Special linear Lie algebra

In mathematics, the special linear Lie algebra of order n (denoted formula_1 or formula_2) is the Lie algebra of formula_3 matrices with trace zero and with the Lie bracket formula_4. 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.
Applications.
The Lie algebra formula_5 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 formula_6 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.
Representation theory of formula_7.
By definition, the Lie algebra formula_8 consists of two-by-two complex matrices with zero trace. There are three standard basis elements, formula_9,formula_10, and formula_11, with
The commutators are
The Lie algebra formula_8 can be viewed as a subspace of its universal enveloping algebra formula_19 and, in formula_20, there are the following commutator relations shown by induction:
Note that, here, the powers formula_23, 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 formula_24 is zero or has formula_11-eigenvalue distinct from the eigenvalues of the others that are nonzero. Saying formula_26 is a formula_27-weight vector is equivalent to saying that it is simultaneously an eigenvector of formula_28; a short calculation then shows that, in that case, the formula_9-eigenvalue of formula_26 is zero: formula_31. Thus, for some integer formula_32, formula_33 and in particular, by the early lemma,
which implies that formula_35. It remains to show formula_36 is irreducible. If formula_37 is a subrepresentation, then it admits an eigenvector, which must have eigenvalue of the form formula_38; thus is proportional to formula_24. By the preceding lemma, we have formula_40 is in formula_41 and thus formula_42. formula_43
As a corollary, one deduces:
The beautiful special case of formula_54 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 formula_54. 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.
Representation theory of formula_56.
When formula_57 for a complex vector space formula_44, each finite-dimensional irreducible representation of formula_59 can be found as a subrepresentation of a tensor power of formula_44.