Clifford Algebras of Odd dimension give the Bn series: B1=A1=C1 - B2=C2 - B3 - ...
Clifford Algebras of Even dimension give the Dn series: D1=A0 - D2 - D3=A3 - D4 - D5 - ...
A-D-E series: A0=D1 - A1=B1=C1 - A2 - A3=D3 - D4 - D5 - E6 - E7 - E8
Cartan Matrices and Commutation Relations
Root Lattice subset of Weight :Lattice
Coxeter Groups that do NOT correspond to Lie Algebras
As is discussed on the Clifford Algebra page, Rotation Groups are double covered by the Lie Groups whose Lie Algebras are the bivector algebras of Clifford Algebras. You can ask WHAT ARE THE FINITE SUBGROUPS OF THE ROTATION GROUPS of rotations in real n-dimensional Euclidean space Rn? The answers are in Dave Rusin's Known Math pages: -------------------------------- If n is allowed to vary, the answer is "all finite groups". If n is assumed to be large but fixed, you can still assume the answer is going to be very messy, since the union of the answers for the various n will include all groups. Actually it's worse than that: it is true that you will sometimes have two subgroups of O(n,R) which are isomorphic as abstract groups but not conjugate (so that as groups of symmetries they are essentially distinct). -------------------------------- All finite simple groups have been classified, leading to the discovery of the Monster group (see Finite Group Theory, by M. Aschbacher, Cambridge 1986.), and including, among others, finite simple groups of Lie type. They are closely related to the Lie groups discussed on this page, and to finite fields, as has been pointed out to me by Julius Bauer (bauer@phyq4.physik.uni-freiburg.de). The finite groups of Lie type are related to the Lie groups of this page through their Weyl-Coxeter groups. which are finite subgroups of rotation groups that are generated by reflections. TO UNDERSTAND THE FINITE REFLECTION WEYL-COXETER GROUPS consider the rotation group in real Euclidean N-space RN. It is double covered by the simply connected Lie group Spin(0,N), which is infinitesimally generated by the Lie algebra Spin(0,N). Each rotation operates in a 2-dimensional subspace of RN. How many basis elements does Spin(0,N) have? It has as many basis elements as there are 2-dim subspaces of RN. Since an orthonormal basis of RN has N elements, the basis for Spin(0,N) has N! / (2!)((N-2)!) = (1/2) N (N-1) elements, which gives the correspondence of the Lie algebra Spin(0,n) to the bivector subalgebra of the Clifford algebra Cl(0,N).
Consider the case of N being an odd number, N = 2n+1, corresponding to the Bn Lie algebra Spin(0,2n+1). If N = 3 or 7, RN corresponds to the imaginary part of Q = quaternions; and O = octonions. The basis of the Lie algebra Spin(0,N) has N ((N-1)/2) elements. They can be regarded as (N-1)/2 copies of RN. n = (N-1)/2 is the rank of the Lie algebra Spin(0,N). Consider the ((N-1)/2)-dimensional real Euclidean space WEYL for which each of the (N-1)/2 axes of an orthonormal basis corresponds to RN in the N ((N-1)/2)-dimensional Lie algebra Spin(0,N). The order of the Weyl Group of Bn is n! 2^n which is 2^n times the order n! of the Weyl Group of An.
Start with the case N = 3 of the B1 Lie algebra
Spin(0,3) = SU(2) = Sp(1) = Lie algebra of S3.
The rank of Spin(0,3) is (3-1)/2 = 1,
so that WEYL is a 1-dimensional real Euclidean space
in which the 3 basis elements of Spin(0,3) must be represented.
Put 1 basis element at the origin.
The other 2 basis elements are then at -1 and +1:
(-1)--------(0)--------(+1)
Here WEYL is the root vector space of Spin(0,3) in which
reflections through the origin 0 act to interchange -1 and +1.
Those reflections form a finite reflection group,
the Weyl group of Spin(0,3), which is Z2.
Since 3 is the dimension of the imaginary quaternions, Spin(0,3) has quaternionic structure and in fact generates the Lie group S3 of imaginary quaternions, and corresponds to the Associative Triangle of Imaginary Quaternions.
Since A1=B1=C1 it has 2 root vectors in a 1-dimensional space, its Coxeter Number, the number of root vectors divided by the rank, or dimensionality of its root vector space, is 2 / 1 = 2.
A1=B1=C1 is the basic building block of all Lie algebras, in that any Lie algebra Gn of rank n can be constructed from the root vector space of a Cartesian product of n copies of A1=B1=C1.
If N is the number of root vectors of Gn, its Coxeter number H is H = N / n.
Since 2n of the root vectors of Gn correspond to the Cartesian product of n copies of A1=B1=C1, the excess number of root vectors for each copy of A1=B1=C1 is ( N - 2n ) / n = H - 2.
