In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Freudenthal magic square (or Freudenthal–Tits magic square) is a construction relating several
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
s (and their associated
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s). It is named after
Hans Freudenthal
Hans Freudenthal (17 September 1905 – 13 October 1990) was a Jewish-German-born Dutch mathematician. He made substantial contributions to algebraic topology and also took an interest in literature, philosophy, history and mathematics education ...
and
Jacques Tits
Jacques Tits () (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory and incidence geometry. He introduced Tits buildings, the Tits alternative, the Tits group, and the Tits metric.
Life and ...
, who developed the idea independently. It associates a Lie algebra to a pair of division algebras ''A'', ''B''. The resulting Lie algebras have
Dynkin diagram
In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the classification of semisimple Lie algebras ...
s according to the table at right. The "magic" of the Freudenthal magic square is that the constructed Lie algebra is symmetric in ''A'' and ''B'', despite the original construction not being symmetric, though
Vinberg's symmetric method gives a symmetric construction.
The Freudenthal magic square includes all of the
exceptional Lie group
In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symm ...
s apart from
''G''2, and it provides one possible approach to justify the assertion that "the exceptional Lie groups all exist because of the
octonions
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have ...
": ''G''
2 itself is the
automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
of the octonions (also, it is in many ways like a
classical Lie group
In mathematics, the classical groups are defined as the special linear groups over the reals , the complex numbers and the quaternions together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or ske ...
because it is the stabilizer of a generic 3-form on a 7-dimensional vector space – see
prehomogeneous vector space In mathematics, a prehomogeneous vector space (PVS) is a finite-dimensional vector space ''V'' together with a subgroup ''G'' of the general linear group GL(''V'') such that ''G'' has an open dense orbit in ''V''. Prehomogeneous vector spaces were i ...
).
Constructions
See
history
History (derived ) is the systematic study and the documentation of the human activity. The time period of event before the History of writing#Inventions of writing, invention of writing systems is considered prehistory. "History" is an umbr ...
for context and motivation. These were originally constructed circa 1958 by Freudenthal and Tits, with more elegant formulations following in later years.
Tits' approach
Tits' approach, discovered circa 1958 and published in , is as follows.
Associated with any normed real
division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.
Definitions
Formally, we start with a non-zero algebra ''D'' over a fie ...
''A'' (i.e., R, C, H or O) there is a
Jordan algebra
In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms:
# xy = yx (commutative law)
# (xy)(xx) = x(y(xx)) ().
The product of two elements ''x'' and ''y'' in a Jordan alg ...
, ''J''
3(''A''), of 3 × 3 ''A''-
Hermitian matrices
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th ...
. For any pair (''A'', ''B'') of such division algebras, one can define a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
:
where
denotes the Lie algebra of
derivation
Derivation may refer to:
Language
* Morphological derivation, a word-formation process
* Parse tree or concrete syntax tree, representing a string's syntax in formal grammars
Law
* Derivative work, in copyright law
* Derivation proceeding, a proc ...
s of an algebra, and the subscript 0 denotes the
trace-free
In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix ().
It can be proved that the trace o ...
part. The Lie algebra ''L'' has
as a subalgebra, and this acts naturally on
. The Lie bracket on
(which is not a subalgebra) is not obvious, but Tits showed how it could be defined, and that it produced the following table of
compact Lie algebra
In the mathematical field of Lie theory, there are two definitions of a compact Lie algebra. Extrinsically and topologically, a compact Lie algebra is the Lie algebra of a compact Lie group; this definition includes tori. Intrinsically and algeb ...
s.
By construction, the row of the table with ''A''=R gives
, and similarly vice versa.
Vinberg's symmetric method
The "magic" of the Freudenthal magic square is that the constructed Lie algebra is symmetric in ''A'' and ''B''. This is not obvious from Tits' construction.
Ernest Vinberg
Ernest Borisovich Vinberg (russian: Эрне́ст Бори́сович Ви́нберг; 26 July 1937 – 12 May 2020) was a Soviet and Russian mathematician, who worked on Lie groups and algebraic groups, discrete subgroups of Lie groups, invar ...
gave a construction which is manifestly symmetric, in . Instead of using a Jordan algebra, he uses an algebra of skew-hermitian trace-free matrices with entries in ''A'' ⊗ ''B'', denoted
. Vinberg defines a Lie algebra structure on
:
When ''A'' and ''B'' have no derivations (i.e., R or C), this is just the Lie (commutator) bracket on
. In the presence of derivations, these form a subalgebra acting naturally on
as in Tits' construction, and the tracefree commutator bracket on
is modified by an expression with values in
.
