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Isomorphism problems and groups of automorphisms for generalized Weyl algebras. (English) Zbl 0961.16016
A generalized Weyl algebra $$A(a)$$ over an algebraically closed field $$k$$ of characteristic zero is generated by elements $$h,x,y$$ subject to the relations $xh=(h-1)x,\quad yh=(h+1)y,\quad xy=a(h-1),\quad yx=a(h),$ where $$a\in k[h]$$. The Weyl algebra $$A_1(k)$$ and the factor algebra $$B_\lambda=U(sl_2)/(C-\lambda)$$, $$C$$ the Casimir element, $$\lambda\in k$$, are particular examples of $$A(a)$$ for some specific $$a$$. An element $$a$$ is reflective if there exists some $$\rho\in k$$ such that $$(\rho-a)=(-1)^{\deg a}a(h)$$. Denote by $$G$$ the subgroup of $$k$$-automorphisms of $$A(a)$$ generated by all automorphisms $\exp(\lambda\text{ ad }x^m),\quad\exp(\lambda\text{ ad }y^m),\qquad m\geq 1,\;\lambda\in k,$ and by the automorphisms $$\Theta_\mu$$, where $$\Theta_\mu(x)=\mu x$$, $$\Theta_\mu(y)=\mu^{-1}y$$, $$\Theta(h)=h$$. If $$a$$ is not reflective then $$G$$ coincides with the automorphism group of $$A(a)$$. If $$a$$ is reflective then the automorphism group of $$A(a)$$ is generated by $$G$$ and a new automorphism $$\Omega$$ such that $$\Omega(x)=y$$, $$\Omega(y)=(-1)^{\deg a}x$$, $$\Omega(h)=1+\rho-h$$. Two algebras $$A(a_1),A(a_2)$$ are isomorphic if and only if $$a_2(h)=\eta a_1(\tau\pm h)$$ for some $$\eta,\tau\in k$$, $$\eta\neq 0$$.
Another class of algebras considered in the paper consists of the algebras $R(f)=\langle A,B,H\mid[H,A]=A,\;[H,B]=-B,\;[A,B]=f(H)\rangle$ where $$f\in k[H]$$. A particular case of $$R(f)$$ is the algebra $$U(sl_2)$$. It is shown that $R(f_1)\simeq R(f_2)\iff f_2(H)=\eta f_1(\tau\pm H)$ for some $$\eta,\tau\in k$$, $$\eta\neq 0$$.

##### MSC:
 16S36 Ordinary and skew polynomial rings and semigroup rings 16W20 Automorphisms and endomorphisms 16W35 Ring-theoretic aspects of quantum groups (MSC2000)
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