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An example of a non-zero Lie coalgebra M for which Loc(M)=0

1990; Elsevier BV; Volume: 68; Issue: 3 Linguagem: Inglês

10.1016/0022-4049(90)90089-z

1873-1376

Walter Michaelis,

Advanced Topics in Algebra

Let K be a field of characteristic 0, let W1=DerK(K[x]), and let M=(W1)∘=the dual Lie coalgebra of W1. By definition (W1)∘ is the largest subspace of (W1)∗ carrying a Lie coalgebra structure induced by the restriction to (W1)∘ of ϕ∗ where ϕ=[,]: W1⊗W1→W1 is the Lie bracket of W1. Then M is an example of a non-zero Lie coalgebra in which no element of M other than 0 lies in a finite-dimensional sub Lie coalgebra. Thus Loc(M)=0 where Loc(M) denotes the sum of all of the finite-dimensional sub Lie coalgebras of M.