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 G_{2}, F_{4}, E_{6}, E_{7}, E_{8}. 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 (A_{1}, A_{2} and D_{4}). In particular, for λ=−1/2, −1/3, −1/5, its values are 78, 133, 248, the dimensions of E_{6}, E_{7}, and E_{8}.

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 E_{7.5}. 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 E_{7}. It is an extension of a Heisenberg algebra associated with E_{7} with the algebra E_{7} itself, and the numerology is

133+1+56 = 190.

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About Peter Cameron

I count all the things that need to be counted.

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