Triality
A more recent construction, due to
Pierre Ramond
Pierre Ramond (; born 31 January 1943) is distinguished professor of physics at University of Florida in Gainesville, Florida. He initiated the development of superstring theory.
Academic career
Ramond completed his BSEE from Newark College of ...
and Bruce Allison and developed by Chris Barton and
Anthony Sudbery, uses
triality
In mathematics, triality is a relationship among three vector spaces, analogous to the duality relation between dual vector spaces. Most commonly, it describes those special features of the Dynkin diagram D4 and the associated Lie group Spin(8) ...
in the form developed by
John Frank Adams
John Frank Adams (5 November 1930 – 7 January 1989) was a British mathematician, one of the major contributors to homotopy theory.
Life
He was born in Woolwich, a suburb in south-east London, and attended Bedford School. He began research ...
; this was presented in , and in streamlined form in . Whereas Vinberg's construction is based on the automorphism groups of a division algebra ''A'' (or rather their Lie algebras of derivations), Barton and Sudbery use the group of automorphisms of the corresponding triality. The triality is the trilinear map
:
obtained by taking three copies of the division algebra ''A'', and using the inner product on ''A'' to dualize the multiplication. The automorphism group is the subgroup of SO(''A''
1) × SO(''A''
2) × SO(''A''
3) preserving this trilinear map. It is denoted Tri(''A''). The following table compares its Lie algebra to the Lie algebra of derivations.
Barton and Sudbery then identify the magic square Lie algebra corresponding to (''A'',''B'') with a Lie algebra structure on the vector space
:
The Lie bracket is compatible with a Z
2 × Z
2 grading, with tri(''A'') and tri(''B'') in degree (0,0), and the three copies of ''A'' ⊗ ''B'' in degrees (0,1), (1,0) and (1,1). The bracket preserves tri(''A'') and tri(''B'') and these act naturally on the three copies of ''A'' ⊗ ''B'', as in the other constructions, but the brackets between these three copies are more constrained.
For instance when ''A'' and ''B'' are the octonions, the triality is that of Spin(8), the double cover of SO(8), and the Barton-Sudbery description yields
:
where V, S
+ and S
− are the three 8-dimensional representations of
(the fundamental representation and the two
spin representation
In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they are two equ ...
s), and the hatted objects are an isomorphic copy.
With respect to one of the Z
2 gradings, the first three summands combine to give
and the last two together form one of its spin representations Δ
+128 (the superscript denotes the dimension). This is a well known
symmetric decomposition of
E8.
The Barton–Sudbery construction extends this to the other Lie algebras in the magic square. In particular, for the exceptional Lie algebras in the last row (or column), the symmetric decompositions are:
:
:
:
:
Generalizations
Split composition algebras
In addition to the
normed division algebra
In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic f ...
s, there are other
composition algebra
In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies
:N(xy) = N(x)N(y)
for all and in .
A composition algebra includes an involution c ...
s over R, namely the
split-complex number
In algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components and , and is written z=x+yj, where j^2=1. The ''conjugate'' of is z^*=x-yj. Since j^2=1, the product of a number wi ...
s, the
split-quaternions
In abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle (lawyer), James Cockle in 1849 under the latter name. They form an associative algebra of dimension four over the real numbers.
A ...
and the
split-octonions In mathematics, the split-octonions are an 8-dimensional nonassociative algebra over the real numbers. Unlike the standard octonions, they contain non-zero elements which are non-invertible. Also the signature (quadratic form), signatures of their ...
. If one uses these instead of the complex numbers, quaternions, and octonions, one obtains the following variant of the magic square (where the split versions of the division algebras are denoted by a prime).
Here all the Lie algebras are the
split real form
In mathematics, the notion of a real form relates objects defined over the Field (algebra), field of Real number, real and Complex number, complex numbers. A real Lie algebra ''g''0 is called a real form of a complex Lie algebra ''g'' if ''g'' is ...
except for so
3, but a sign change in the definition of the Lie bracket can be used to produce the split form so
2,1. In particular, for the exceptional Lie algebras, the maximal compact subalgebras are as follows:
A non-symmetric version of the magic square can also be obtained by combining the split algebras with the usual division algebras. According to Barton and Sudbery, the resulting table of Lie algebras is as follows.
