|
In the theory of Lie groups, Lie algebras and their representation theory, a Lie algebra extension is an enlargement of a given Lie algebra by another Lie algebra . Extensions arise in several ways. There is the trivial extension obtained by taking a direct sum of two Lie algebras. Other types are the split extension and the central extension. Extensions may arise naturally, for instance, when forming a Lie algebra from projective group representations. Such a Lie algebra will contain central charges. Starting with a polynomial loop algebra over finite-dimensional simple Lie algebra and performing two extensions, a central extension and an extension by a derivation, one obtains a Lie algebra which is isomorphic with an untwisted affine Kac–Moody algebra. Using the centrally extended loop algebra algebra one may construct a current algebra in two spacetime dimensions. The Virasoro algebra is the universal central extension of the Witt algebra.〔 Central extensions are needed in physics, because the symmetry group of a quantized system usually is a central extension of the classical symmetry group, and in the same way the corresponding symmetry Lie algebra of the quantum system is, in general, a central extension of the classical symmetry algebra. Kac–Moody algebras have been conjectured to be a symmetry groups of a unified superstring theory.〔 (The Beacon of Kac–Moody Symmetry for Physics. (free access) )〕 The centrally extended Lie algebras play a dominant role in quantum field theory, particularly in conformal field theory, string theory and in M-theory. A large portion towards the end is devoted to background material for applications of Lie algebra extensions, both in mathematics and in physics, in areas where they are actually useful. A parenthetical link, (background material), is provided where it might be beneficial. ==History== Due to the Lie correspondence, the theory, and consequently the history of Lie algebra extensions, is tightly linked to the theory and history of group extensions. A systematic study of group extensions was performed by the Austrian mathematician Otto Schreier in 1923 in his PhD. thesis and later published.〔 Otto Schreier (1901 - 1929) was a pioneer in the theory of extension of groups. Along with his rich research papers, his lecture notes were posthumously published (edited by Emanuel Sperner) under the name ''Einführung in die analytische Geometrie und Algebra'' (Vol I 1931, Vol II 1935), later in 1951 translated to English in (Introduction to Modern Algebra and Matrix Theory ). See for further reference.〕 The problem posed for his thesis by Otto Hölder was "given two groups and , find all groups having a normal subgroup isomorphic to such that the factor group is isomorphic to ". Lie algebra extensions are most interesting and useful for infinite-dimensional Lie algebras. In 1967, Victor Kac and Robert Moody independently generalized the notion of classical Lie algebras, resulting in a new theory of infinite-dimensional Lie algebras algebras, now called Kac–Moody algebras. They generalize the finite-dimensional simple Lie algebras and can often concretely be constructed as extensions. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Lie algebra extension」の詳細全文を読む スポンサード リンク
|