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Zero divisors in amalgamated free products of Lie algebras. (Russian, English) Zbl 1059.17003

Sib. Mat. Zh. 45, No. 1, 229-238 (2004); translation in Sib. Math. J. 45, No. 1, 188-195 (2004).
Let \(L\) be a Lie algebra over a field. Nonzero elements \(x,y\in L\) are called zero divisors iff \(\langle x \rangle\) commutes with \(\langle y\rangle\), where \(\langle l \rangle\) denotes the ideal in \(L\) generated by an element \(l\). A Lie algebra \(A\) is called a domain if it lacks zero divisors. The authors investigate the question when an amalgamated free product of two domains is a domain. The commuting elements of an amalgamated free product of two Lie algebras are classified. The center of an amalgamated free product of Lie algebras is computed. The results are necessary for constructing the algebraic geometry over Lie algebras that was suggested in [A. Myasnikov and V. Remeslennikov, J. Algebra 234, No. 1, 225–276 (2000; Zbl 0970.20017)].

MSC:

17B01 Identities, free Lie (super)algebras
14A22 Noncommutative algebraic geometry

Citations:

Zbl 0970.20017
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