The real exceptional Lie algebras appearing here can again be described by their maximal compact subalgebras.
Arbitrary fields
The split forms of the composition algebras and Lie algebras can be defined over any
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
K. This yields the following magic square.
There is some ambiguity here if K is not algebraically closed. In the case K = C, this is the complexification of the Freudenthal magic squares for R discussed so far.
More general Jordan algebras
The squares discussed so far are related to the Jordan algebras ''J''
3(''A''), where ''A'' is a division algebra. There are also Jordan algebras ''J
n''(''A''), for any positive integer ''n'', as long as ''A'' is associative. These yield split forms (over any field K) and compact forms (over R) of generalized magic squares.
For ''n'' = 2, J
''2''(''O'') is also a Jordan algebra. In the compact case (over R) this yields a magic square of orthogonal Lie algebras.
The last row and column here are the orthogonal algebra part of the isotropy algebra in the symmetric decomposition of the exceptional Lie algebras mentioned previously.
These constructions are closely related to
hermitian symmetric space
In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian s ...
s – cf.
prehomogeneous vector space In mathematics, a prehomogeneous vector space (PVS) is a finite-dimensional vector space ''V'' together with a subgroup ''G'' of the general linear group GL(''V'') such that ''G'' has an open dense orbit in ''V''. Prehomogeneous vector spaces were i ...
s.
Symmetric spaces
Riemannian symmetric space
In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, ...
s, both compact and non-compact, can be classified uniformly using a magic square construction, in . The irreducible compact symmetric spaces are, up to finite covers, either a compact simple Lie group, a Grassmannian, a
Lagrangian Grassmannian, or a
double Lagrangian Grassmannian
A double is a look-alike or doppelgänger; one person or being that resembles another.
Double, The Double or Dubble may also refer to:
Film and television
* Double (filmmaking), someone who substitutes for the credited actor of a character
* Th ...
of subspaces of
for normed division algebras A and B. A similar construction produces the irreducible non-compact symmetric spaces.
History
Rosenfeld projective planes
Following
Ruth Moufang
Ruth Moufang (10 January 1905 – 26 November 1977) was a German mathematician.
Biography
Born to German chemist Eduard Moufang and Else Fecht Moufang. Eduard Moufang was the son of Friedrich Carl Moufang (1848-1885) from Mainz, and Elisab ...
's discovery in 1933 of the
Cayley projective plane or "octonionic projective plane" P
2(O), whose symmetry group is the exceptional Lie group
F4, and with the knowledge that
''G''2 is the automorphism group of the octonions, it was proposed by that the remaining exceptional Lie groups
''E''6,
''E''7, and
E8 are isomorphism groups of projective planes over certain algebras over the octonions:
* the , C ⊗ O,
* the , H ⊗ O,
* the , O ⊗ O.
This proposal is appealing, as there are certain exceptional compact
Riemannian symmetric space
In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, ...
s with the desired symmetry groups and whose dimension agree with that of the putative projective planes (dim(P
2(K ⊗ K′)) = 2 dim(K)dim(K′)), and this would give a uniform construction of the exceptional Lie groups as symmetries of naturally occurring objects (i.e., without an a priori knowledge of the exceptional Lie groups). The Riemannian symmetric spaces were classified by Cartan in 1926 (Cartan's labels are used in sequel); see
classification Classification is a process related to categorization, the process in which ideas and objects are recognized, differentiated and understood.
Classification is the grouping of related facts into classes.
It may also refer to:
Business, organizat ...
for details, and the relevant spaces are:
* the
octonionic projective plane
In mathematics, the Cayley plane (or octonionic projective plane) P2(O) is a projective plane over the octonions.Baez (2002).
The Cayley plane was discovered in 1933 by Ruth Moufang, and is named after Arthur Cayley for his 1845 paper describing ...
– FII, dimension 16 = 2 × 8, F
4 symmetry,
Cayley projective plane P
2(O),
* the bioctonionic projective plane – EIII, dimension 32 = 2 × 2 × 8, E
6 symmetry, complexified Cayley projective plane, P
2(C ⊗ O),
* the ""
– EVI, dimension 64 = 2 × 4 × 8, E
7 symmetry, P
2(H ⊗ O),
* the ""
– EVIII, dimension 128 = 2 × 8 × 8, E
8 symmetry, P
2(O ⊗ O).
