## E7.5

I’m at the workshop on Algebra, Combinatorics, Dynamics and Applications in Belfast this week. Despite the extremely friendly title, this is a meeting that takes me well out of my comfort zone. An excellent thing: I learned about categorification and about cluster algebras here in the past, and this time I feel myself better informed about Poisson algebras and semi-classical limits of quantum algebras.

But the one small thing I want to talk about is some numerology which I learned from a talk by Dmitriy Rumynin. There is much more to it than numerology, as the talk showed; but I will stick to that.

As I have discussed in my series of posts about the ADE affair, it is well known that there are four infinite series of simple Lie algebras over the complex numbers and a finite family of sporadic examples going by the names G2, F4, E6, E7, E8. Pierre Deligne proposed that they are not sporadic but actually part of a family. Indeed, he gave a formula, namely

−2(λ+5)(λ−6)/λ(λ−1),

which for various negative rational values of λ evaluates to the dimensions of all the exceptional Lie algebras and a few small classical ones too for good measure (A1, A2 and D4). In particular, for λ=−1/2, −1/3, −1/5, its values are 78, 133, 248, the dimensions of E6, E7, and E8.

Now someone pointed out that there is an obvious gap here, namely λ=−1/4, for which the formula evaluates to 190. The mythical Lie algebra of dimension 190 has been named E7.5. Dmitriy didn’t want to talk about this, but when someone asked him “Does it exist?” he replied, “It is more real than I am: it has a Wikipedia page, I don’t!”.

Indeed, there is a Lie algebra of dimension 190; it is not simple, but is related to E7. It is an extension of a Heisenberg algebra associated with E7 with the algebra E7 itself, and the numerology is

133+1+56 = 190.