E8 (MATHEMATICS)
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(Redirected from E₈ (mathematics))
In mathematics, 'E8' is the name given to a family of closely related structures. In particular, it is the name of some exceptional simple Lie algebras as well as that of the associated simple Lie groups. It is also the name given to the corresponding root system, root lattice, and Weyl/Coxeter group, and to some finite simple Chevalley groups. It was discovered by Wilhelm Killing (1888-1890).
The designation E8 comes from Wilhelm Killing and Élie Cartan's classification of the complex simple Lie algebras, which fall into four infinite families labeled A''n'', B''n'', C''n'', D''n'', and five exceptional cases labeled E6, E7, E8, F4, and G2. The E8 algebra is the largest and most complicated of these exceptional cases, and is often the last case of various theorems to be proved.
E8 has rank 8 (the maximum number of mutually commutative degrees of freedom), and dimension 248 (as a manifold). The vectors of the root system are in eight dimensions, and are specified later in this article. The Weyl group of E8, which acts as a symmetry group of the maximal torus by means of the conjugation operation from the whole group, is of order 696729600.
E8 is unique among simple Lie groups in that its non-trivial representation of smallest dimension is the adjoint representation (of dimension 248) acting on the Lie algebra E8 itself.
There is a Lie algebra En for every integer ''n''≥3, which is infinite dimensional if ''n'' is greater than 8.
The complex Lie group E8 of complex dimension 248 can be considered as a simple real Lie group of (real) dimension 496, which is simply connected, has maximal compact subgroup the compact form of E8, and has an outer automorphism group of order 2 generated by complex conjugation.
As well as the complex Lie group of type E8, there are three real forms of the group, 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'/2'Z'), 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.
The coefficients of the character formulas for infinite dimensional irreducible representations of E8 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 Vogan (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 E8 (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 ''E''8 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.
One can construct the (compact form of the) E8 group as the automorphism group of the corresponding 'e'8 Lie algebra. This algebra has a 120-dimensional subalgebra 'so'(16) generated by ''J''''ij'' as well as 128 new generators ''Q''''a'' that transform as a Weyl-Majorana spinor of 'spin'(16). These statements determine the commutators
:
as well as
:
while the remaining commutator (not anticommutator!) is defined as
:
It is then possible to check that the Jacobi identity is satisfied.
The compact real form of E8 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 (see J.M. Landsberg, L. Manivel, (2001)).
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 'R'8. It is irreducible in the sense that it cannot be built from root systems of smaller rank. Each of the root vectors in E8 have equal length. It is convenient for many purposes to normalize them to have length √2.
In the so-called ''even coordinate system'' E8 is given as the set of all vectors in 'R'8 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 D8 root system. The E8 root system also contains a copy of A8 (which has 72 roots) as well as E6 and E7 (in fact, the latter two are usually ''defined'' as subsets of E8).
In the ''odd coordinate system'' E8 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.
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 E8 (by no means unique) is given by the rows of the following matrix:
:
The Dynkin diagram for E8 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.
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 ''α''''i'' are the simple roots. The entries are independent of the choice of simple roots (up to ordering).
The Cartan matrix for E8 is given by
:
The determinant of this matrix is equal to 1.
Main articles: E8 lattice, l1=E8 lattice
The integral span of the E8 root system forms a lattice in 'R'8 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.
The smaller exceptional groups E7 and E6 sit inside E8. In the compact group, both (E7×SU(2)) / ('Z'/2'Z') and (E6×SU(3)) / ('Z'/3'Z') are maximal subgroups of
E8.
The 248-dimensional adjoint representation of E8 may be considered in terms of its restricted representation to the first of these subgroups. It transforms under SU(2)×E7 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 E8, when similarly restricted, transforms under SU(3)×E6 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 E8 Lie group has applications in theoretical physics, in particular in string theory and supergravity. The group E8×E8 (the Cartesian product of two copies of E8) 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.
