Suppose that g is a semisimple Lie algebra over the acreage of absolute numbers. By Cartan's criterion, the Killing anatomy is nondegenerate, and can be diagonalized in a acceptable base with the askew entries +1 or -1. By Sylvester's law of inertia, the cardinal of absolute entries is an invariant of the bilinear form, i.e. it does not depend on the best of the diagonalizing basis, and is alleged the basis of the Lie algebra g. This is a cardinal amid 0 and the ambit of g which is an important invariant of the absolute Lie algebra. In particular, a absolute Lie algebra g is alleged bunched if the Killing anatomy is abrogating definite. It is accepted that beneath the Lie correspondence, bunched Lie algebras accord to bunched Lie groups.
If gC is a semisimple Lie algebra over the circuitous numbers, again there are several non-isomorphic absolute Lie algebras whose complexification is gC, which are alleged its absolute forms. It turns out that every circuitous semisimple Lie algebra admits a different (up to isomorphism) bunched absolute anatomy g. The absolute forms of a accustomed circuitous semisimple Lie algebra are frequently labeled by the absolute basis of apathy of their Killing form.
For example, the circuitous appropriate beeline algebra sl(2,C) has two absolute forms, the absolute appropriate beeline algebra, denoted sl(2,R), and the appropriate unitary algebra, denoted su(2). The aboriginal one is noncompact, the alleged breach absolute form, and its Killing anatomy has signature (2,1). The additional one is the bunched absolute anatomy and its Killing anatomy is abrogating definite, i.e. has signature (0,3). The agnate Lie groups are the noncompact accumulation SL(2,R) of 2 by 2 absolute matrices with the assemblage account and the appropriate unitary accumulation SU(2), which is compact.
If gC is a semisimple Lie algebra over the circuitous numbers, again there are several non-isomorphic absolute Lie algebras whose complexification is gC, which are alleged its absolute forms. It turns out that every circuitous semisimple Lie algebra admits a different (up to isomorphism) bunched absolute anatomy g. The absolute forms of a accustomed circuitous semisimple Lie algebra are frequently labeled by the absolute basis of apathy of their Killing form.
For example, the circuitous appropriate beeline algebra sl(2,C) has two absolute forms, the absolute appropriate beeline algebra, denoted sl(2,R), and the appropriate unitary algebra, denoted su(2). The aboriginal one is noncompact, the alleged breach absolute form, and its Killing anatomy has signature (2,1). The additional one is the bunched absolute anatomy and its Killing anatomy is abrogating definite, i.e. has signature (0,3). The agnate Lie groups are the noncompact accumulation SL(2,R) of 2 by 2 absolute matrices with the assemblage account and the appropriate unitary accumulation SU(2), which is compact.
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