Automorphisms and conjugacy of compact real forms of the classical infinite dimensional matrix Lie algebras (Q2753167)

The paper deals with infinite dimensional matrix Lie algebras \(\mathfrak g=\mathfrak s\mathfrak l(J,\mathbb{K}), \mathfrak g=\mathfrak s\mathfrak p(J,\mathbb{K}), \mathfrak g=\mathfrak o(J,J,\mathbb{K})\). Here \(\mathbb{K}\) is a field of zero characteristic and J is an infinite set. Any element of these algebras is by definition finitary, i.e. has only finitely many non-zero entries. Besides, the elements of \(\mathfrak s\mathfrak l(J,\mathbb{K})\) are matrices with trace zero, and the elements of \(\mathfrak s\mathfrak p(J,\mathbb{K})\), resp. \(\mathfrak o(J,J,\mathbb{K})\) leave a skew-symmetric, resp. symmetric bilinear form invariant. The main result of the paper is the specification of all automorphisms and the automorphism groups of \(\mathfrak g\). If \(\text{card }J\geq 5\), then it is derived that every automorphism of \(\mathfrak s\mathfrak p(J,\mathbb{K})\) and \(\mathfrak o(J,J,\mathbb{K})\) is a conjugation \(\pi_{A}\) by a special kind of matrix \(A\). And for algebra \(\mathfrak s\mathfrak l(J,\mathbb{K})\) is either a conjugation \(\pi_{A}\) or of the form \(\pi(x)=\pi_{A}^{-T}(x)=-Ax^{T}A^{-1}\) for any \(x\in \mathfrak s\mathfrak l(J,\mathbb{K})\). It is proved when a conjugation \(\pi_{A}\) defines an automorphism of \(\mathfrak g\). For example, if \(\mathfrak g=\mathfrak s\mathfrak l(J,\mathbb{K})\), then \(A\in GL^{f}(J,\mathbb{K})\), where \(GL^{f}(J,\mathbb{K})\) is a group of all invertible \(J\times J\) matrices with the property that \(x\) and \(x^{-1}\) have finitary rows and columns.