In mathematics, the special linear Lie algebra of order n over a field F, denoted or, is the Lie algebra of all the n \times n matrices (with entries in F) with trace zero and with the Lie bracket given by the commutator. 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 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

Representation theory of sl2C

The Lie algebra is a three-dimensional complex Lie algebra. Its defining feature is that it contains a basis e, h, f 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. The Lie algebra can be viewed as a subspace of its universal enveloping algebra and, in U, there are the following commutator relations shown by induction: Note that, here, the powers f^k, 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 v_j is zero or has h-eigenvalue distinct from the eigenvalues of the others that are nonzero. Saying v is a -weight vector is equivalent to saying that it is simultaneously an eigenvector of h and e; a short calculation then shows that, in that case, the e-eigenvalue of v is zero:. Thus, for some integer N \ge 0, and in particular, by the early lemma, which implies that \lambda = N. It remains to show is irreducible. If is a subrepresentation, then it admits an eigenvector, which must have eigenvalue of the form N - 2j; thus is proportional to v_j. By the preceding lemma, we have v = v_0 is in W and thus W' = W. \square 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.

Representation theory of slnC

When for a complex vector space V of dimension n, each finite-dimensional irreducible representation of \mathfrak g can be found as a subrepresentation of a tensor power of V. The Lie algebra can be explicitly realized as a matrix Lie algebra of traceless n\times n matrices. This is the fundamental representation for. Set M_{i,j} to be the matrix with one in the i,j entry and zeroes everywhere else. Then Form a basis for. This is technically an abuse of notation, and these are really the image of the basis of in the fundamental representation. Furthermore, this is in fact a Cartan–Weyl basis, with the H_i spanning the Cartan subalgebra. Introducing notation if j > i, and, also if j > i, the E_{i,j} are positive roots and F_{i,j} are corresponding negative roots. A basis of simple roots is given by E_{i,i+1} for.