E8 (mathematics)
| Group theory |
 |
Group theory
| [show]Finite groups and classification of finite simple groups |
Cyclic group Zn
Symmetric group, Sn
Dihedral group, Dn
Alternating group An
Mathieu groups M11, M12, M22, M23, M24
Conway groups Co1, Co2, Co3
Janko groups J1, J2, J3, J4
Fischer groups F22, F23, F24
Baby Monster group B
Monster group M |
|
| |
In
mathematics,
E8 is any of several closely related
exceptional simple Lie groups, linear
algebraic groups and Lie algebras of
dimension 248; the same notation is used for the corresponding
root lattice, which has
rank 8. The designation E
8 comes from the Cartan–Killing classification of the complex
simple Lie algebras, which fall into four infinite series labeled A
n, B
n, C
n, D
n, and
five exceptional cases labeled
E6,
E7, E
8,
F4, and
G2. The E
8 algebra is the largest and most complicated of these exceptional cases.
Wilhelm Killing (
1888,
1888,
1889,
1890) discovered the complex Lie algebra E
8 during his classification of simple compact Lie algebras, though he did not prove its existence, which was first shown by
Élie Cartan. Cartan determined that a complex simple Lie algebra of type E
8 admits three real forms. Each of them gives rise to a
simply-connected simple
Lie group of dimension 248, exactly one of which is
compact.
Algebraic groups and Lie algebras of type E
8 over other
fields have also been considered: for example, in the case of
finite fields they lead to an infinite family of
finite simple groups of Lie type.
[edit] Basic description
The compact Lie group E
8 has dimension 248. Its
rank, which is the dimension of its maximal torus, is 8. Therefore the vectors of the root system are in eight-dimensional Euclidean space: they are described explicitly later in this article. The
Weyl group of E
8, which is the
group of symmetries of the maximal torus which are induced by
conjugations in the whole group, has order 696729600.
The compact group E
8 is unique among simple compact Lie groups in that its non-
trivial representation of smallest dimension is the
adjoint representation (of dimension 248) acting on the Lie algebra E
8 itself; it is also the unique one which has the following four properties: trivial center, compact, simply connected, and simply laced (all roots have the same length).
There is a Lie algebra
En for every integer
n ≥ 3, which is infinite dimensional if
n is greater than 8.
[edit] Real and complex forms
There is a unique complex Lie algebra of type E
8, corresponding to a complex group of complex dimension 248. The complex Lie group E
8 of
complex dimension 248 can be considered as a simple real Lie group of real dimension 496. This is simply connected, has maximal
compact subgroup the compact form (see below) of E
8, and has an outer automorphism group of order 2 generated by complex conjugation.
As well as the complex Lie group of type E
8, there are three real forms of the Lie algebra, three real forms of the group with trivial center (two of which have non-algebraic double covers, giving two further real forms), all of real dimension 248, as follows:
- A compact form (which is usually the one meant if no other information is given), which is simply connected and has trivial outer automorphism group.
- A split form, which has maximal compact subgroup Spin(16)/(Z/2Z), fundamental group of order 2, and a non-algebraic double cover and has trivial outer automorphism group.
- A third form, which has maximal compact subgroup E7×SU(2)/(−1×−1), fundamental group of order 2, and a non-algebraic double cover and has trivial outer automorphism group.
For a complete list of real forms of simple Lie algebras, see the
list of simple Lie groups.
There is also at least one algebraic group and Lie algebra of type E
8 over any field (or even commutative ring), called the split form. Over algebraically closed fields this is unique, but over other fields there are often many other forms of E
8.
[edit] Representation theory
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
A121732 in
OEIS):
- 1, 248, 3875, 27000, 30380, 147250, 779247, 1763125, 2450240, 4096000, 4881384, 6696000, 26411008, 70680000, 76271625, 79143000, 146325270, 203205000, 281545875, 301694976, 344452500, 820260000, 1094951000, 2172667860, 2275896000, 2642777280, 2903770000, 3929713760, 4076399250, 4825673125, 6899079264, 8634368000, 8634368000, 12692520960…
The 248-dimensional representation is the
adjoint representation. There are two non-isomorphic irreductible representations of dimension 8634368000. The
fundamental representations are those with dimensions 3875, 6696000, 6899079264, 146325270, 2450240, 30380, 248 and 147250 (corresponding to the eight nodes in the
Dynkin diagram in the order chosen for the
Cartan matrix below, i.e., the nodes are read in the seven-node chain first, with the last node being connected to the third).