Next consider the case N = 5 of the B2 Lie algebra
Spin(0,5) = Sp(2).
The rank of Spin(0,5) is (5-1)/2 = 2,
so that WEYL is a 2-dimensional real Euclidean space
in which the 10 basis elements of Spin(0,5) must be represented.
Since we have 2 axes, put 2 basis elements at the origin.
4 of the other 8 basis elements are then
at -1 and +1 on each of the two axes.
However, here we have 4 more elements to place in WEYL.
Symmetry indicates that they should be at (+/- 1 , +/- 1):
(-1,+1) (0,+1) (+1,+1)
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(-1,0)--------(0,0)--------(+1,0)
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(-1,-1) (0,-1) (+1,-1)
Here WEYL is the root vector space of Spin(0,4) in which
reflections through the origin (0,0) act:
on the first axis to interchange the
three points of the form (-1,X) and (+1,X);
on the second axis to interchange the
three points of the form (X,-1) and (X,+1);
and
on the two diagonal axes to interchange
the three upper left corner and lower right corner points,
and the three upper right and lower left corner points.
Those reflections form a finite reflection group
generated by all reflections through hyperplanes
through the origin and perpendicular to a pair of outer points,
which finite reflection group is
the Weyl group of Spin(0,5), which is (Z2)^2 x S2
of order 2^2 x 2! = 8.
A set of Cartan matrix ratios of the inner products of the
positive roots A1 = (+1,0) , A2 = (-1,+1) is:
C12 = 2 (A1,A2) / (A2,A2) = 2 (-1+0) / (1+1) = -1
C21 = 2 (A2,A1) / (A1,A1) = 2 (-1+0) / (1+0) = -2
A Cartan matrix is:
2 -1
-2 2
It has N = 8 root vectors, so its Coxeter number H is H = N / n = 8 / 2 = 4.
The excess number of root vectors for each of the 2 copies of A1=B1=C1 is 4 - 2 = 2.
the 3-dimensional polytope formed by
the root vectors of the B3 Lie algebra Spin(0,7) is
the 12-vertex cuboctahedron plus the 6-vertex octahedron
with Weyl group (Z2^3) x 3! of order 48
(since 7 is the dimension of the imaginary octonions,
Spin(0,7) has octonionic structure and
in fact generates the Lie group Spin(0,7) with fibration
S7 --} Spin(0,7) --} G2
since G2 is the group of automorphisms of the 7-sphere S7,
and so can be seen as
the set of pairs (x,y) where x,y are in S7,
Spin(0,7) is in a sense made up of 3 copies of
the imaginary octonions S7);
A Cartan matrix is:
2 -1 0
-2 2 -1
0 -1 2
It has N = 18 root vectors, so its Coxeter number H is H = N / n = 18 / 3 = 6.
The excess number of root vectors for each of the 3 copies of A1=B1=C1 is 6 - 2 = 4.
Consider the case of N being an even number, N = 2n, corresponding to the Dn Lie algebra Spin(0,2n). RN can be considered to have 1 "real axis" dimension and (N-1) "imaginary axis" dimensions. If N = 4 or 8, the axes correspond to the real and imaginary axes of Q = quaternions or O = octonions. The basis of the Lie algebra Spin(0,2n) = Spin(0,N) has (N/2) (N-1) elements. They can be regarded as N/2 copies of the (N-1)-dimensional imaginary subspace of RK. n = N/2 is the rank of the Lie algebra Spin(0,2n). Consider the (N/2)-dimensional real Euclidean space WEYL for which each of the N/2 axes of an orthonormal basis corresponds to one of the N-1 imaginary subspaces of RN in the (N/2)(N-1)-dimensional Lie algebra Spin(0,N). The order of the Weyl Group of Dn is n! 2^(n-1) which is 2^(n-1) times the order n! of the Weyl Group of An.
You can construct the Root Vector Diagram of D(n+1) from the Root Vector Diagram of Dn by
noting that the Root Vector Diagram of Dn has (N/2)(N-1) = n(2n - 1) - n = 2n(n-1) vertices
and then adding to those 2n(n-1) vertices
a 2n-vertex n-dim hyperoctahedron cross polytope above in the (n+1)-st dimension, and
a 2n-vertex n-dim hyperoctahedron cross polytope below in the (n+1)-st dimension,
to get a Root Vector Diagram for D(n+1) that has 2n(n-1) + 2n + 2n = 2n^2 -2n + 4n = 2n^2 + 2n =
2(n+1)(n) = 2(n+1)(n+1 - 1) vertices.
It has N = 2n(n - 1) root vectors, so its Coxeter number H is H = 2n(n - 1) / n = 2(n - 1).
