Fixed rings of twisted generalized Weyl algebras

Twisted generalized Weyl algebras (TGWAs) are a large family of that includes several interest for ring theory and representation theory, such as algebras, primitive quotients $U(\mathfrak{sl}_2)$, multiparameter quantized algebras. In this work, we study invariants TGWAs under diagonal automorphisms. Under certain conditions, able to show the fixed TGWA by an automorphism is again TGWA. particular, true $\Bbbk$-finitistic type $(A_1)^n$ $A_2$. We apply theorem properties ring, noetherian property simplicity. also look at behavior simple weight modules when restricted action ring. As auxiliary result, in order tensor products TGWAs, prove class regular, $\mu$-consistent closed products.