The coefficients of the character formulas for infinite dimensional irreducible
representations of E
8 depend on some large square matrices consisting of polynomials, the
Lusztig–Vogan polynomials, an analogue of
Kazhdan–Lusztig polynomials introduced for
reductive groups in general by
George Lusztig and
David Kazhdan (1983). The values at 1 of the Lusztig–Vogan polynomials give the coefficients of the matrices relating the standard representations (whose characters are easy to describe) with the irreducible representations.
These matrices were computed after four years of collaboration by a
group of 18 mathematicians and computer scientists, led by
Jeffrey Adams, with much of the programming done by
Fokko du Cloux. The most difficult case (for exceptional groups) is the split
real form of E
8 (see above), where the largest matrix is of size 453060×453060. The Lusztig–Vogan polynomials for all other exceptional simple groups have been known for some time; the calculation for the split form of
E8 is far longer than any other case. The announcement of the result in March 2007 received extraordinary attention from the media (see the external links), to the surprise of the mathematicians working on it.
The representations of the E
8 groups over finite fields are given by
Deligne–Lusztig theory.
[edit] Constructions
One can construct the (compact form of the) E
8 group as the
automorphism group of the corresponding
e8 Lie algebra. This algebra has a 120-dimensional subalgebra
so(16) generated by
Jij as well as 128 new generators
Qa that transform as a
Weyl–Majorana spinor of
spin(16). These statements determine the commutators
![[J_{ij},J_{k\ell}]=\delta_{jk}J_{i\ell}-\delta_{j\ell}J_{ik}-\delta_{ik}J_{j\ell}+\delta_{i\ell}J_{jk}](http://upload.wikimedia.org/math/1/2/2/1228e886cdc52aebf0972174498179f7.png)
as well as
![[J_{ij},Q_a] = \frac 14
(\gamma_i\gamma_j-\gamma_j\gamma_i)_{ab} Q_b,](http://upload.wikimedia.org/math/8/7/1/8715b3afb5587c891c9a869679c42978.png)
while the remaining commutator (not anticommutator!) is defined as
![[Q_a,Q_b]=\gamma^{[i}_{ac}\gamma^{j]}_{cb}
J_{ij}.](http://upload.wikimedia.org/math/5/c/7/5c7d805ebe3f0a51722c0054393cea37.png)
It is then possible to check that the
Jacobi identity is satisfied.
[edit] Geometry
The compact real form of E
8 is the
isometry group of a 128-dimensional
Riemannian manifold known informally as the 'octo-octonionic projective plane' because it can be built using an algebra that is the tensor product of the
octonions with themselves. This can be seen systematically using a construction known as the
magic square, due to
Hans Freudenthal and
Jacques Tits (
Landsberg & Manivel 2001).
[edit] E8 root system
Zome model of the E
8 root system.
A
root system of rank
r is a particular finite configuration of vectors, called
roots, which span an
r-dimensional
Euclidean space and satisfy certain geometrical properties. In particular, the root system must be invariant under
reflection through the hyperplane perpendicular to any root.
The
E8 root system is a rank 8 root system containing 240 root vectors spanning
R8. It is
irreducible in the sense that it cannot be built from root systems of smaller rank. All the root vectors in E
8 have the same length. It is convenient for many purposes to normalize them to have length √2.
[edit] Construction
Graph of E8 projected into the Coxeter plane
In the so-called
even coordinate system E
8 is given as the set of all vectors in
R8 with length squared equal to 2 such that coordinates are either all
integers or all
half-integers and the sum of the coordinates is even.
Explicitly, there are 112 roots with integer entries obtained from
by taking an arbitrary combination of signs and an arbitrary
permutation of coordinates, and 128 roots with half-integer entries obtained from
by taking an even number of minus signs (or, equivalently, requiring that the sum of all the eight coordinates be even). There are 240 roots in all.
The 112 roots with integer entries form a D
8 root system. The E
8 root system also contains a copy of A
8 (which has 72 roots) as well as
E6 and
E7 (in fact, the latter two are usually
defined as subsets of E
8).
In the
odd coordinate system E
8 is given by taking the roots in the even coordinate system and changing the sign of any one coordinate. The roots with integer entries are the same while those with half-integer entries have an odd number of minus signs rather than an even number.