E8 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 in the heterotic string includes the symmetry breaking of E8 to its maximal subalgebra
SU(3)×E6.
Michael Freedman showed in 1982 that the E8 manifold was one of the strangest examples of a 4-manifold, though this only uses the lattice and not the Lie algebra or Lie group.
★ J. F. Adams, ''Lectures on Exceptional Lie Groups'' (Chicago Lectures in Mathematics) ISBN 0226005275
★ Killing, ''Die Zusammensetzung der stetigen/endlichen Transformationsgruppen'' Mathematische Annalen, Volume 31, Number 2 June, 1888, Pages 252-290 , Volume 33, Number 1 March, 1888, Pages 1-48 , Volume 34, Number 1 March, 1889, Pages 57-122 , Volume 36, Number 2 June, 1890,Pages 161-189
★ J.M. Landsberg, L. Manivel, (2001)), ''The projective geometry of Freudenthal's magic square'', Journal of Algebra (2001)
★
Links related to the calculation of the Lusztig-Vogan polynomials in 2007 with mathematical content:
★ Home page of Jeffrey Adams
★ Kazhdan-Lusztig-Vogan Polynomials for E8
★ D. Vogan, Narrative of the Project to compute Kazhdan-Lusztig Polynomials for E8
★ D. Vogan, ''The Character Table for E8, or How We Wrote Down a 453,060 x 453,060 Matrix and Found Happiness'' Slides for a popular talk on E8.
★
★ The n-Category Café — University of Texas blog posting by John Baez on E8
Links related to the calculation of the Lusztig-Vogan polynomials in 2007 without mathematical content:
★ Math research team maps E8 — from MIT
★ BBC News story "248-dimension maths puzzle solved"
★ Journey to the 57th Dimension — from National Science Foundation
★ New Scientist
★ NPR broadcast "Team Solves Mammoth, Century-Old Math Problem"
Other external links:
★ ; also available here
★ Graphic representation of E8 root system
In mathematics, 'E8' is the name given to a family of closely related structures. In particular, it is the name of some exceptional simple Lie algebras as well as that of the associated simple Lie groups. It is also the name given to the corresponding root system, root lattice, and Weyl/Coxeter group, and to some finite simple Chevalley groups. It was discovered by Wilhelm Killing (1888-1890).
The designation E8 comes from Wilhelm Killing and Élie Cartan's classification of the complex simple Lie algebras, which fall into four infinite families labeled A''n'', B''n'', C''n'', D''n'', and five exceptional cases labeled E6, E7, E8, F4, and G2. The E8 algebra is the largest and most complicated of these exceptional cases, and is often the last case of various theorems to be proved.
Basic description
E8 has rank 8 (the maximum number of mutually commutative degrees of freedom), and dimension 248 (as a manifold). The vectors of the root system are in eight dimensions, and are specified later in this article. The Weyl group of E8, which acts as a symmetry group of the maximal torus by means of the conjugation operation from the whole group, is of order 696729600.
E8 is unique among simple Lie groups in that its non-trivial representation of smallest dimension is the adjoint representation (of dimension 248) acting on the Lie algebra E8 itself.
There is a Lie algebra En for every integer ''n''≥3, which is infinite dimensional if ''n'' is greater than 8.
Real forms
The complex Lie group E8 of complex dimension 248 can be considered as a simple real Lie group of (real) dimension 496, which is simply connected, has maximal compact subgroup the compact form of E8, and has an outer automorphism group of order 2 generated by complex conjugation.
As well as the complex Lie group of type E8, there are three real forms of the group, 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'/2'Z'), 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.
Representation theory
The coefficients of the character formulas for infinite dimensional irreducible representations of E8 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 Vogan (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 E8 (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 ''E''8 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.
Constructions
One can construct the (compact form of the) E8 group as the automorphism group of the corresponding 'e'8 Lie algebra. This algebra has a 120-dimensional subalgebra 'so'(16) generated by ''J''''ij'' as well as 128 new generators ''Q''''a'' that transform as a Weyl-Majorana spinor of 'spin'(16). These statements determine the commutators
:
as well as
:
while the remaining commutator (not anticommutator!) is defined as
:
It is then possible to check that the Jacobi identity is satisfied.
