Lie automorphisms of incidence algebras

Abstract

Let X X be a finite connected poset and K K a field. We give a full description of the Lie automorphisms of the incidence algebra I ( X , K ) I(X,K) . In particular, we show that they are in general not proper.

Potent preservers of incidence algebras

“…Let X be a finite poset and F be a field with char(F ) = 2 and |F | > 2. Then any bijective idempotent preserver ϕ : I(X, F ) → I(X, F ) is a Lie automorphism of I(X, F ), so, whenever X is connected, ϕ admits the description given in [20].…”

“…In view of [20,Theorem 4.15] we may take ϕ to be elementary. Then ϕ(e xy ) = σ(x, y)θ(e xy ) for all x < y, where θ(e xy ) ∈ B and σ(x, y) ∈ F * by [20,Lemma 4.3]. By (iii) we have ϕ(e x ) 2 = ϕ(e x ) = ϕ(e 2…”

“…. , x n } be a finite connected poset, F a field with char(F ) = 2, ϕ an elementary Lie automorphism of I(X, F ) and ϕ = τ θ,σ,c its description as in [20,Theorem 5.18]. Then for all…”

“…The following bijective linear map was used in [20] as an example of a nonproper [21] Lie automorphism (i.e., a Lie automorphism which is not the sum of an automorphism or the negative of an anti-automorphism and a central-valued linear map) of I(X, F ). It turns out to be an idempotent preserver of I(X, F ).…”

“…We start Section 3 by observing in Remark 3.1 that Proposition 2.1 also holds over fields of characteristic 2 with more than 2 elements. In the latter case the description of bijective linear idempotent preservers of I(X, F ) reduces to the description of idempotent-preserving elementary Lie automorphisms [20] of I(X, F ), which is accomplished in Theorem 3.5. In Section 4 we characterize Jordan automorphisms of the incidence algebra I(X, F ) of a finite connected poset X over a field F (see Theorem 4.4), which together with the result of Section 2 gives the description of bijective linear idempotent preservers of I(X, F ), whenever char(F ) = 2 (see Corollary 4.5).…”