[edit] Simple roots
A set of
simple roots for a root system Φ is a set of roots that form a
basis for the Euclidean space spanned by Φ with the special property that each root has components with respect to this basis that are either all nonnegative or all nonpositive.
One choice of simple roots for E
8 is given by the rows of the following matrix:
![\left [\begin{smallmatrix}
\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&\frac{1}{2}\\
-1&1&0&0&0&0&0&0
\\
0&-1&1&0&0&0&0&0 \\
0&0&-1&1&0&0&0&0 \\
0&0&0&-1&1&0&0&0 \\
0&0&0&0&-1&1&0&0 \\
0&0&0&0&0&-1&1&0 \\
1&1&0&0&0&0&0&0 \\
\end{smallmatrix}\right ].](http://upload.wikimedia.org/math/7/5/b/75bce2aa3f595732bd54baa61e503070.png)
The set of simple roots is by no means unique (the number of possible choices of positive roots is the order of the Weyl group); however, the particular choice displayed above has the unique property that the positive roots are then exactly those whose rightmost nonzero coordinate is positive.
[edit] Dynkin diagram
The
Dynkin diagram for E
8 is given by
This diagram gives a concise visual summary of the root structure. Each node of this diagram represents a simple root. A line joining two simple roots indicates that they are at an angle of 120° to each other. Two simple roots which are not joined by a line are
orthogonal.
[edit] Weyl group
The
Weyl group of E
8 is of order 696729600, and can be described as O
8+(2): it is of the form 2.
G.2 (that is, a
stem extension by the cyclic group of order 2 of an extension of the cyclic group of order 2 by a group
G) where
G is the unique
simple group of order 174182400 (which can be described as PSΩ
8+(2)).
[1]
[edit] Cartan matrix
The
Cartan matrix of a rank
r root system is an
r ×
r matrix whose entries are derived from the simple roots. Specifically, the entries of the Cartan matrix are given by
where (−,−) is the Euclidean
inner product and
αi are the simple roots. The entries are independent of the choice of simple roots (up to ordering).
The Cartan matrix for E
8 is given by
![\left [
\begin{smallmatrix}
2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\
-1 & 2 & -1& 0 & 0 & 0 & 0 & 0 \\
0 & -1 & 2 & -1 & 0 & 0 & 0 & -1 \\
0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\
0 & 0 & -1 & 0 & 0 & 0 & 0 & 2
\end{smallmatrix}\right ].](http://upload.wikimedia.org/math/f/f/7/ff7a158d069c956aff0b61218652263d.png)
The
determinant of this matrix is equal to 1.
[edit] E8 root lattice
The integral span of the E
8 root system forms a
lattice in
R8 naturally called the
E8 root lattice. This lattice is rather remarkable in that it is the only (nontrivial) even,
unimodular lattice with rank less than 16.
[edit] Simple subalgebras of E8
An incomplete simple subgroup tree of E
8 The Lie algebra E8 contains as subalgebras all the
exceptional Lie algebras as well as many other important Lie algebras in mathematics and physics. The height of the Lie algebra on the diagram approximately corresponds to the rank of the algebra. A line from an algebra down to a lower algebra indicates that the lower algebra is a subalgebra of the higher algebra. Some algebras are more obvious such as SU(n) is a subalgebra of O(2n) and some are less obvious especially the exceptional algebras G2, F4, E6 & E7. The
orthogonal and
unitary[disambiguation needed] subalgebras are particularly important in physics as they are used to represent
space-time and
bosonic symmetries respectively. Some of the smaller algebras are equivalent e.g. O(3)~SU(2).
[edit] Chevalley groups of type E8
As an algebraic group, E
8 can be defined over the integers (by means of a
Chevalley basis for the Lie algebra) hence, in particular, over any commutative ring. Its points over a
finite field with
q elements form a finite
Chevalley group, generally written E
8(
q), which is simple for any
q.
[2][3] Its number of elements is given by the formula (sequence
A008868 in
OEIS):
- q120(q30 − 1)(q24 − 1)(q20 − 1)(q18 − 1)(q14 − 1)(q12 − 1)(q8 − 1)(q2 − 1)
The first term in this sequence, the order of E
8(2), namely 337804753143634806261388190614085595079991692242467651576160959909068800000 ≈ 3.38×10
74, is already larger than the size of the
Monster group; Coxeter and Moser
[4] remark that it is comparable (actually about 5×10
5 times smaller) to Eddington's historical
estimation of the number of protons in the Universe. This group E
8(2) is the last one described (but without its character table) in the
ATLAS of Finite Groups[5].
