# Beautiful descriptions of exceptional groups

I'm curious about the beautiful descriptions of exceptional simple complex Lie groups and algebras (and maybe their compact forms). By beautiful I mean: simple (not complicated - it means that we need not so many words to describe this).

For $G_2$ we know the automorphisms of octonions and the rolling distribution (and also the intersection of three $Spin_7$-s in $Spin_8$).

For $F_4$ we know the automorphisms of Jordan algebra $H_3(\mathbb O)$ and Lie algebra of commutators of right multiplications in this algebra (see Chevalley-Schafer's paper for details).

For $E_6$ we know the automorphisms of determinant in $H_3(\mathbb O)$ and Lie algebra linearly spanned by right multiplications and $\mathfrak f_4$.

For $\mathfrak f_4$, $\mathfrak e_6$, $\mathfrak e_7$, $\mathfrak e_8$ we know the Vinberg-Freudenthal Magic Square.

What do we know (expressing in a simple form) about $E_7$ and $E_8$?

• You're presumably aware of Baez's paper on the octonions, and of math.ucr.edu/home/baez/octonions Jun 15 '12 at 19:25
• @Andre: of course I read this Jun 15 '12 at 19:26
• A related question is «what are ugly descriptions of exceptional groups?» Jul 6 '12 at 13:17
• @Mariano: I think that "Freudenthal Space Representation" is a very ugly description of $E_7$ Jul 8 '12 at 13:27
• For me descriptions in Adams book "Exceptional Lie Groups" are ugly. He uses arguments that something exists but does not give explicit construction.
– user21230
Jul 30 '12 at 10:13

It is not always clear what one means by 'the simplest description' of one of the exceptional Lie groups. In the examples you've given above, you quote descriptions of these groups as automorphisms of algebraic structures, and that's certainly a good way to do it, but that's not the only way, and one can argue that they are not the simplest in terms of a very natural criterion, which I'll now describe:

Say that you want to describe a subgroup $G\subset \text{GL}(V)$ where $V$ is a vector space (let's not worry too much about the ground field, but, if you like, take it to be $\mathbb{R}$ or $\mathbb{C}$ for the purposes of this discussion). One would like to be able to describe $G$ as the stabilizer of some element $\Phi\in\text{T}(V{\oplus}V^\ast)$, where $\mathsf{T}(W)$ is the tensor algebra of $W$. The tensor algebra $\mathsf{T}(V{\oplus}V^\ast)$ is reducible under $\text{GL}(V)$, of course, and, ideally, one would like to be able to chose a 'simple' defining $\Phi$, i.e., one that lies in some $\text{GL}(V)$-irreducible submodule $\mathsf{S}(V)\subset\mathsf{T}(V{\oplus}V^\ast)$.

Now, all of the classical groups are defined in this way, and, in some sense, these descriptions are as simple as possible. For example, if $V$ with $\dim V = 2m$ has a symplectic structure $\omega\in \Lambda^2(V^\ast)$, then the classical group $\text{Sp}(\omega)\subset\text{GL}(V)$ has codimension $m(2m{-}1)$ in $\text{GL}(V)$, which is exactly the dimension of the space $\Lambda^2(V^\ast)$. Thus, the condition of stabilizing $\omega$ provides exactly the number of equations one needs to cut out $\text{Sp}(\omega)$ in $\text{GL}(V)$. Similarly, the standard definitions of the other classical groups as subgroups of linear transformations that stabilize an element in a $\text{GL}(V)$-irreducible subspace of $\mathsf{T}(V{\oplus}V^\ast)$ are as 'efficient' as possible.

In another direction, if $V$ has the structure of an algebra, one can regard the multiplication as an element $\mu\in \text{Hom}\bigl(V\otimes V,V\bigr)= V^\ast\otimes V^\ast \otimes V$, and the automorphisms of the algebra $A = (V,\mu)$ are, by definition, the elements of $\text{GL}(V)$ whose extensions to $V^\ast\otimes V^\ast \otimes V$ fix the element $\mu$. Sometimes, if one knows that the multiplication is symmetric or skew-symmetric and/or traceless, one can regard $\mu$ as an element of a smaller vector space, such as $\Lambda^2(V^\ast)\otimes V$ or even the $\text{GL}(V)$-irreducible module $\bigl[\Lambda^2(V^\ast)\otimes V\bigr]_0$, i.e., the kernel of the natural contraction mapping $\Lambda^2(V^\ast)\otimes V\to V^\ast$.

