We prove that among all finite-dimensional simple Lie algebras of ADE type (the simply-laced Dynkin diagrams), the exceptional algebra E7 is uniquely characterized by the property that the sum of its dimension and the maximal Kac label of its untwisted affine extension equals 137. The proof proceeds by exhaustive computation over the ADE classification. We note two ancillary number-theoretic properties of 137 as remarks.
The classification of finite-dimensional simple Lie algebras over C by Dynkin diagrams is one of the central results of 20th-century mathematics. The simply-laced (ADE) families comprise the infinite series An (n ≥ 1) and Dn (n ≥ 4), together with the three exceptional algebras E6, E7, and E8.
Each simple Lie algebra g admits an untwisted affine extension g(1), whose extended Dynkin diagram carries a set of positive integers called Kac labels (also known as marks or numerical labels). These labels encode the coefficients of the imaginary root in the basis of simple roots of the affine algebra, and they determine the structure of the affine Weyl group.
We define a numerical invariant:
For a finite-dimensional simple Lie algebra g, let
where (a0, a1, …, an) are the Kac labels of the untwisted affine extension g(1).
We prove that S(g) = 137 has a unique solution among all ADE algebras, and that solution is g = E7.
2.1. Let g be a finite-dimensional simple Lie algebra over C with Dynkin diagram of ADE type. The dimension of g is:
| An | dim = n(n + 2), for n ≥ 1 |
| Dn | dim = n(2n − 1), for n ≥ 4 |
| E6 | dim = 78 |
| E7 | dim = 133 |
| E8 | dim = 248 |
2.2. The untwisted affine extension g(1) is obtained by adjoining an affine node (labeled 0) to the Dynkin diagram. The Kac labels (a0, a1, …, an) are the unique positive integers, with gcd = 1, satisfying the null eigenvector equation of the extended Cartan matrix:
These labels are standard and appear in Kac [1, Ch. 4, Table Aff 1] and Fuchs & Schweigert [6, Table B.3]. We adopt the convention that a0 is the label of the affine node.
2.3. For the ADE types, the Kac labels are:
| An(1) | All labels equal 1. That is, ai = 1 for i = 0, 1, …, n. |
| Dn(1) | a0 = 1, a1 = 1, ai = 2 for 2 ≤ i ≤ n − 2, an−1 = 1, an = 1. |
| E6(1) | (1, 1, 2, 3, 2, 1, 2). Max = 3. |
| E7(1) | (1, 2, 3, 4, 3, 2, 1, 2). Max = 4. |
| E8(1) | (1, 2, 3, 4, 5, 6, 4, 2, 3). Max = 6. |
Among all finite-dimensional simple Lie algebras of ADE type, E7 is the unique algebra satisfying
where (ai) are the Kac labels of the untwisted affine extension.
The proof is by exhaustive computation over the ADE classification.
For An(1), all Kac labels equal 1, so maxi(ai) = 1. Thus:
Setting (n + 1)2 = 137: since 112 = 121 and 122 = 144, there is no integer solution. Hence no An algebra achieves S = 137.
For Dn(1), the maximal Kac label is 2 (attained on the interior nodes). Thus:
Setting n(2n − 1) + 2 = 137 gives n(2n − 1) = 135. Testing:
| n | n(2n − 1) | S(Dn) |
|---|---|---|
| 4 | 28 | 30 |
| 5 | 45 | 47 |
| 6 | 66 | 68 |
| 7 | 91 | 93 |
| 8 | 120 | 122 |
| 9 | 153 | 155 |
Since n(2n − 1) is strictly increasing and jumps from 120 to 153, skipping 135, there is no integer solution. Hence no Dn algebra achieves S = 137.
dim(E6) = 78. The Kac labels of E6(1) are (1, 1, 2, 3, 2, 1, 2), with maximum 3. Thus:
dim(E7) = 133. The Kac labels of E7(1) are (1, 2, 3, 4, 3, 2, 1, 2), with maximum 4. Thus:
dim(E8) = 248. The Kac labels of E8(1) are (1, 2, 3, 4, 5, 6, 4, 2, 3), with maximum 6. Thus:
The following table collects all results:
| Algebra | dim(g) | max Kac label | S(g) |
|---|---|---|---|
| An | n(n+2) | 1 | (n+1)2 |
| Dn | n(2n−1) | 2 | n(2n−1)+2 |
| E6 | 78 | 3 | 81 |
| E7 | 133 | 4 | 137 |
| E8 | 248 | 6 | 254 |
For An, S is always a perfect square, and 137 is not a perfect square. For Dn, S = n(2n−1)+2 skips 137 between n = 8 and n = 9. Among the exceptionals, only E7 yields S = 137.
This completes the proof.
The number 137 is the unique prime p < 106 such that ordp(10) = bit_length(p), where ordp(10) is the multiplicative order of 10 modulo p and bit_length(p) = ⌊log2(p)⌋ + 1.
That this is unique among primes below 106 can be verified computationally. We omit the search but note that the condition simultaneously constrains the prime in two unrelated numeral systems (decimal and binary), making coincidences sparse.
Write 137 = 20 + 23 + 27. The exponent set is {0, 3, 7}. Incrementing each by 1 gives {1, 3, 7} — but more directly, the digits of 137 in decimal are {1, 3, 7}. In the standard labeling of the Fano plane PG(2, F2) with points {1, 2, 3, 4, 5, 6, 7}, the set {1, 3, 7} is a line (one of the seven lines of the projective plane over F2).
To verify: the seven lines of PG(2, F2) in one standard labeling are {1,2,4}, {2,3,5}, {3,4,6}, {4,5,7}, {5,6,1}, {6,7,2}, {7,1,3}. The set {1,3,7} = {7,1,3} is indeed the last of these.
The Fano plane is the incidence geometry of the octonions and governs the multiplication table of the imaginary units e1, …, e7. The exceptional Lie algebras E6, E7, E8 are intimately connected to the octonions through the Freudenthal–Tits magic square and the construction of their root systems. That the decimal digits of S(E7) = 137 form a Fano line is a concrete — if coincidental — echo of this structural relationship.
The invariant S(g) = dim(g) + max(Kac label) has, to the author’s knowledge, not appeared previously in the literature. It combines a global invariant (dimension) with a local invariant (the largest coefficient in the null root of the affine extension). That E7 is the unique ADE algebra where this sum equals a number with the arithmetic properties described above is, at minimum, a curiosity worth recording.