The orthogonal Lie algebra of operators: Ideals and derivations

Abstract

We study in this paper the infinite-dimensional orthogonal Lie algebra O C which consists of all bounded linear operators T on a separable, infinite-dimensional, complex Hilbert space H satisfying CT C = −T * , where C is a conjugation on H. By employing results from the theory of complex symmetric operators and skew-symmetric operators, we determine the Lie ideals of O C and their dual spaces. We study derivations of O C and determine their spectra. These results complete some results of P. de la Harpe and provide interesting contrasts between O C and the algebra B(H) of all bounded linear operators on H. IntroductionThe study of skew-symmetric matrices (i.e., those matrices M satisfying M + M tr = 0, where M tr denotes the transpose of M ) can be traced back to the work of L.-K. Hua on automorphic functions [27,28], N. Jacobson on projective geometry [30], and C. Siegel on symplectic geometry [48]. Skew-symmetric matrices arise naturally in partial differential equations [42], differential geometry [13], algebraic geometry [15] and many other mathematical disciplines. In the finite-dimensional Hilbert space of even dimension, there is a natural one to one correspondence between the set of all non-singular skew-symmetric matrices and the set of all sympletic forms on the Hilbert space ([32, page 16]). As a classical finite-dimensional Lie algebra over the complex field C, the orthogonal Lie algebra so(n, C) consists of all n × n skew-symmetric complex matrices, that is, so(n, C) = {X ∈ M n (C) : X + X tr = 0}, where M n (C) denotes the set of all n × n complex matrices. It is known that so(n, C) is the Lie algebra of the complex orthogonal Lie group SO(n, C) = {X ∈ M n (C) : XX tr = I, detX = 1} (see [23, page 41] or [25, page 341] for more details), and plays important roles in the classification of semi-simple Lie algebras.Skew-symmetric matrices are playing important roles in quantum physics. In particular, they are closely related to the noncommutative spacetimes. A noncommutative spacetime [49] is defined by the Hermitian generatorsx i of a noncommutative C * -algebra of "functions on spacetime" which obey the commutation