This is the now-traditional definition of $G_2$, the simple Lie group of dimension $14$: One takes $V = \text{Im}\mathbb{O}\simeq \mathbb{R}^7$ and defines $G_2\subset \text{GL}(V)$ as the stabilizer of the vector cross-product $\mu\in \bigl[\Lambda^2(V^\ast)\otimes V\bigr]_0\simeq \mathbb{R}^{140}$. Note that the condition of stabilizing $\mu$ is essentially $140$ equations on elements of $\text{GL}(V)$ (which has dimension $49$), so this is many more equations than one would really need. (If you don't throw away the subspace defined by the identity element in $\mathbb{O}$, the excess of equations needed to define $G_2$ as a subgroup of $\text{GL}(\mathbb{O})$ is even greater.)

However, as was discovered by Engel and Reichel more than 100 years ago, one can define $G_2$ over $\mathbb{R}$ much more efficiently: Taking $V$ to have dimension $7$, there is an element $\phi\in \Lambda^3(V^\ast)$ such that $G_2$ is the stabilizer of $\phi$. In fact, since $G_2$ has codimension $35$ in $\text{GL}(V)$, which is exactly the dimension of $\Lambda^3(V^\ast)$, one sees that this definition of $G_2$ is the most efficient that it can possibly be. (Over $\mathbb{C}$, the stabilizer of the generic element of $\Lambda^3(V^\ast)$ turns out to be $G_2$ crossed with the cube roots of unity, so the identity component is still the right group, you just have to require in addition that it fix a volumme form on $V$, so that you wind up with $36$ equations to define the subgroup of codimension $35$.)

For the other exceptional groups, there are similarly more efficient descriptions than as automorphisms of algebras. Cartan himself described $F_4$, $E_6$, and $E_7$ in their representations of minimal dimsension as stabilizers of homogeneous polynomials (which he wrote down explicitly) on vector spaces of dimension $26$, $27$, and $56$ of degrees $3$, $3$, and $4$, respectively. There is no doubt that, in the case of $F_4$, this is much more efficient (in the above sense) than the traditional definition as automorphisms of the exceptional Jordan algebra. In the $E_6$ case, this is the standard definition. I think that, even in the $E_7$ case, it's better than the one provided by the 'magic square' construction.

In the case of $E_8\subset\text{GL}(248)$, it turns out that $E_8$ is the stabilizer of a certain element $\mu\in \Lambda^3\bigl((\mathbb{R}^{248})^\ast\bigr)$, which is essentially the Cartan $3$-form on on the Lie algebra of $E_8$. I have a feeling that this is the most 'efficient' description of $E_8$ there is (in the above sense).

This last remark is a special case of a more general phenomenon that seems to have been observed by many different people, but I don't know where it is explicitly written down in the literature: If $G$ is a simple Lie group of dimension bigger than $3$, then $G\subset\text{GL}({\frak{g}})$ is the identity component of the stabilizer of the Cartan $3$-form $\mu_{\frak{g}}\in\Lambda^3({\frak{g}}^\ast)$. Thus, you can recover the Lie algebra of $G$ from knowledge of its Cartan $3$-form alone.

On 'rolling distributions': You mentioned the description of $G_2$ in terms of 'rolling distributions', which is, of course, the very first description (1894), by Cartan and Engel (independently), of this group. They show that the Lie algebra of vector fields in dimension $5$ whose flows preserve the $2$-plane field defined by $$dx_1 - x_2\ dx_0 = dx_2 - x_3\ dx_0 = dx_4 - {x_3}^2\ dx_0 = 0$$ is a $14$-dimensional Lie algebra of type $G_2$. (If the coefficients are $\mathbb{R}$, this is the split $G_2$.) It is hard to imagine a simpler definition than this. However, I'm inclined not to regard it as all that 'simple', just because it's not so easy to get the defining equations from this and, moreover, the vector fields aren't complete. In order to get complete vector fields, you have to take this $5$-dimensional affine space as a chart on a $5$-dimensional compact manifold. (Cartan actually did this step in 1894, as well, but that would take a bit more description.) Since $G_2$ does not have any homogeneous spaces of dimension less than $5$, there is, in some sense, no 'simpler' way for $G_2$ to appear.

