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4.10 Regular subalgebras of Kac–Moody algebras

This section is based on [96Jump To The Next Citation Point].

4.10.1 Definitions

Let 𝔤 be a Kac–Moody algebra, and let ¯𝔤 be a subalgebra of 𝔤 with triangular decomposition ¯𝔤 = ¯𝔫 − ⊕ ¯𝔥 ⊕ ¯𝔫+. We assume that ¯𝔤 is canonically embedded in 𝔤, i.e., that the Cartan subalgebra ¯𝔥 of ¯𝔤 is a subalgebra of the Cartan subalgebra 𝔥 of 𝔤, ¯𝔥 ⊂ 𝔥, so that ¯𝔥 = ¯𝔤 ∩ 𝔥. We shall say that ¯𝔤 is regularly embedded in 𝔤 (and call it a “regular subalgebra”) if and only if two conditions are fulfilled: (i) The root generators of ¯𝔤 are root generators of 𝔤, and (ii) the simple roots of ¯𝔤 are real roots of 𝔤. It follows that the Weyl group of ¯𝔤 is a subgroup of the Weyl group of 𝔤 and that the root lattice of ¯𝔤 is a sublattice of the root lattice of 𝔤.

The second condition is automatic in the finite-dimensional case where there are only real roots. It must be separately imposed in the general case. Consider for instance the rank 2 Kac–Moody algebra 𝔤 with Cartan matrix

( ) 2 − 3 . − 3 2


-1-- x = √3--[e1,e2], (4.73 ) y = √1--[f1,f2], (4.74 ) 3 z = − (h1 + h2). (4.75 )
It is easy to verify that x,y,z define an A1 subalgebra of 𝔤 since [z,x] = 2x, [z,y] = − 2y and [x,y] = z. Moreover, the Cartan subalgebra of A1 is a subalgebra of the Cartan subalgebra of 𝔤, and the step operators of A1 are step operators of 𝔤. However, the simple root α = α + α 1 2 of A 1 (which is an A 1-real root since A 1 is finite-dimensional), is an imaginary root of 𝔤: α1 + α2 has norm squared equal to − 2. Even though the root lattice of A1 (namely, { ±α }) is a sublattice of the root lattice of 𝔤, the reflection in α is not a Weyl reflection of 𝔤. According to our definition, this embedding of A1 in 𝔤 is not a regular embedding.

4.10.2 Examples – Regular subalgebras of E10

We shall describe some regular subalgebras of E10. The Dynkin diagram of E10 is displayed in Figure 21View Image.

A9 ⊂ 𝓑 ⊂ E10

A first, simple, example of a regular embedding is the embedding of A 9 in E 10 which will be used to define the level when trying to reformulate eleven-dimensional supergravity as a nonlinear sigma model. This is not a maximal embedding since one can find a proper subalgebra ℬ of E10 that contains A9. One may take for ℬ the Kac–Moody subalgebra of E10 generated by the operators at levels 0 and ±2, which is a subalgebra of the algebra containing all operators of even level15. It is regularly embedded in E10. Its Dynkin diagram is shown in Figure 22View Image.

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Figure 21: The Dynkin diagram of E10. Labels 1, ⋅⋅⋅ ,7 and 10 enumerate the nodes corresponding the regular E8 subalgebra discussed in the text.
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Figure 22: The Dynkin diagram of +++ ℬ ≡ E 7. The root without number is the root denoted ¯α10 in the text.

In terms of the simple roots of E10, the simple roots of ℬ are α1 through α9 and ¯α10 = 2 α10 + α1 + 2 α2 + 3α3 + 2α4 + α5. The algebra ℬ is Lorentzian but not hyperbolic. It can be identified with the “very extended” algebra E++7+ [86].

DE10 ⊂ E10

In [67], Dynkin has given a method for finding all maximal regular subalgebras of finite-dimensional simple Lie algebras. The method is based on using the highest root and is not generalizable as such to general Kac–Moody algebras for which there is no highest root. Nevertherless, it is useful for constructing regular embeddings of overextensions of finite-dimensional simple Lie algebras. We illustrate this point in the case of E 8 and its overextension E ≡ E++ 10 8. In the notation of Figure 21View Image, the simple roots of E8 (which is regularly embedded in E10) are α1, ⋅⋅⋅ ,α7 and α10.

Applying Dynkin’s procedure to E8, one easily finds that D8 can be regularly embedded in E8. The simple roots of D8 ⊂ E8 are α2,α3,α4, α5,α6,α7, α10 and β ≡ − θE8, where

θE8 = 3α10 + 6α3 + 4 α2 + 2α1 + 5α4 + 4α5 + 3α6 + 2 α7 (4.76 )
is the highest root of E 8. One can replace this embedding, in which a simple root of D 8, namely β, is a negative root of E8 (and the corresponding raising operator of D8 is a lowering operator for E8), by an equivalent one in which all simple roots of D8 are positive roots of E8.

