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Concept
Special linear Lie algebra
Résumé
In mathematics, the special linear Lie algebra of order n (denoted or ) is the Lie algebra of matrices with trace zero and with the Lie bracket . This algebra is well studied and understood, and is often used as a model for the study of other Lie algebras. The Lie group that it generates is the special linear group.
The Lie algebra is central to the study of special relativity, general relativity and supersymmetry: its fundamental representation is the so-called spinor representation, while its adjoint representation generates the Lorentz group SO(3,1) of special relativity.
The algebra plays an important role in the study of chaos and fractals, as it generates the Möbius group SL(2,R), which describes the automorphisms of the hyperbolic plane, the simplest Riemann surface of negative curvature; by contrast, SL(2,C) describes the automorphisms of the hyperbolic 3-dimensional ball.
Representation theory of semisimple Lie algebras
The Lie algebra is a three-dimensional complex Lie algebra. Its defining feature is that it contains a basis satisfying the commutation relations
, and .
This is a Cartan-Weyl basis for .
It has an explicit realization in terms of two-by-two complex matrices with zero trace:
, .
This is the fundamental or defining representation for .
The Lie algebra can be viewed as a subspace of its universal enveloping algebra and, in , there are the following commutator relations shown by induction:
Note that, here, the powers , etc. refer to powers as elements of the algebra U and not matrix powers. The first basic fact (that follows from the above commutator relations) is:
From this lemma, one deduces the following fundamental result:
The first statement is true since either is zero or has -eigenvalue distinct from the eigenvalues of the others that are nonzero. Saying is a -weight vector is equivalent to saying that it is simultaneously an eigenvector of ; a short calculation then shows that, in that case, the -eigenvalue of is zero: . Thus, for some integer , and in particular, by the early lemma,
which implies that .
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