What doesn't seem to be often mentioned is that Cartan also described the other exceptional groups as automorphisms of plane fields in this way as well. For example, he shows that the Lie algebra of $F_4$ is realized as the vector fields whose flows preserve a certain 8-plane field in 15-dimensional space. There are corresponding descriptions of the other exceptional algebras as stabilizers of plane fields in other dimensions. K. Yamaguchi has classified these examples and, in each case, writing down explicit formulae turns out to be not difficult at all. Certainly, in each case, writing down the defining equations in this way takes less time and space than any of the algebraic methods known.

Further remark: Just so this won't seem too mysterious, let me describe how this goes in general: Let $G$ be a simple Lie group, and let $P\subset G$ be a parabolic subgroup. Let $M = G/P$. Then the action of $P$ on the tangent space of $M$ at $[e] = eP\in M$ will generally preserve a filtration $$(0) = V_0 \subset V_1\subset V_2\subset \cdots \subset V_{k-1} \subset V_k = T_{[e]}M$$ such that each of the quotients $V_{i+1}/V_i$ is an irreducible representation of $P$. Corresponding to this will be a set of $G$-invariant plane fields $D_i\subset TM$ with the property that $D_i\bigl([e]\bigr) = V_i$. What Yamaguchi shows is that, in many cases (he determines the exact conditions, which I won't write down here), the group of diffeomorphisms of $M$ that preserve $D_1$ is $G$ or else has $G$ as its identity component.

What Cartan does is choose $P$ carefully so that the dimension of $G/P$ is minimal among those that satisfy these conditions to have a nontrivial $D_1$. He then takes a nilpotent subgroup $N\subset G$ such that $T_eG = T_eP \oplus T_eN$ and uses the natural immersion $N\to G/P$ to pull back the plane field $D_1$ to be a left-invariant plane field on $N$ that can be described very simply in terms of the multiplication in the nilpotent group $N$ (which is diffeomorphic to some $\mathbb{R}^n$). Then he verifies that the Lie algebra of vector fields on $N$ that preserve this left-invariant plane field is isomorphic to the Lie algebra of $G$. This plane field on $N$ is bracket generating, i.e., 'non-holonomic' in the classical terminology. This is why it gets called a 'rolling distribution' in some literature. In the case of the exceptional groups $G_2$ and $F_4$, the parabolic $P$ is of maximal dimension, but this is not so in the case of $E_6$, $E_7$, and $E_8$, if I remember correctly.