[edit] Subgroups
The smaller exceptional groups
E7 and
E6 sit inside E
8. In the compact group, both (E
7×SU(2)) / (
Z/2
Z) and (E
6×SU(3)) / (
Z/3
Z) are
maximal subgroups of E
8.
The 248-dimensional adjoint representation of E
8 may be considered in terms of its
restricted representation to the first of these subgroups. It transforms under SU(2)×E
7 as a sum of
tensor product representations, which may be labelled as a pair of dimensions as
(Since there is a quotient in the product, these notations may strictly be taken as indicating the infinitesimal (Lie algebra) representations.) Since the adjoint representation can be described by the roots together with the generators in the
Cartan subalgebra, we may see that decomposition by looking at these. In this description:
- The (3,1) consists of the roots (0,0,0,0,0,0,1,−1), (0,0,0,0,0,0,−1,1) and the Cartan generator corresponding to the last dimension.
- The (1,133) consists of all roots with (1,1), (−1,−1), (0,0), (−1/2,−1/2) or (1/2,1/2) in the last two dimensions, together with the Cartan generators corresponding to the first 7 dimensions.
- The (2,56) consists of all roots with permutations of (1,0), (−1,0) or (1/2,−1/2) in the last two dimensions.
The 248-dimensional adjoint representation of E
8, when similarly restricted, transforms under SU(3)×E
6 as:
We may again see the decomposition by looking at the roots together with the generators in the Cartan subalgebra. In this description:
- The (8,1) consists of the roots with permutations of (1,−1,0) in the last three dimensions, together with the Cartan generator corresponding to the last two dimensions.
- The (1,78) consists of all roots with (0,0,0), (−1/2,−1/2,−1/2) or (1/2,1/2,1/2) in the last three dimensions, together with the Cartan generators corresponding to the first 6 dimensions.
- The (3,27) consists of all roots with permutations of (1,0,0), (1,1,0) or (−1/2,1/2,1/2) in the last three dimensions.
- The (3,27) consists of all roots with permutations of (−1,0,0), (−1,−1,0) or (1/2,−1/2,−1/2) in the last three dimensions.
The finite quasisimple groups that can embed in (the compact form of) E
8 were found by
Griess & Ryba (1999)
[edit] Invariant polynomial
E8 is the automorphism group of an octic polynomial invariant, thought to be the lowest order symmetric polynomial invariant of E8.
[6]
[edit] Applications
The E
8 Lie group has applications in
theoretical physics, in particular in
string theory and
supergravity. The group E
8×E
8 (the
Cartesian product of two copies of E
8) serves as the
gauge group of one of the two types of
heterotic string and is one of two
anomaly-free gauge groups that can be coupled to the
N = 1 supergravity in 10 dimensions. E
8 is the
U-duality group of supergravity on an eight-torus (in its split form).
One way to incorporate the
standard model of particle physics into heterotic string theory is the
symmetry breaking of E
8 to its maximal subalgebra SU(3)×E
6.
In 1982,
Michael Freedman used the
E8 lattice to construct an example of a
topological 4-manifold, the
E8 manifold, which has no
smooth structure.
R. Coldea, D. A. Tennant, and E. M. Wheeler et al. (
2010) reported that in an experiment with a
cobalt-
niobium crystal, under certain physical conditions the
electron spins in it exhibited two of the 8 peaks related to E
8 predicted by
Zamolodchikov (1989) .
Phytagoras lahir pada tahun 570 SM, di pulau Samos, di daerah Ionia. Pythagoras (582 SM – 496 SM, bahasa Yunani: Πυθαγόρας) adalah seorang matematikawan dan filsuf Yunani yang paling dikenal melalui teoremanya.Dikenal sebagai "Bapak Bilangan", dia memberikan sumbangan yang penting terhadap filsafat dan ajaran keagamaan pada akhir abad ke-6 SM. Kehidupan dan ajarannya tidak begitu jelas akibat banyaknya legenda dan kisah-kisah buatan mengenai dirinya.
