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Infinite dimensional Lie group

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In mathematics, F_{4} is the name of a Lie group and also its Lie algebra f_{4}. It is one of the five exceptional simple Lie groups. F_{4} has rank 4 and dimension 52. The compact form is simply connected and its outer automorphism group is the trivial group. Its fundamental representation is 26dimensional.
The compact real form of F_{4} is the isometry group of a 16dimensional Riemannian manifold known as the octonionic projective plane OP^{2}. This can be seen systematically using a construction known as the magic square, due to Hans Freudenthal and Jacques Tits.
There are 3 real forms: a compact one, a split one, and a third one. They are they isometry groups of the three real Albert algebras.
The F_{4} Lie algebra may be constructed by adding 16 generators transforming as a spinor to the 36dimensional Lie algebra so(9), in analogy with the construction of E_{8}.
In older books and papers, F_{4} is sometimes denoted by E_{4}.
Contents
Algebra
Dynkin diagram
The Dynkin diagram for F_{4} is .
Weyl/Coxeter group
Its Weyl/Coxeter group is the symmetry group of the 24cell: it is a solvable group of order 1152.
Cartan matrix
 \left[ \begin{array}{rrrr} 2&1&0&0\\ 1&2&2&0\\ 0&1&2&1\\ 0&0&1&2 \end{array} \right]
F_{4} lattice
The F_{4} lattice is a fourdimensional bodycentered cubic lattice (i.e. the union of two hypercubic lattices, each lying in the center of the other). They form a ring called the Hurwitz quaternion ring. The 24 Hurwitz quaternions of norm 1 form the vertices of a 24cell centered at the origin.
Roots of F_{4}
The 48 root vectors of F_{4} can be found as the vertices of the 24cell in two dual configurations:
 24 roots by (±1,±1,0,0), permuting coordinate positions
 8 roots by (±1, 0, 0, 0), permuting coordinate positions
 16 roots by (±½, ±½, ±½, ±½).
Simple roots
One choice of simple roots for F_{4}, , is given by the rows of the following matrix:
 \begin{bmatrix} 0&1&1&0 \\ 0&0&1&1 \\ 0&0&0&1 \\ \frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\\ \end{bmatrix}
F_{4} polynomial invariant
Just as O(n) is the group of automorphisms which keep the quadratic polynomials x^{2} + y^{2} + ... invariant, F_{4} is the group of automorphisms of the following set of 3 polynomials in 27 variables. (The first can easily be substituted into other two making 26 variables).
 C_1 = x+y+z
 C_2 = x^2+y^2+z^2+2X\overline{X}+2Y\overline{Y}+2Z\overline{Z}
 C_3 = xyz  xX\overline{X}  yY\overline{Y}  zZ\overline{Z} + XYZ + \overline{XYZ}
Where x, y, z are real valued and X, Y, Z are octonion valued. Another way of writing these invariants is as (combinations of) Tr(M), Tr(M^{2}) and Tr(M^{3}) of the hermitian octonion matrix:
 M = \begin{bmatrix} x & \overline{Z} & Y \\ Z & y & \overline{X} \\ \overline{Y} & X & z \end{bmatrix}
Representations
The characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all given by the Weyl character formula. The dimensions of the smallest irreducible representations are (sequence A121738 in OEIS):
 1, 26, 52, 273, 324, 1053 (twice), 1274, 2652, 4096, 8424, 10829, 12376, 16302, 17901, 19278, 19448, 29172, 34749, 76076, 81081, 100776, 106496, 107406, 119119, 160056 (twice), 184756, 205751, 212992, 226746, 340119, 342056, 379848, 412776, 420147, 627912…
The 52dimensional representation is the adjoint representation, and the 26dimensional one is the tracefree part of the action of F_{4} on the exceptional Albert algebra of dimension 27.
There are two nonisomorphic irreducible representations of dimensions 1053, 160056, 4313088, etc. The fundamental representations are those with dimensions 52, 1274, 273, 26 (corresponding to the four nodes in the Dynkin diagram in the order such that the double arrow points from the second to the third).
See also
References
 Adams, J. Frank (1996), Lectures on exceptional Lie groups, Chicago Lectures in Mathematics,
 John Baez, The Octonions, Section 4.2: F_{4}, (2002), 14520539Bull. Amer. Math. Soc. . Online HTML version at http://math.ucr.edu/home/baez/octonions/node15.html.
 Chevalley C, Schafer RD (February 1950). "The Exceptional Simple Lie Algebras F(4) and E(6)". Proc. Natl. Acad. Sci. U.S.A. 36 (2): 137–41.
