In mathematics, the special linear Lie algebra of order
) 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.
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.
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.
is a three-dimensional complex Lie algebra.
Its defining feature is that it contains a basis
satisfying the commutation relations This is a Cartan-Weyl basis for
It has an explicit realization in terms of 2-by-2 complex matrices with zero trace: This is the fundamental or defining representation for
can be viewed as a subspace of its universal enveloping algebra
, there are the following commutator relations shown by induction:[1]
The first basic fact (that follows from the above commutator relations) is:[1] Lemma — Let
, From this lemma, one deduces the following fundamental result:[2] Theorem — Let
-eigenvalue distinct from the eigenvalues of the others that are nonzero.
-weight vector is equivalent to saying that it is simultaneously an eigenvector of
; a short calculation then shows that, in that case, the
and in particular, by the early lemma, which implies that
is a subrepresentation, then it admits an eigenvector, which must have eigenvalue of the form
As a corollary, one deduces: The beautiful special case of
shows a general way to find irreducible representations of Lie algebras.
Namely, we divide the algebra to three subalgebras "h" (the Cartan subalgebra), "e", and "f", which behave approximately like their namesakes in
Namely, in an irreducible representation, we have a "highest" eigenvector of "h", on which "e" acts by zero.
The basis of the irreducible representation is generated by the action of "f" on the highest eigenvectors of "h".
See the theorem of the highest weight.
for a complex vector space
can be found as a subrepresentation of a tensor power of
[4] The Lie algebra can be explicitly realized as a matrix Lie algebra of traceless
This is technically an abuse of notation, and these are really the image of the basis of
spanning the Cartan subalgebra.
A basis of simple roots is given by