• @robot: Yes, of course, this is very easy: If $\tau\in V^\ast$ and $\delta\in\mathsf{S}^3(V^\ast)$ are given as you describe, one shows that there is an $e\in V$ such that $\tau(v) = \delta(e,e,v)$. Then, defining $\gamma\in\mathsf{S}^2(V^\ast)$ by $\gamma(v,w)=\delta(e,v,w)$ gives a nondegenerate quadratic form on $V$, thereby establishing an isomorphism between $V$ and $V^*$. Then the algebra multiplication $\mu\in\mathsf{S}^2(V^\ast)\otimes V$ is just $\delta$ itself under the mapping $\mathsf{S}^3(V^\ast)\to \mathsf{S}^2(V^\ast)\otimes V^\ast \to \mathsf{S}^2(V^\ast)\otimes V$. Jun 17 '12 at 13:53
• @robot: Cartan's $E_7$-defining quartic is not in terms of octonions: Let $W$ be $8$-dimensional with volume form $\nu\in\Lambda^8(W^\ast)$. Set $V = \Lambda^2(W)\oplus\Lambda^2(W^\ast)$. For $(x,\xi)\in V$, let $x\cdot\xi\in \text{End}(W)$ be the image under the contraction $\Lambda^2(W)\otimes\Lambda^2(W^\ast)\to W\otimes W^\ast = \text{End}(W)$. Set $q(x,\xi) = \text{Pf}(x) + \text{tr}\bigl((x\cdot\xi)^2\bigr) + \text{Pf}(\xi)$. Then the identity component of the stabilizer of $q\in\mathsf{S}^4(V^\ast)$ is $E_7$. ($\nu$ is used to turn $\text{Pf}(x)$ and $\text{Pf}(\xi)$ into scalars). Jun 17 '12 at 14:19
• @zroslav: Sure. To my knowledge, the most modern reference with complete proofs is by Keizo Yamaguchi, Differential systems associated with simple graded Lie algebras. Progress in differential geometry, 413–494, Adv. Stud. Pure Math., 22, Math. Soc. Japan, Tokyo, 1993. This proves general results about Lie groups defined as automorphisms of homogeneous plane fields, but he doesn't write out the formulae as explicitly as one might like sometimes. In particular, one doesn't see the simple explicit formulae that Cartan wrote down, but one can work it out without much difficulty in each case. Jun 18 '12 at 15:04
• @zroslav: Irrelevant side remark: One can consider these distributions $D_i$ (plus some appropriate natural conditions) on a general manifold $M$ and then it can be proven that they induce a regular normal Cartan geometry on $M$ of type $(G,P)$. To put it a little bit more precisely: "The category of manifolds with certain filtered bundles is equivalent to the category of normal regular Cartan geometries of type $(G,P)$." Jun 18 '12 at 16:48
• continued: Thus you can approach the plane fields as Cartan geometries over the plane. If you view the plane as the big affine cell in G/P (image of the opposite unipotent radical to the unipotent radical of P under exponentiation and projection)and pull-back the Maurer-Cartan form you get precisely what professor Bryant describes in his last comment. So you can see these plane fields as special (flat) examples of more general geometric structures. For details see recent book Parabolic Geometries by Cap & Slovák. Jun 18 '12 at 16:52

There's a nice construction of the $E_8$ Lie algebra due to Borcherds based on methods from vertex operator algebras, but with no understanding of vertex algebras needed. See p. 152 of these notes from a course by Borcherds and others. See also section 7.4 of notes by Johnson-Freyd. The idea is to start with the root system and root lattice, and construct the Lie algebra using Serre's relations. But with the relations there is a sign ambiguity, so one passes to a 2-fold cover of the lattice to resolve the sign issues, and check that everything works. Once you have $E_8$, you can find $E_7$ sitting inside it. Since the lattice is self-dual (simply-connected), you can just exponentiate to get the Lie group.

• Incidentally, the sign ambiguity can be traced to the fact that the Weyl group is not a subgroup of $G$ but only a subquotient, and "hence" there can't be a functorial construction of groups from their Dynkin diagrams. The double cover of the lattice carries an action of the actual subgroup of $G$ one uses instead of the Weyl group. Jun 16 '12 at 14:40
• @Allen: There is a functorial construction of Lie groups from their Dynkin diagrams (via the Serre relations). There isn't a functorial construction of Lie groups from their root systems. Jun 17 '12 at 10:20
• Oops, of course, the choice of simple roots exactly breaks the Weyl group symmetry that one must break. Thanks André. Jun 26 '12 at 2:31
• As of July 2016, the link to Geraschenko's notes from Borcherd's 2006 course is now broken. My notes lift directly from that one. The current link is math.northwestern.edu/~theojf/LieQuantumGroups.pdf, but this is likely to change in the next couple years. Borcherds has a fantastic (competing?) collection of notes available at math.berkeley.edu/~reb/courses/261/index.html. Jul 14 '16 at 23:06
• @მამუკა ჯიბლაძე : here is the new location perimeterinstitute.ca/personal/tjohnsonfreyd/…
– BS.
Aug 26 '17 at 15:58

If you start from basics, then J.Tits' "Local approach to buildings" [1] would certainly win, as you won't even need a definition of a group to describe the natural geometries for the exceptional Lie groups.

[1] Tits, J. "A local approach to buildings", The geometric vein: The Coxeter Festschrift, Springer-Verlag, 1981, pp. 519–547

Here is a description that is new and you can judge whether it is beautiful. Given any simple complex Lie group $G$ and almost any irreducible representation $V$, the stabilizer of almost any $G$-invariant polynomial $f$ on $V$ has identity component $G$.