Geometry
The compact real form of E8 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 (see J.M. Landsberg, L. Manivel, (2001)).
E8 root system
Zome Model of the E8 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 'R'8. It is irreducible in the sense that it cannot be built from root systems of smaller rank. Each of the root vectors in E8 have equal length. It is convenient for many purposes to normalize them to have length √2.
Construction
In the so-called ''even coordinate system'' E8 is given as the set of all vectors in 'R'8 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 D8 root system. The E8 root system also contains a copy of A8 (which has 72 roots) as well as E6 and E7 (in fact, the latter two are usually ''defined'' as subsets of E8).
In the ''odd coordinate system'' E8 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.
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 E8 (by no means unique) is given by the rows of the following matrix:
:
Dynkin diagram
The Dynkin diagram for E8 is given by
:
Dynkin diagram of E8
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.
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 ''α''''i'' are the simple roots. The entries are independent of the choice of simple roots (up to ordering).
The Cartan matrix for E8 is given by
:
The determinant of this matrix is equal to 1.
E8 root lattice
Main articles: E8 lattice, l1=E8 lattice
The integral span of the E8 root system forms a lattice in 'R'8 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.
Significant maximal subgroups
The smaller exceptional groups E7 and E6 sit inside E8. In the compact group, both (E7×SU(2)) / ('Z'/2'Z') and (E6×SU(3)) / ('Z'/3'Z') are maximal subgroups of
E8.
The 248-dimensional adjoint representation of E8 may be considered in terms of its restricted representation to the first of these subgroups. It transforms under SU(2)×E7 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 E8, when similarly restricted, transforms under SU(3)×E6 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.
Applications
The E8 Lie group has applications in theoretical physics, in particular in string theory and supergravity. The group E8×E8 (the Cartesian product of two copies of E8) 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.
E8 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 in the heterotic string includes the symmetry breaking of E8 to its maximal subalgebra
SU(3)×E6.
Michael Freedman showed in 1982 that the E8 manifold was one of the strangest examples of a 4-manifold, though this only uses the lattice and not the Lie algebra or Lie group.
References
★ J. F. Adams, ''Lectures on Exceptional Lie Groups'' (Chicago Lectures in Mathematics) ISBN 0226005275
★ Killing, ''Die Zusammensetzung der stetigen/endlichen Transformationsgruppen'' Mathematische Annalen, Volume 31, Number 2 June, 1888, Pages 252-290 , Volume 33, Number 1 March, 1888, Pages 1-48 , Volume 34, Number 1 March, 1889, Pages 57-122 , Volume 36, Number 2 June, 1890,Pages 161-189
★ J.M. Landsberg, L. Manivel, (2001)), ''The projective geometry of Freudenthal's magic square'', Journal of Algebra (2001)
★
External links
Links related to the calculation of the Lusztig-Vogan polynomials in 2007 with mathematical content:
★ Home page of Jeffrey Adams
★ Kazhdan-Lusztig-Vogan Polynomials for E8
★ D. Vogan, Narrative of the Project to compute Kazhdan-Lusztig Polynomials for E8
★ D. Vogan, ''The Character Table for E8, or How We Wrote Down a 453,060 x 453,060 Matrix and Found Happiness'' Slides for a popular talk on E8.
★
★ The n-Category Café — University of Texas blog posting by John Baez on E8
Links related to the calculation of the Lusztig-Vogan polynomials in 2007 without mathematical content:
★ Math research team maps E8 — from MIT
★ BBC News story "248-dimension maths puzzle solved"
★ Journey to the 57th Dimension — from National Science Foundation
★ New Scientist
★ NPR broadcast "Team Solves Mammoth, Century-Old Math Problem"
Other external links:
★ ; also available here
★ Graphic representation of E8 root system
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