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X-inner automorphisms of enveloping rings. (English) Zbl 0744.16017

Summary: We determine the \(X\)-inner automorphisms of the smash product \(R\# U(L)\) of a prime ring \(R\) by the universal enveloping algebra \(U(L)\) of a characteristic 0 Lie algebra \(L\). Specifically, we show that any such automorphism \(\sigma\) stabilizing \(R\) can be written as a product \(\sigma=\sigma_ 1\sigma_ 2\), where \(\sigma_ 1\) is induced by conjugation by a unit of \(Q_ s(R)\), the symmetric Martindale ring of quotients of \(R\), and \(\sigma_ 2\) is induced by conjugation by a unit of \(Q_ s(T)\). Here \(S=Q_ l(R)\) is the left Martindale ring of quotients of \(R\) and \(T\) is the centralizer of \(S\) in \(S\#U(L)\supseteq R\#U(L)\). One of the subtleties of the proof is that we must work in several unrelated overrings of \(R\#U(L)\).

MSC:

16S40 Smash products of general Hopf actions
17B35 Universal enveloping (super)algebras
16W20 Automorphisms and endomorphisms
16N60 Prime and semiprime associative rings
16S30 Universal enveloping algebras of Lie algebras
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
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References:

[1] Bergen, J.; Montgomery, S., Smash products and outer derivations, Israel J. Math., 53, 321-345 (1986) · Zbl 0602.16007
[2] Montgomery, S., \(X\)-inner automorphisms of filtered algebras, (Proc. Amer. Math. Soc., 83 (1981)), 263-268 · Zbl 0474.16003
[3] Montgomery, S., \(X\)-inner automorphisms of filtered algebras, II, (Proc. Amer. Math. Soc., 87 (1983)), 569-575 · Zbl 0509.16018
[4] Passman, D. S., Prime ideals in enveloping rings, Trans. Amer. Math. Soc., 302, 535-560 (1987) · Zbl 0628.16020
[5] Passman, D. S., Infinite Crossed Products (1989), Academic Press: Academic Press San Diego, CA · Zbl 0519.16010
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