This is done as follows. It is reasonable to guess that the searched-for Weyl element that maps the “old” D8 on the “new” D8 is some product of the Weyl reflections in the four E8-roots orthogonal to the simple roots α3, α4, α5, α6 and α7, expected to be shared (as simple roots) by E8, the old D8 and the new D8 – and therefore to be invariant under the searched-for Weyl element. This guess turns out to be correct: Under the action of the product of the commuting E8-Weyl reflections in the E8-roots μ1 = 2α1 + 3α2 + 5α3 + 4α4 + 3α5 + 2α6 + α7 + 3α10 and μ2 = 2α1 + 4α2 + 5α3 + 4α4 + 3α5 + 2α6 + α7 + 2α10, the set of D8-roots {α2,α3, α4,α5,α6, α7,α10,β} is mapped on the equivalent set of positive roots {α10, α3,α4,α5, α6,α7,α2, ¯β}, where

β¯= 2α1 + 3α2 + 4α3 + 3α4 + 2α5 + α6 + 2α10. (4.77 )
In this equivalent embedding, all raising operators of D8 are also raising operators of E8. What is more, the highest root of D8,
θD8 = α10 + 2α3 + 2α4 + 2α5 + 2α6 + 2α7 + α2 + β¯ (4.78 )
is equal to the highest root of E8. Because of this, the affine root α8 of the untwisted affine extension + E 8 can be identified with the affine root of + D 8, and the overextended root α9 can also be taken to be the same. Hence, DE10 can be regularly embedded in E10 (see Figure 23View Image).
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Figure 23: DE10 ≡ D++8 regularly embedded in E10. Labels 2,⋅⋅⋅ ,10 represent the simple roots α ,⋅⋅⋅ ,α 2 10 of E 10 and the unlabeled node corresponds to the positive root ¯ β = 2α1 + 3α2 + 4α3 + 3α4 + 2 α5 + α6 + 2α10.

The embedding just described is in fact relevant to string theory and has been discussed from various points of view in previous papers [125Jump To The Next Citation Point23Jump To The Next Citation Point]. By dimensional reduction of the bosonic sector of eleven-dimensional supergravity on a circle, one gets, after dropping the Kaluza–Klein vector and the 3-form, the bosonic sector of pure 𝒩 = 1 ten-dimensional supergravity. The simple roots of DE10 are the symmetry walls and the electric and magnetic walls of the 2-form and coincide with the positive roots given above [45Jump To The Next Citation Point]. A similar construction shows that ++ A 8 can be regularly embedded in E10, and that DE10 can be regularly embedded in BE10 ≡ B++8. See [106] for a recent discussion of DE10 in the context of Type I supergravity.

4.10.3 Further properties

As we have just seen, the raising operators of ¯𝔤 might be raising or lowering operators of 𝔤. We shall consider here only the case when the positive (respectively, negative) root generators of ¯𝔤 are also positive (respectively, negative) root generators of 𝔤, so that ¯𝔫 − = 𝔫− ∩ ¯𝔤 and ¯𝔫+ = 𝔫+ ∩ ¯𝔤 (“positive regular embeddings”). This will always be assumed from now on.

In the finite-dimensional case, there is a useful criterion to determine regular algebras from subsets of roots. This criterion, which does not use the highest root, has been generalized to Kac–Moody algebras in [76Jump To The Next Citation Point]. It covers also non-maximal regular subalgebras and goes as follows:

Theorem: Let Φ+real be the set of positive real roots of a Kac–Moody algebra 𝔤. Let γ1,⋅⋅⋅ ,γn ∈ Φ+real be chosen such that none of the differences γi − γj is a root of 𝔤. Assume furthermore that the γi’s are such that the matrix C = [C ] = [2(γ |γ )∕(γ |γ)] ij i j i i has non-vanishing determinant. For each 1 ≤ i ≤ n, choose non-zero root vectors Ei and Fi in the one-dimensional root spaces corresponding to the positive real roots γi and the negative real roots − γi, respectively, and let Hi = [Ei,Fi ] be the corresponding element in the Cartan subalgebra of 𝔤. Then, the (regular) subalgebra of 𝔤 generated by {Ei,Fi,Hi }, i = 1, ⋅⋅⋅ ,n, is a Kac–Moody algebra with Cartan matrix [C ] ij.

Proof: The proof of this theorem is given in [76]. Note that the Cartan integers (γi|γj) 2(γi|γi) are indeed integers (because the γi’s are positive real roots), which are non-positive (because γi − γj is not a root), so that [C ] ij is a Cartan matrix.


  1. When the Cartan matrix is degenerate, the corresponding Kac–Moody algebra has nontrivial ideals [116Jump To The Next Citation Point]. Verifying that the Chevalley–Serre relations are fulfilled is not sufficient to guarantee that one gets the Kac–Moody algebra corresponding to the Cartan matrix [C ] ij since there might be non-trivial quotients. Situations in which the algebra generated by the set {Ei, Fi,Hi} is the quotient of the Kac–Moody algebra with Cartan matrix [Cij] by a non-trivial ideal were discussed in [96Jump To The Next Citation Point].
  2. If the matrix [Cij] is decomposable, say C = D ⊕ E with D and E indecomposable, then the Kac–Moody algebra 𝕂 𝕄 (C) generated by C is the direct sum of the Kac–Moody algebra 𝕂𝕄 (D ) generated by D and the Kac–Moody algebra 𝕂 𝕄 (E) generated by E. The subalgebras 𝕂 𝕄 (D ) and 𝕂 𝕄 (E ) are ideals. If C has non-vanishing determinant, then both D and E have non-vanishing determinant. Accordingly, 𝕂𝕄 (D ) and 𝕂 𝕄 (E ) are simple [116Jump To The Next Citation Point] and hence, either occur faithfully or trivially. Because the generators Ei are linearly independent, both 𝕂 𝕄 (D) and 𝕂 𝕄 (E) occur faithfully. Therefore, in the above theorem the only case that requires special treatment is when the Cartan matrix C has vanishing determinant.

As we have mentioned above, it is convenient to universally normalize the Killing form of Kac–Moody algebras in such a way that the long real roots have always the same squared length, conveniently taken equal to two. It is then easily seen that the Killing form of any regular Kac–Moody subalgebra of E 10 coincides with the invariant form induced from the Killing form of E10 through the embedding since E10 is “simply laced”. This property does not hold for non-regular embeddings as the example given in Section 4.1 shows: The subalgebra A1 considered there has an induced form equal to minus the standard Killing form.

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