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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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