subject to the defining relations \[ X_\pm a=\sigma^{\pm 1}(a)X_\pm,\quad X_+X_-=h+\xi X_-X_+. \] Examples of these algebras are generalized Weyl algebras in the sense of V. Bavula, \(U(sl_2)\), \(U_q(sl_2)\), down-up algebras. There is found a criterion for the existence of a special type of Hopf algebra structure on \(A\). For any maximal ideal \(\mathfrak m\) in \(A\) the Verma module \(M(\mathfrak m)\) is defined as \(A/(AX_++A\mathfrak m)\). Let \(L(\mathfrak m)\) be the unique simple quotient of \(M(\mathfrak m)\). Any finite dimensional simple module over \(A\) is isomorphic to some \(L(\mathfrak m)\). One of the main results of the paper is the calculation of the dimension of \(L(\mathfrak m)\). Another result is a Clebsch-Gordon decomposition theorem of tensor products of two simple modules.