NONLINEAR ANTI-COMMUTING MAPS OF STRICTLY TRIANGULAR MATRIX LIE ALGEBRAS

Let N(F) be the Lie algebra consisting of all strictly upper triangular (n + 1) x (n + 1) matrices over a field F. A map phi on N(F) is called to be anti-commuting if [phi(x), y] = -[ x, phi(y)] for any x, y is an element of N(F). We show that for n >= 4, a nonlinear map phi : N(F) -> N(F) is anti-commuting if and only if there exist b, b(1), b(2) is an element of F and a nonlinear function f : N(F) -> F such that phi = ad (bE(2n))+ mu((n,n+1))(b2) + mu((12))(b1) + phi(f), where ad (bE(2n)) is an inner anti-commuting map, mu((n,n+1))(b2), mu((12))(b1) are extremal anti-commuting maps, phi(f) is a central anti-commuting map.