About the definition of E8, and Rosenfeld's "Geometry of Lie groups"

Question

I've been searching the literature for a direct definition of the group $E_8$ (over a general field, but even a definition of just one incarnation would be great). I knew (from talking to people) that there's probably nothing available, but I'm confused about one point.

By "direct definition", I mean something other than a definition of $E_8$ as a group of automorphisms of its own Lie algebra.

Something promising is the "octo-octonionic projective space" $(\mathbb{O} \otimes \mathbb{O} )\mathbb{P}^2$ -- the group of isometries of the latter is meant to be a form of $E_8$. In his paper on the octonions, John Baez mentions this, but warns that $(\mathbb{O} \otimes \mathbb{O} )\mathbb{P}^2$ can only be defined in terms of $E_8$, so this is circular, and he adds "alas, nobody seems to know how to define [it] without first defining $E_8$. Thus this group remains a bit enigmatic."

The existence of the book Geometry of Lie groups by Boris Rosenfeld confuses me. In it, he claims to construct the plane, calling it $(\mathbb{O} \otimes \mathbb{O} ) {\simeq \atop S^2}$ (I cannot even reproduce it well in Latex). See Theorem 7.16 in particular.

The problem is that each object, in this book, is claimed to be definable "by direct analogy" with some other object, itself usually not quite defined in full, and so on. I'm having an awful lot of trouble reading Rosenfeld's book. In the end (see 7.7.3) he claims that everything can be carried out over a finite field, yielding a definition of $E_8(q)$. But I cannot find the details anywhere -- the thing is, I don't even know if I'm reading a survey, or a complete treatment with proofs that escape me.

Is anybody on MO familiar with Rosenfeld's book? Or is there an alternative reference for this mysterious "octo-octonionic plane"?

Answer 1

Here's an easy, direct definition of $E_8$.

The compact Lie group $E_8$ is the colimit in the category of topological groups of the following diagram of groups $$ {\scriptstyle\begin{matrix} &SU(2)&\\[-1mm] &\downarrow&\\[-1mm] &SU(3)&\\[-1mm] &\uparrow&\\[-1mm] SU(2)\to SU(3) \leftarrow SU(2)&\!\!\!\!\!\to SU(3) \leftarrow SU(2)\to SU(3) \leftarrow\!\!\!& SU(2)\to SU(3) \leftarrow SU(2)\to SU(3) \leftarrow SU(2)\to SU(3) \leftarrow SU(2)\\ \end{matrix}}, $$ modulo the normal subgroup $N$ generated by commutators of non-adjacent $SU(2)$'s.

Namely: $$ E_8=\mathrm{colim}\left(\scriptstyle\begin{matrix} SU(2)&\!\!\!\!\!\to SU(3) \leftarrow SU(2)\to SU(3) \leftarrow\!\!\!& SU(2)&&SU(2)\\ \downarrow&\downarrow&\downarrow&&\downarrow\\ SU(3)&SU(3)&SU(3)&&SU(3)\\ \uparrow&\uparrow&\uparrow&&\uparrow\\ SU(2)&SU(2)&SU(2)&\!\!\!\!\!\to SU(3) \leftarrow\!\!\!& SU(2) \end{matrix}\right)/N. $$

Here, whenever we see a subdiagram $SU(2)\to SU(3) \leftarrow SU(2)$, the two maps are given by $\big(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\big)\mapsto \big(\begin{smallmatrix}a&b&0\\c&d&0\\0&0&1\end{smallmatrix}\big)$ and $\big(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\big)\mapsto \big(\begin{smallmatrix}1&0&0\\0&a&b\\0&c&d\end{smallmatrix}\big)$.

If you want the complex Lie group $E_8$, use $SL(2)$'s and $SL(3)$'s instead of $SU(2)$'s and $SU(3)$'s.

If you want the group of $k$-points of the algebraic group $E_8$, use $SL(2,k)$'s and $SL(3,k)$'s ($k$ an arbitrary ring). [Edit: this doesn't work for arbitrary rings — see the comments below]

Answer 2

The algebraic group $E_8$ is the group of automorphisms of the $E_8$ lattice vertex algebra, by Frenkel-Kac and Segal. This vertex algebra has a self-dual integral form, so the construction works over arbitrary commutative rings. To construct the vertex algebra, one only needs the rank 8 unimodular $E_8$ lattice, not the 248-dimensional $E_8$ Lie algebra.

