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Constructing simply laced Lie algebras from extremal elements. (English) Zbl 1169.17013
An extremal element Lie algebra \(L\) over a field \(k\) is an element \(x\) in \(L\) such that \([x,[x,L]]\subseteq kx\). Following the authors, a Lie algebra \(L\) is said to be realized by a given graph if the following conditions holds: a) \(L\) is generated by a set of extremal elements which are the vertices of the given graph and b) two vertices of the graph are joined by an edge if and only if the corresponding elements do not commute in \(L\). In this paper, the authors construct an algebraic variety that parametrizes Lie algebras realized by finite simple graphs without loops or multiple edges. Over fields of characteristic different from 2, the varieties related to graphs corresponding to simply laced Dynkin diagrams of affine type turn to be affine spaces and Chevalley algebras of the corresponding finite-type diagrams can be parametrized through them.

17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
17B01 Identities, free Lie (super)algebras
14R10 Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
57M15 Relations of low-dimensional topology with graph theory
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