The excess number of root vectors for each of the n copies of A1=B1=C1 is 2n - 4.
Start with the case N = 4 of the D2 Lie algebra
Spin(0,4) = Spin(0,3) x Spin(0,3) = SU(2) x SU(2) =
= Sp(1) x Sp(1) = Lie algebra of S3 x S3.
The rank of Spin(0,4) is 4/2 = 2,
so that WEYL is a 2-dimensional real Euclidean space
in which the 6 basis elements of Spin(0,4) must be represented.
Since we have 2 axes, put 2 basis elements at the origin.
The other 4 basis elements are then
at -1 and +1 on each of the two axes:
(0,+1)
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(-1,0)--------(0,0)--------(+1,0)
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(0,-1)
Here WEYL is the root vector space of Spin(0,4) in which
reflections through the origin (0,0) act
on the first axis to interchange (-1,0) and (+1,0)
and
on the second axis to interchange (0,-1) and (0,+1) .
Those reflections form a finite reflection group,
generated by all reflections through hyperplanes
through the origin and perpendicular to a pair of outer points,
which finite reflection group is
the Weyl group of Spin(0,4), which is Z2 x Z2, or (Z2)^2,
of order 4,
and
the Root Vector Diagram is a 2-dim square:
*--* | | *--*
A Cartan matrix, which determines the commutation relations
of the Lie algebra, is determined by the D2 root vector space
is determined by ratios of the inner products of the
positive roots. By convention, the diagonal entries
are set equal to 2:
Cii = 2 (Ai,Ai) / (Ai,Ai)
where the Ai are root vectors and Cii is the Cartan matrix entry.
Since the two positive root vectors are orthogonal to each other,
the off-diagonal Cij are zero.
A Cartan matrix is:
2 0
0 2
Since 4-1 = 3 is the dimension of the imaginary quaternions,
Spin(0,4) has quaternionic structure and
in fact generates two copies of the Lie group S3,
one for the group of Lorentz rotations and
another for the group of Lorentz boosts.
It has N = 4 root vectors, so its Coxeter number H is H = 4 / 2 = 2.
The excess number of root vectors for each of the 2 copies of A1=B1=C1 is 2 - 2 = 0, which is consistent with the fact that the D2 Lie algebra is reducible to 2 copies of A1=B1=C1.
Del Pezzo Surfaces also correspond to the A-D-E series,
as does the McKay Correspondence.
Cartan matrices are only unique up to equivalence classes, because different choices of ordering of basis elements will produce different, but equivalent, matrices. How does a Cartan matrix, which is only K x K where K is the rank of the Lie algebra, determine all the commutation relations between all the elements? For example, a D series Lie algebra Spin(0,N) has rank N/2, so the Cartan matrix has N^2 / 4 entries, while the Lie algebra has (1/2) N (N-1) = N^2 / 2 - N/2 elements. If N = 4, the Cartan matrix has 4 entries to determine commutation relations among 6 elements, and the ratio gets worse as N gets larger (for N = 8, it is 16 entries for 28 elements). Roughly, the Lie algebra is made up of the elements that we identified with each axis of the root vector space. For the case of D4, we had 4 axes, each with 7 elements of the Spin(0,8) Lie algebra. However, the entire Lie algebra is algebraically generated by only 3 elements for each axis: the one we put at the origin, and one on each side, with all 3 forming a B2 Spin(0,3) = SU(2) = Sp(1) subalgebra for each axis of the root vector space. The ones at the origin commute with everything, so all the ones at the origin together form the maximal commutative subalgebra, or Cartan subalgebra, that generates the maximal torus of the Lie group (so-called because it is topologically the product of circles, and so is a torus). The commutators of the 2 elements on opposite sides of the origin give the Cartan subalgebra element at the origin. These relations correspond to the diagonal 2 entries in the Cartan matrix. The commutation relations between the Cartan subalgebra element at the origin and each of the 2 elements on each side are determined by the upper and lower triangular entries of the Cartan matrix. What about the commutation relations among the elements other than the 3 for each axis? The structural requirements of a Lie algebra, such as the Jacobi identity, provide the additional information you need. Varadarajan devotes section 4.8 of his book to showing in detail how this works. For more details, see the books of Bourbaki, Varadarajan, or Humphreys, listed in references, or the free book by Cahn. The Cartan matrices are fundamental to the McKay Correspondence between A-D-E Lie algebras and finite subroups of Spin(3) = SU(2) = S3.
According to Fulton and Harris, Representation Theory, Springer-Verlag 1991 and Bourbaki, Groupes et Algebres de Lie, Chapitres 4, 5, et 6, for all Lie algebras the weight lattice /\w of irreducible finite-dimensional representations includes as a sublattice the root lattice /\r generated by the root vectors of the Lie algebra. The group /\w / /\r is finite, of order equal to the determinant of the Cartan matrix.