Answer 3

Recently Lusztig gave a much simpler definition of $E_8$ (and all the simple Lie algebras/groups) that avoids the usual sign issue with the standard Chevalley/Serre construction. See Lusztig - On conjugacy classes in the group $E_8$ and Geck - On the construction of semisimple Lie algebras and Chevalley groups. I think this construction of Lusztig's is not as well known as it should be.

Answer 4

Here's one of my favorite definitions, that gives a construction of (many forms) of $E_8$ as well as a plethora of other exceptional groups. I hope it's ok to construct the Lie algebras here, and let the groups be their automorphism groups.

Begin with a field $k$ of characteristic zero. Let $B$ and $C$ be composition algebras over $k$. I.e., $B$ and $C$ are $k$-algebras, isomorphic to $\bar k$ or $(\bar k \times \bar k)$ or $M_2(\bar k)$ or $O_{\bar k}$ (split octonion algebra over $\bar k$) after passing to an algebraic closure $\bar k / k$. For the split $E_8$, take $B = C = O_k$.

Let $A = B \otimes_k C$, viewed as a $k$-algebra with involution, typically nonassociative. For $E_8$, $A = O_k \otimes_k O_k$. This is an important example of a "structurable algebra," defined and studied by Bruce Allison.

Allison, B. N., Tensor products of composition algebras, Albert forms and some exceptional simple Lie algebras, Trans. Am. Math. Soc. 306, No. 2, 667-695 (1988). ZBL0649.17006.

Now I'll review the construction of a Lie algebra from $A$, following section 1 of the above reference. If $x \in A$, define $L_x \in End_k(A)$ by $L_x(y) = xy$. Define $R_x \in End_k(A)$ by $R_x(y) = yx$. Define $V_{x,y} \in End_k(A)$ by $$V_{x,y}(z) = (x \bar y)z + (z \bar y)x - (z \bar x)y.$$

Let ${\mathfrak g}_0$ be the $k$-subspace of $End_k(A)$ generated by the endomorphisms $V_{x,y}$ for all $x,y \in A$. A key identity (for all structurable algebras) implies that ${\mathfrak g}_0$ is not just a $k$-subspace... it's a Lie algebra!

Lie ${\mathfrak g}_{\pm 1} = A$. Let ${\mathfrak g}_{\pm 2} = A_0 = \{ a \in A : a + \bar a = 0 \}$. Define $${\mathfrak g} = {\mathfrak g}_{-2} \oplus {\mathfrak g}_{-1} \oplus {\mathfrak g}_{0} \oplus {\mathfrak g}_1 \oplus {\mathfrak g}_2.$$ Allison defines a 5-graded Lie algebra structure on this direct sum, directly from the algebra-with-involution structure on $A$. We've already seen the Lie bracket on ${\mathfrak g}_0$. The bracket $[X,Y]$ with $X \in {\mathfrak g}_0$ and $Y \in {\mathfrak g}_1$ is the obvious one. For the rest, I'll refer to Section 1 of Allison's paper, since it's a bit tedious (and takes up a half-page).

Anyways, for $A = O \otimes O$, the 64-dimensional structurable algebra, this gives an explicit construction of a Lie algebra of type $E_8$. In general, one gets a nice construction of 5-graded Lie algebras... it's very helpful for understanding "Heisenberg parabolics" for example.

Answer 5

The most direct definition is probably as (the identity component of) the stabilizer of a tensor. More precisely, in The octic $E_8$ invariant by Cederwall and Palmkvist an explicit symmetric 8-form $f$ on $\mathbb{R}^{248}$ is constructed, having the compact $E_8$ (times $\{\pm1\}$) as the stabilizer in $GL(248, \mathbb{R})$. It is shown, moreover, in Simple groups stabilizing polynomials by Garibaldi and Guralnick, that any $E_8$-invariant 8-form that is not a scalar multiple of the fourth power of the Killing form $\kappa$ will do, and that the stabilizer in $GL(248, \mathbb{C})$ is the complex form (times $\mu_8$). The invariant 8-forms can be produced from arbitrary 8-forms by averaging with respect to the Haar measure (and almost all of them are not proportional to $\kappa^4$).

Answer 6

There seems to be a construction of $\mathfrak{e}_8$ using some sort of geometrical objects in Configurations of lines and models of Lie algebras by Laurent Manivel.