Special linear Lie algebra explained
, denoted
or
, is the
Lie algebra of all the
matrices (with entries in
) 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 ball.
Representation theory
See also: Representation theory of semisimple Lie algebras.
Representation theory of sl2C
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 2-by-2 complex matrices with zero trace:
E=\begin{bmatrix}
0&1\\
0&0
\end{bmatrix}
,
F=\begin{bmatrix}
0&0\\
1&0
\end{bmatrix}
,
H=\begin{bmatrix}
1&0\\
0&-1
\end{bmatrix}
.
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:
,
[e,fk]=-k(k-1)fk-1+kfk-1h
.
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
and
; 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,
0=e ⋅ vN+1=(λ-(N+1)+1)vN,
which implies that
. It remains to show
W=\operatorname{span}\{vj\midj\ge0\}
is irreducible. If
is a subrepresentation, then it admits an eigenvector, which must have eigenvalue of the form
; thus is proportional to
. By the preceding lemma, we have
is in
and thus
.
As a corollary, one deduces:
has finite dimension and is irreducible, then
-eigenvalue of
v is a nonnegative integer
and
has a basis
.
-eigenvalue of
is a nonnegative integer and
is irreducible, then
has a basis
; in particular has finite dimension.
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
akg=ak{sl}nC=\operatorname{ak{sl}}V
for a complex vector space
of dimension
, each finite-dimensional irreducible representation of
can be found as a subrepresentation of a
tensor power of
.
The Lie algebra can be explicitly realized as a matrix Lie algebra of traceless
matrices. This is the fundamental representation for
.
Set
to be the matrix with one in the
entry and zeroes everywhere else. Then
Hi:=Mi,i-Mi+1,i+1,with1\leqi\leqn-1
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
spanning the Cartan subalgebra. Introducing notation
if
, and
, also if
, the
are positive roots and
are corresponding negative roots.
A basis of simple roots is given by
for
.
References
- Etingof, Pavel. "Lecture Notes on Representation Theory".
- Book: Kac, Victor. Victor Kac. Integrable Representations of Kac–Moody Algebras and the Weyl Group. Infinite dimensional Lie algebras. 3rd . Cambridge University Press. 10.1017/CBO9780511626234.004 . 1990. 0-521-46693-8.
- A. L. Onishchik, E. B. Vinberg, V. V. Gorbatsevich, Structure of Lie groups and Lie algebras. Lie groups and Lie algebras, III. Encyclopaedia of Mathematical Sciences, 41. Springer-Verlag, Berlin, 1994. iv+248 pp. (A translation of Current problems in mathematics. Fundamental directions. Vol. 41, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990. Translation by V. Minachin. Translation edited by A. L. Onishchik and E. B. Vinberg)
- V. L. Popov, E. B. Vinberg, Invariant theory. Algebraic geometry. IV. Linear algebraic groups. Encyclopaedia of Mathematical Sciences, 55. Springer-Verlag, Berlin, 1994. vi+284 pp. (A translation of Algebraic geometry. 4, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989. Translation edited by A. N. Parshin and I. R. Shafarevich)
- .
See also
