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| {{DISPLAYTITLE:F<sub>4</sub> (mathematics)}}
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| {{Group theory sidebar |Topological}}
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| {{Lie groups |Simple}}
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| In [[mathematics]], '''F<sub>4</sub>''' is the name of a [[Lie group]] and also its [[Lie algebra]] '''f'''<sub>4</sub>. It is one of the five exceptional [[simple Lie group]]s. F<sub>4</sub> 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 26-dimensional.
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| The compact real form of F<sub>4</sub> is the [[isometry group]] of a 16-dimensional [[Riemannian manifold]] known as the [[octonionic projective plane]] '''OP'''<sup>2</sup>. This can be seen systematically using a construction known as the [[Freudenthal magic square|''magic square'']], due to [[Hans Freudenthal]] and [[Jacques Tits]].
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| There are [[list of simple Lie groups|3 real forms]]: a compact one, a split one, and a third one.
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| The F<sub>4</sub> Lie algebra may be constructed by adding 16 generators transforming as a [[spinor]] to the 36-dimensional Lie algebra '''so'''(9), in analogy with the construction of [[E8 (mathematics)|E<sub>8</sub>]].
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| In older books and papers, F<sub>4</sub> is sometimes denoted by E<sub>4</sub>.
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| == Algebra ==
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| ===Dynkin diagram===
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| The [[Dynkin diagram]] for F<sub>4</sub> is [[Image:Dynkin diagram F4.png|Dynkin diagram of F_4]].
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| === Weyl/Coxeter group ===
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| Its [[Weyl group|Weyl]]/[[Coxeter group|Coxeter]] group is the [[symmetry group]] of the [[24-cell]]: it is a [[solvable group]] of order 1152.
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| === Cartan matrix ===
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| :<math>\begin{bmatrix}
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| 2&-1&0&0\\
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| -1&2&-2&0\\
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| 0&-1&2&-1\\
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| 0&0&-1&2
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| \end{bmatrix}</math>
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| === F<sub>4</sub> lattice === | |
| The F<sub>4</sub> [[lattice (group)|lattice]] is a four dimensional [[body-centered cubic]] lattice (i.e. the union of two [[hypercubic lattice]]s, each lying in the center of the other). They form a [[ring (mathematics)|ring]] called the [[Hurwitz quaternion]] ring. The 24 Hurwitz quaternions of norm 1 form the vertices of a [[24-cell]] centered at the origin.
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| === Roots of F<sub>4</sub> ===
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| [[File:F4 roots by 24-cell duals.svg|left|thumb|The 24 vertices of [[24-cell]] (red) and 24 vertices of its dual (yellow) represent the 48 root vectors of F<sub>4</sub> in this [[Coxeter plane]] projection]]
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| The 48 root vectors of F<sub>4</sub> can be found as the vertices of the [[24-cell]] in two dual configurations:
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| '''24-cell vertices:''' {{CDD|node_1|3|node|4|node|3|node}}
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| * 24 roots by (±1,±1,0,0), permutating coordinate positions
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| '''Dual 24-cell vertices:''' {{CDD|node|3|node|4|node|3|node_1}}
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| * 8 roots by (±1, 0, 0, 0), permutating coordinate positions
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| * 16 roots by (±½, ±½, ±½, ±½).
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| ====Simple roots====
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| One choice of [[Simple root (root system)|simple root]]s for F<sub>4</sub>, {{Dynkin2|node_n1|3|node_n2|4b|node_n3|3|node_n4}}, is given by the rows of the following matrix:
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| :<math>\begin{bmatrix} | |
| 0&1&-1&0 \\
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| 0&0&1&-1 \\
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| 0&0&0&1 \\
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| \frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}\\
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| \end{bmatrix}</math>
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| [[File:F4HassePoset.svg|thumb|300px|[[Hasse diagram]] of F4 [[Root_system#The_root_poset|root poset]] with edge labels identifying added simple root position]] | |
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| === F<sub>4</sub> polynomial invariant ===
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| Just as O(''n'') is the group of automorphisms which keep the quadratic polynomials ''x''<sup>2</sup> + ''y''<sup>2</sup> + ... invariant, F<sub>4</sub> 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).
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| :<math>C_1 = x+y+z</math>
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| :<math>C_2 = x^2+y^2+z^2+2X\overline{X}+2Y\overline{Y}+2Z\overline{Z} </math>
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| :<math>C_3 = xyz - xX\overline{X} - yY\overline{Y} - zZ\overline{Z} + XYZ + \overline{XYZ} </math>
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| 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''<sup>2</sup>) and Tr(''M''<sup>3</sup>) of the [[hermitian matrix|hermitian]] [[octonion]] [[matrix (mathematics)|matrix]]:
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| :<math> M = \begin{bmatrix}
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| x & \overline{Z} & Y \\
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| Z & y & \overline{X} \\
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| \overline{Y} & X & z
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| \end{bmatrix} </math>
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| F<sub>4</sub> is the only exceptional lie group which gives the automorphisms of a set of real commutative polynomials. (The other exceptional lie groups require anti-commutative polynomial invariants).
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| ==Representations==
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| 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 {{OEIS|id=A121738}}:
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| :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…
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| The 52-dimensional representation is the [[Adjoint representation of a Lie algebra|adjoint representation]], and the 26-dimensional one is the trace-free part of the action of F<sub>4</sub> on the exceptional [[Albert algebra]] of dimension 27.
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| There are two non-isomorphic irreducible representations of dimensions 1053, 160056, 4313088, etc. The [[fundamental representation]]s are those with dimensions 52, 1274, 273, 26 (corresponding to the four nodes in the [[#Dynkin diagram|Dynkin diagram]] in the order such that the double arrow points from the second to the third).
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| ==See also==
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| * [[Cayley plane]]
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| * [[Dynkin diagram]]
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| * [[Exceptional Jordan algebra]]
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| * [[Fundamental representation]]
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| * [[Simple Lie group]]
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| ==References==
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| * [[John Baez]], ''The Octonions'', Section 4.2: F<sub>4</sub>, [http://www.ams.org/bull/2002-39-02/S0273-0979-01-00934-X/home.html Bull. Amer. Math. Soc. '''39''' (2002), 145-205]. Online HTML version at
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| http://math.ucr.edu/home/baez/octonions/node15.html.
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| *{{Citation | last1=Adams | first1=J. Frank | title=Lectures on exceptional Lie groups | url=http://books.google.com/books?isbn=0226005275 | publisher=[[University of Chicago Press]] | series=Chicago Lectures in Mathematics | isbn=978-0-226-00526-3 | mr=1428422 | year=1996}}
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| * {{Cite book
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| | edition = 1
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| | publisher = CRC Press
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| | isbn = 0-8247-1326-5
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| | last = Jacobson
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| | first = Nathan
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| | authorlink = Nathan Jacobson
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| | title = Exceptional Lie Algebras
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| | date = 1971-06-01
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| }}
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| {{Exceptional_Lie_groups}}
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| [[Category:Algebraic groups]]
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| [[Category:Lie groups]]
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