The Quantum Structure of Space and Time (293 pages)

Mathematical Structures

4.7 Prepared Comments

153

4.7.1 Hewnann Nicolai: El0 and K (Elo) : prospects and chal-
lenges

Definition of Elo: The maximal rank hyperbolic Kac-Moody algebra el0 =
Lie(E10) (in split real form) is defined via the so-called Chevalley Serre presen-
tation in terms of generators hi, ei, fi (i = 1,... ,lo) with relations [l]

[hi, hj] = 0, [ei, fj] = 6ijhi, [hi, ej] = Aijej, [hi, fj] = -Aijfj,

(adei)l-Aijej = 0, (adfi)l-Aijfj = 0.

where {hi} span the Cartan subalgebra b. The entries of the Cartan matrix Aij can
be read off from the Dynkin diagram displayed in Figure 1 below. With all other
Kac-Moody algebras, el0 shares the following key properties:

Root space decomposition: for any root a E Q(E10) = II1,g (= the unique even

self-dual Lorentzian lattice in ten dimensions), we have

(elo)a = {z E el0 : [h,z] = a(h)z forallh E b}

One distinguishes real roots (a2 = 2) and imaginary roots (a’” 5 0); the latter

can be further subdivided into lightlike (null) roots (a2 = 0) and timelike roots

Triangular decomposition: this is a generalization of the well known decompo-

sition of finite dimensional matrices into (strictly) upper and lower triangular

(n*), and diagonal matrices (b), respectively.

(a2 < 0).

el0 = n- @ b @ n+ , with nk := @a?O(elO)a

This is the feature that ensures computability in the present context, via choice

of a triangular gauge for the ‘vielbein’ V(t) E Elo/K(Elo).

Existence of an invariant bilinear form:

(hilhj) = Aij , (~Ifj) = Jij , ([~,YIIz) = (xI[7J,ZI).

This is the feature which, in the present context, allows for the formulation of

an action principle. Because dim el0 = 00, this quadratic form is, in fact, the

only polynomial Casimir invariant, ensuring the (essential) uniqueness of the

a-model action below [2, 31.

Compact subalgebra telo: The Chevalley involution is defined by

w(ei) = -fi, w(fi) = -ei, w(hi) = -hi

and extends to all of el0 by w(w(z),w(y),w(y)]. The fixed point set eel0 = {z E
el0 : w(z) = z} is a subalgebra of elo, which is called the compact subalgebra. Note

that eel0 is not a Kac-Moody algebra [4].

Level decomposition: No closed formulas exist for the dimensions of the root

spaces, although the root multiplicities are in principle computable recursively [l].