For the various Lie algebras, the determinants of their Cartan matrices are:
For An the quotient /\w / /\r is of order n+1 and can therefore be identified with the (n+1)-dim fundamental representation of the An Lie algebra.
All Coxeter Groups correspond to Singularities in the sense of Arnold.
Most Coxeter Groups correspond to Lie Algebras, but some do not:
H2, H3, and H4 correspond to Penrose tilings of 2, 3, and 4 dimensions, so, they are "really" crystallographic, in the sense that such Penrose tilings can be realized as Irrational Slices of regular E6 or E8 lattices.
H4 is "really" a creature of the E8 lattice, as are H3 and H2. Compare the discussion of icosians in Conway and Sloane, Sphere Packings, Lattices and Groups, 3rd ed, Springer 1999, at pages 207-211, which discusses not only the E8 lattice but also the 24-dim Leech lattice.
Since my intuition for visualization is more Lie group oriented than singularity oriented, I like this because it brings H4, H3, and H2 into the Lie structures through E8. However, the relationships should also produce some interesting corresponding relationships among singularities.
Singularities seem to have an order/structure that is consistent with the Sefirot.
Higher order singularities tend to involve moduli spaces, whose infinities are reminiscent of the divisors of zero that appear when Cayley-Dickson algebras are extended to sedenions and beyond. A similar significant increase in complexity also occurs in Clifford algebras, because above 8-dim Cl(8) the dimension 2^(N/2) of the spinor space Cl(N) grows to be much larger than its vector space of dimension N. However, even for very large dimension, say 8N, the Clifford periodicity factorization
enables you to see that the Cl(8) structure is "really there" at ALL levels. That is possibly the "reason why" the "infinities of zero divisors" of higher Cayley-Dickson algebras seem to have octonionic structure (such as Moreno's G2). Maybe that is also related to moduli spaces, and maybe the moduli spaces will also some day be seen to have some octonionic structure.
For further details, see
the book Geometric Differentiation for the intelligence of curves and surfaces (Cambridge 1994) by Ian R. Porteous;
the thesis of Alex Flegmann, Evolutes, Involutes, and the Coxeter Group H4 (Publication de l'Institute de Recherche Mathematique Advancee, Universite Louis Pasteur, Strasbourg 1985), in which Alex Flegmann
the book The Theory of Singularities and Its Applications, by V. I. Arnold (Pisa 1991);
the book Catastrophe Theory, by V. I. Arnold (3rd edition, Springer-Verlag 1992);
the book Catastrophe Theory for Scientists and Engineers, by Robert Gilmore, (Dover 1993 republication of Wiley 1981 edition); and
the book Singularities of Differentiable Maps, Volume I, by V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko (Birkhauser 1985), according to which:
Such singularities may be useful in describing Branching of the Worlds of the Many-Worlds, related to 26-dimensional Bosonic String Theory and a 27-dimensional M-Theory.
Coxeter, Regular Polytopes, Dover 1973 (reprint from 1963). Coxeter, Regular Complex Polytopes, 2nd ed, Cambridge 1991. Kaleidoscopes: Selected Writings of H. S. M. Coxeter, introduced and compiled by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, and Asia Ivic Weiss, John Wiley and Sons, 1995. Synergetics, by R. Buckminster Fuller, Macmillan 1982. Differential Geometry, Gauge Theories, and Gravity, by Gockeler and Schucker, Cambridge 1987. Nonassociative Algebras in Physics, by Lohmus, Paal, and Sorgsepp, Hadronic Press 1994. Topological Geometry, 2nd ed, (new edition to be titled Clifford Algebras and the Classical Groups) by Porteous, Cambridge 1981. J. Math. Phys. 14 (1973) 1651-1667, by Gunaydin and Gursey. Lie Groups, Lie Algebras, and Their Representations, by V. S. Varadarajan, Springer Grad. Text Math. No. 102, 1984. Reflection Groups and Coxeter Groups, by Humphreys, Cambridge 1990. Introduction to Lie Algebras and Representation Theory, by Humphreys, Springer-Verlag 1972. Groupes et Algebres de Lie Chapitres 1; 2 et 3; 4, 5 et 6; 7 et 8; 9 by Bourbaki HERE IS AN EXCELLENT FREE BOOK: Semi-Simple Lie Algebras and their Representations by Robert N. Cahn. The original publisher of the 1984 book, Benjamin-Cummings, gave him permission to put the entire book on his WWW site in the form of freely downloadable postscript files. Thanks to both Robert Cahn and Benjamin-Cummings for making such good material freely available to everybody.
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