Lie derivations of incidence algebras

Abstract

Abstract. Let X be a locally finite preordered set, R a commutative ring with identity and I(X, R) the incidence algebra of X over R. In this note we prove that each Lie derivation of I(X, R) is proper, provided that R is 2-torsion free.

Lie triple derivations of incidence algebras

“…In particular, , the third, fourth and sixth terms of the right-hand side of (25) coincide with the corresponding terms of the right-hand side of (26). From the definition of the restriction of f , it is clear that f (x, y) = f | y…”

“…Observe that the sum above is finite, and hence f | y x ∈Ĩ(X, R). The following fact is [26,Lemma 3.3].…”

“…Notice that in the theory of operator algebras, the incidence algebra I(X, R) of a finite poset X is referred as a digraph algebra or a finite dimensional CSL algebra. Hence the second author of this note [22], Khrypchenko [11], and Zhang-Khrypchenko [26] studied the Herstein's program on incidence algebras in a linear and combinatorial manner. Our main motivation of this article is, following the trace of [22,11,26], to connect the Herstein's program to operator algebras depending on the methods of linear algebra.…”

“…Let D :Ĩ(X, R) → I(X, R) be a derivation. We definê D(f )(x, y) := D(f | y x )(x, y) for all f ∈ I(X, R), x y. ThenD is a linear extension of D and is a derivation of I(X, R) by [26,Remark 3.7]. Let E be a derivation of I(X, R) satisfying E(g) = D(g) for all g ∈Ĩ(X, R).…”

Commuting Maps on Certain Incidence Algebras

“…Recently, the second author [16] studied the Herstein's Lie-type mapping research program (see [4]) on incidence algebras and proved that every Jordan derivation of I(X, R) degenerates to a derivation, provided that R is 2-torsion free. Since then more and more Lie-type maps were considered on incidence algebras, see [8,15,18] etc. Roughly speaking, all the known Lie-type maps of I(X, R) are proper or of the standard form.…”