The difficulty with this proposal is that while the octonions are a division algebra, and thus a projective plane is defined over them, the bioctonions, quateroctonions and octooctonions are not division algebras, and thus the usual definition of a projective plane does not work. This can be resolved for the bioctonions, with the resulting projective plane being the complexified Cayley plane, but the constructions do not work for the quateroctonions and octooctonions, and the spaces in question do not obey the usual axioms of projective planes,
hence the quotes on "(putative) projective plane". However, the tangent space at each point of these spaces can be identified with the plane (H ⊗ O)
2, or (O ⊗ O)
2 further justifying the intuition that these are a form of generalized projective plane.
Accordingly, the resulting spaces are sometimes called Rosenfeld projective planes and notated as if they were projective planes. More broadly, these compact forms are the Rosenfeld elliptic projective planes, while the dual non-compact forms are the Rosenfeld hyperbolic projective planes. A more modern presentation of Rosenfeld's ideas is in , while a brief note on these "planes" is in .
[This Week's Finds in Mathematical Physics – Week 106]
, John Baez
John Carlos Baez (; born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California. He has worked on spin foams in loop quantum gravity, appl ...
July 23, 1997
The spaces can be constructed using Tits' theory of buildings, which allows one to construct a geometry with any given algebraic group as symmetries, but this requires starting with the Lie groups and constructing a geometry from them, rather than constructing a geometry independently of a knowledge of the Lie groups.
Magic square
While at the level of manifolds and Lie groups, the construction of the projective plane P
2(K ⊗ K′) of two normed division algebras does not work, the corresponding construction at the level of Lie algebras ''does'' work. That is, if one decomposes the Lie algebra of infinitesimal isometries of the projective plane P
2(K) and applies the same analysis to P
2(K ⊗ K′), one can use this decomposition, which holds when P
2(K ⊗ K′) can actually be defined as a projective plane, as a ''definition'' of a "magic square Lie algebra" ''M''(K,K′). This definition is purely algebraic, and holds even without assuming the existence of the corresponding geometric space. This was done independently circa 1958 in and by Freudenthal in a series of 11 papers, starting with and ending with , though the simplified construction outlined here is due to .
See also
*
E6 (mathematics)
In mathematics, E6 is the name of some closely related Lie groups, linear algebraic groups or their Lie algebras \mathfrak_6, all of which have dimension 78; the same notation E6 is used for the corresponding root lattice, which has rank&nbs ...
*
E7 (mathematics)
In mathematics, E7 is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7, all of which have dimension 133; the same notation E7 is used for the corresponding root lattice, which has rank 7. ...
*
E8 (mathematics)
In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8. The designation ...
*
F4 (mathematics)
In mathematics, F4 is the name of a Lie group and also its Lie algebra f4. It is one of the five exceptional simple Lie groups. F4 has rank 4 and dimension 52. The compact form is simply connected and its outer automorphism group is the trivial ...
*
G2 (mathematics)
In mathematics, G2 is the name of three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras \mathfrak_2, as well as some algebraic groups. They are the smallest of the five exceptional simple Lie gr ...
*
Euclidean Hurwitz algebra
In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital algebra, unital real numbers, real non-associative algebras endowed with a posi ...
*
Euclidean Jordan algebra
In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms:
# xy = yx (commutative law)
# (xy)(xx) = x(y(xx)) ().
The product of two elements ''x'' and ''y'' in a Jordan alg ...
*
Jordan triple system
In algebra, a triple system (or ternar) is a vector space ''V'' over a field F together with a F-trilinear map
: (\cdot,\cdot,\cdot) \colon V\times V \times V\to V.
The most important examples are Lie triple systems and Jordan triple systems. The ...
Notes
References
*
*
*
4.3: The Magic Square*
*
*
*
*
*
*
*
*
*
*
* (reprint of 1951 article)
*
*
*
*
*
*
*
*
*
* {{cite journal , first=Ichiro , last=Yokota , title=Non-symmetry of the Freudenthal's magic square , journal= J. Fac. Sci. Shinshu Univ. , volume= 20 , year = 1985 , pages = 13
Lie groups
Representation theory