Special linear Lie algebra

Lie groups and Lie algebras
Classical groups General linear GL(n) Special linear SL(n) Orthogonal O(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n)
Simple Lie groups Classical An Bn Cn Dn Exceptional G2 F4 E6 E7 E8
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Lie algebras Lie group–Lie algebra correspondence Exponential map Adjoint representation Killing form Index Simple Lie algebra Loop algebra Affine Lie algebra
Semisimple Lie algebra Dynkin diagrams Cartan subalgebra Root system Weyl group Real form Complexification Split Lie algebra Compact Lie algebra
Representation theory Lie group representation Lie algebra representation Representation theory of semisimple Lie algebras Representations of classical Lie groups Theorem of the highest weight Borel–Weil–Bott theorem
Lie groups in physics Particle physics and representation theory Lorentz group representations Poincaré group representations Galilean group representations
Scientists Sophus Lie Henri Poincaré Wilhelm Killing Élie Cartan Hermann Weyl Claude Chevalley Harish-Chandra Armand Borel
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In mathematics, the special linear Lie algebra of order ${\displaystyle n}$ over a field ${\displaystyle F}$, denoted ${\displaystyle {\mathfrak {sl}}_{n}F}$ or ${\displaystyle {\mathfrak {sl}}(n,F)}$, is the Lie algebra of all the ${\displaystyle n\times n}$ matrices (with entries in ${\displaystyle F}$) with trace zero and with the Lie bracket ${\displaystyle :=XY-YX}$ 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.

The Lie algebra ${\displaystyle {\mathfrak {sl}}_{2}\mathbb {C} }$ 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 ${\displaystyle {\mathfrak {sl}}_{2}\mathbb {R} }$ 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.

The Lie algebra ${\displaystyle {\mathfrak {sl}}_{2}\mathbb {C} }$ is a three-dimensional complex Lie algebra. Its defining feature is that it contains a basis ${\displaystyle e,h,f}$ satisfying the commutation relations

${\displaystyle =h}$, ${\displaystyle =-2f}$, and ${\displaystyle =2e}$.

This is a Cartan-Weyl basis for ${\displaystyle {\mathfrak {sl}}_{2}\mathbb {C} }$. It has an explicit realization in terms of 2-by-2 complex matrices with zero trace:

${\displaystyle E={\begin{bmatrix}0&1\\0&0\end{bmatrix}}}$, ${\displaystyle F={\begin{bmatrix}0&0\\1&0\end{bmatrix}}}$, ${\displaystyle H={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}}$.

This is the fundamental or defining representation for ${\displaystyle {\mathfrak {sl}}_{2}\mathbb {C} }$.

The Lie algebra ${\displaystyle {\mathfrak {sl}}_{2}\mathbb {C} }$ can be viewed as a subspace of its universal enveloping algebra ${\displaystyle U=U({\mathfrak {sl}}_{2}\mathbb {C} )}$ and, in ${\displaystyle U}$, there are the following commutator relations shown by induction:^[1]

${\displaystyle =-2kf^{k},\,=2ke^{k}}$,

${\displaystyle =-k(k-1)f^{k-1}+kf^{k-1}h}$.

Note that, here, the powers ${\displaystyle f^{k}}$, 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:^[1]

From this lemma, one deduces the following fundamental result:^[2]

The first statement is true since either ${\displaystyle v_{j}}$ is zero or has ${\displaystyle h}$-eigenvalue distinct from the eigenvalues of the others that are nonzero. Saying ${\displaystyle v}$ is a ${\displaystyle {\mathfrak {b}}}$-weight vector is equivalent to saying that it is simultaneously an eigenvector of ${\displaystyle h}$ and ${\displaystyle e}$; a short calculation then shows that, in that case, the ${\displaystyle e}$-eigenvalue of ${\displaystyle v}$ is zero: ${\displaystyle e\cdot v=0}$. Thus, for some integer ${\displaystyle N\geq 0}$, ${\displaystyle v_{N}\neq 0,v_{N+1}=v_{N+2}=\cdots =0}$ and in particular, by the early lemma,

