Weil Algebra, 3-Lie Algebra and B.R.S. Algebra

Abstract: We consider the 3-Lie algebra induced by a Lie algebra with the help of an analog of a trace. We propose the extension of the Weil algebra of a Lie algebra to the Weil algebra of induced 3-Lie algebra by introducing in addition to an analog of connection and its curvature new elements and defining their differential by means of structure constants of 3-Lie algebra. We also propose a new approach to the universal B.R.S. algebra based on the quantum triple Nambu bracket.