On the automorphism group of cones and wedges
A subset W of a finite dimensional real vector space L is called a wedge iff it satisfies the following conditions: (1) \(W+W=W\); (2) \(\lambda\) \(W\subset W\) for all \(\lambda\in {\mathbb{R}}\), \(\lambda\geq 0\); (3) \(\bar W=W\). We say that the wedge W is generating iff \(L=W-W\). The tangent space \(T_ x\) of W at x is the set of all vectors \(y\in L\) for which there are sequences of points \(x_ n,x'_ n\in W\) converging to x and sequences \(r_ n\) and \(r'_ n\) of positive real numbers such that \(y=\lim r_ n(x_ n-x)=\lim (-r'_ n)(x'_ n-x).\) For a wedge W in a finite dimensional real vector space L we call a point \(x\in W\) a \(C^ 1\)-boundary point of W iff it is a boundary point for which \(T_ x\) is a hyperplane. The set of all these points will be denoted by \(C^ 1(W)\). The automorphism grup Aut W is the group of all those vector space automorphisms g of W-W which satisfy \(gW=W.\) Theorem 1. If W is a generating wedge in a finite dimensional real vector space L, then a linear map \(X: L\to L\) belongs to the Lie algebra \({\mathfrak g}(Aut W)\) of the automorphism group of W if and only if it satisfies any of the following equivalent conditions: (1) For each point \(x\in W\) the vector \(X_ x\) belongs to the tangent space \(T_ x\). (2) For each \(C^ 1\)-boundary point \(x\in C^ 1(W)\) the vector \(X_ x\) belongs to the tangent hyperplane \(T_ x.\) Theorem 2. For a generating wedge W in a finite dimensional real Lie algebra L we have the following statements: (1) W is an invariant wedge, i.e. \(e^{ad x}W=W\) for all \(x\in L\), if and only if for each \(C^ 1\)- boundary point \(x\in W\) we have \([x,L]\subset T_ x\). (2) W is a Lie semialgebra if and only if for each \(C^ 1\)-boundary point \(x\in W\) we have \([x,T_ x]\subset T_ x.\) As a consequence the authors derive a complete description of the Lie semialgebras in compact Lie algebras.
scientific article; zbMATH DE number 3975162