${\displaystyle 0=e\cdot v_{N+1}=(\lambda -(N+1)+1)v_{N},}$

which implies that ${\displaystyle \lambda =N}$. It remains to show ${\displaystyle W=\operatorname {span} \{v_{j}\mid j\geq 0\}}$ is irreducible. If ${\displaystyle 0\neq W'\subset W}$ is a subrepresentation, then it admits an eigenvector, which must have eigenvalue of the form ${\displaystyle N-2j}$; thus is proportional to ${\displaystyle v_{j}}$. By the preceding lemma, we have ${\displaystyle v=v_{0}}$ is in ${\displaystyle W}$ and thus ${\displaystyle W'=W}$. ${\displaystyle \square }$

As a corollary, one deduces:

• If ${\displaystyle V}$ has finite dimension and is irreducible, then ${\displaystyle h}$-eigenvalue of v is a nonnegative integer ${\displaystyle N}$ and ${\displaystyle V}$ has a basis ${\displaystyle v,fv,f^{2}v,\cdots ,f^{N}v}$.
• Conversely, if the ${\displaystyle h}$-eigenvalue of ${\displaystyle v}$ is a nonnegative integer and ${\displaystyle V}$ is irreducible, then ${\displaystyle V}$ has a basis ${\displaystyle v,fv,f^{2}v,\cdots ,f^{N}v}$; in particular has finite dimension.

The beautiful special case of ${\displaystyle {\mathfrak {sl}}_{2}}$ 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 ${\displaystyle {\mathfrak {sl}}_{2}}$. 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.

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When ${\displaystyle {\mathfrak {g}}={\mathfrak {sl}}_{n}\mathbb {C} =\operatorname {\mathfrak {sl}} V}$ for a complex vector space ${\displaystyle V}$ of dimension ${\displaystyle n}$, each finite-dimensional irreducible representation of ${\displaystyle {\mathfrak {g}}}$ can be found as a subrepresentation of a tensor power of ${\displaystyle V}$.^[4]

The Lie algebra can be explicitly realized as a matrix Lie algebra of traceless ${\displaystyle n\times n}$ matrices. This is the fundamental representation for ${\displaystyle {\mathfrak {sl}}_{n}\mathbb {C} }$.

Set ${\displaystyle M_{i,j}}$ to be the matrix with one in the ${\displaystyle i,j}$ entry and zeroes everywhere else. Then

${\displaystyle H_{i}:=M_{i,i}-M_{i+1,i+1},{\text{ with }}1\leq i\leq n-1}$

${\displaystyle M_{i,j},{\text{ with }}i\neq j}$

Form a basis for ${\displaystyle {\mathfrak {sl}}_{n}\mathbb {C} }$. This is technically an abuse of notation, and these are really the image of the basis of ${\displaystyle {\mathfrak {sl}}_{n}\mathbb {C} }$ in the fundamental representation.

Furthermore, this is in fact a Cartan–Weyl basis, with the ${\displaystyle H_{i}}$ spanning the Cartan subalgebra. Introducing notation ${\displaystyle E_{i,j}=M_{i,j}}$ if ${\displaystyle j>i}$, and ${\displaystyle F_{i,j}=M_{i,j}^{T}=M_{j,i}}$, also if ${\displaystyle j>i}$, the ${\displaystyle E_{i,j}}$ are positive roots and ${\displaystyle F_{i,j}}$ are corresponding negative roots.

A basis of simple roots is given by ${\displaystyle E_{i,i+1}}$ for ${\displaystyle 1\leq i\leq n-1}$.

• Etingof, Pavel. "Lecture Notes on Representation Theory".
• Kac, Victor (1990). "Integrable Representations of Kac–Moody Algebras and the Weyl Group". Infinite dimensional Lie algebras (3rd ed.). Cambridge University Press. pp. 30–46. doi:10.1017/CBO9780511626234.004. ISBN 0-521-46693-8.
• Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer
• 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) ISBN 3-540-54683-9
• 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) ISBN 3-540-54682-0
• Serre, Jean-Pierre (2001), Algèbres de Lie semi-simples complexes [Complex Semisimple Lie Algebras], Springer Monographs in Mathematics, translated by Jones, G. A., Springer, doi:10.1007/978-3-642-56884-8, ISBN 978-3-540-67827-4.

• Affine Weyl group
• Finite Coxeter group
• Hasse diagram
• Linear algebraic group
• Nilpotent orbit
• Root system
• sl2-triple
• Weyl group