In
mathematics, the special linear Lie algebra of order n (denoted
or
) is the
Lie algebra of
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
with
trace zero and with the
Lie bracket
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
. 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
In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the ge ...
.
Applications
The Lie algebra
is central to the study of
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws ...
,
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
and
supersymmetry: its
fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible representation, irreducible finite-dimensional representation of a semisimple Lie algebra, semisimple Lie group
or Lie algebra whose highest weig ...
is the so-called
spinor representation, while its
adjoint representation
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is G ...
generates the
Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
SO(3,1) of special relativity.
The algebra
plays an important role in the study of
chaos
Chaos or CHAOS may refer to:
Arts, entertainment and media Fictional elements
* Chaos (''Kinnikuman'')
* Chaos (''Sailor Moon'')
* Chaos (''Sesame Park'')
* Chaos (''Warhammer'')
* Chaos, in ''Fabula Nova Crystallis Final Fantasy''
* Cha ...
and
fractals, as it generates the
Möbius group SL(2,R), which describes the automorphisms of the
hyperbolic plane, the simplest
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
of negative curvature; by contrast,
SL(2,C)
SL may refer to:
Arts and entertainment
* SL (rapper), a rapper from London
* ''Second Life'', a multi-user 3D virtual world
* Sensei's Library, an Internet site dedicated to the game of Go
* Subdominant leittonwechselklänge
* Leica SL, a mirro ...
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
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
Fundamental may refer to:
* Foundation of reality
* Fundamental frequency, as in music or phonetics, often referred to as simply a "fundamental"
* Fundamentalism, the belief in, and usually the strict adherence to, the simple or "fundamental" idea ...
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
. It remains to show
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:
*If
has finite dimension and is irreducible, then
-eigenvalue of ''v'' is a nonnegative integer
and
has a basis
.
*Conversely, if the
-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
In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by ...
), "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
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
:
:
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
.
Notes
References
* Etingof, Pavel.
Lecture Notes on Representation Theory.
*
*
* 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)
*{{Citation , url=https://books.google.com/books?id=7AHsSUrooSsC&pg=PA3, title=Algèbres de Lie semi-simples complexes, last=Serre, first=Jean-Pierre, date=2000, publisher=Springer, trans-title=Complex Semisimple Lie Algebras, isbn=978-3-540-67827-4, language=en, ref={{Harvid, Serre, translator-last=Jones, translator-first=G. A..
See also
*
Affine Weyl group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...
*
Finite Coxeter group
*
Hasse diagram
In order theory, a Hasse diagram (; ) is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set ''(S, ≤)'' one represents ...
*
Linear algebraic group
*
Nilpotent orbit
*
Root system
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representatio ...
*
sl2-triple
*
Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections ...
Lie groups