Cartan's examples

• $G = E_6$, $V$ of dimension 27, and $f$ cubic; or
• $G = E_7$, $V$ of dimension 56, and $f$ quartic

are very special cases of this general principle. (They are very special because in these cases the ring of $G$-invariant polynomials on $V$ is generated by $f$.)

In the case of the group $E_8$, you can take $V$ to be the Lie algebra $\mathfrak{e}_8$. Then the ring of invariant polynomial functions is a polynomial ring with generators of degree 2 (the Killing quadratic form), 8, 12, 14, 18, 20, 24, 30. The new result says: If you take $f$ to be any of the generators besides the Killing form, then $E_8$ is the identity component of the stabilizer of $f$.

This is a very concrete description of $E_8$, because an explicit formula for the degree 8 polynomial is already in the literature (Cederwall and Palmkvist - The octic $E_8$ invariant (arXiv)).

Alternatively, there is a commutative, nonassociative, and $E_8$-invariant product on its 3875-dimensional irreducible representation, and the automorphism group of this nonassociative ring is $E_8$.

There is also a variation on the result I mentioned at the beginning that may be worth mentioning: you can also realize each simple complex Lie group $G$, up to isogeny, as the stabilizer of a cubic form on some representation. For $E_8$, you can take the cubic form to be the one defining the multiplication on the 3875-dimensional representation.

The new results mentioned here are from Garibaldi and Guralnick - Simple groups stabilizing polynomials (MSN, arXiv).

• These are indeed beautiful results. Meanwhile, note that the exterior $3$-form $\mu\in\Lambda^3(\frak{g}^*)$ that defines $\mathrm{E}_8$ in $\mathrm{End}(\frak{g})$ lies in a much smaller vector space than the symmetric octic polynomial $p_8\in S^8(\frak{g}^*)$, the Lie bracket $\lambda\in\frak{g}\otimes\Lambda^2(\frak{g}^*)$, or the commutative, nonassociative product $\pi\in V\otimes S^2(V^*)$, where $\dim V = 3875$ that you mentioned. Among these four definitions, stabilizing $\mu$ defines $\mathrm{E}_8$ via the smallest number of equations. How close to $248\cdot 247$ can one get? Nov 12 '20 at 19:42

Personally I like the definition in Barton, Sudbery paper (thank you, Bruce for adding the reference):

It uses triality algebra based on $\mathbb R, \mathbb C, \mathbb H, \mathbb O$ composition algebras. Using this one can construct all compact and non-compact exceptional Lie algebras.

Tits-Freudenthal magic square correspond to square of algebras:

$\begin{matrix} R\otimes R & R\otimes C & R\otimes H & R\otimes O \\ C\otimes R & C\otimes C & C\otimes H & C\otimes O \\ H\otimes R & H\otimes C & H\otimes H & H\otimes O \\ O\otimes R & O\otimes C & O\otimes H & O\otimes O \\ \end{matrix}$

You can replace composition algebra $A$ with split version $\tilde A$ to obtain non compact version.

Lie algebra in position $A\otimes B$ is $TriA + TriB + A\otimes B + A\otimes B + A\otimes B$. The triality Lie algebra is equal to $Der A+2A'$ which is equal to $0,so_2+so_2, so_3+so_3+so_3, so_8$ for the four composition algebras listed above. The bracket is defined in mentioned paper. To obtain $f_4$ with compact $spin_9$ we should change sign in last two $A\otimes B$.

Explanation

I would like to add few sentences why I think this is beautiful description of exceptional Lie groups. It is rather description of exceptional Lie algebras, not groups. The groups can be obtained from Lie algebras by using exponential map.

First reason is that all four exceptional Lie algebras: $f_4$, $e_6$, $e_7$, $e_8$ are obtained in uniform way. Second reason is that it is elegant and fairly easy to understand the bracket. You should comprehend the notion of triality in composition algebra. Third reason is that you can easily see the symmetry of Freudenthal-Tits "magic" square of Lie algebras. It is no more "magic" as it was in original construction of Tits and Freudenthal where Jordan algebra was used.

We can look at $n=2$ algebras which is "younger brother" of magic square for $n=3$. The exceptional symmetric spaces are obtained as quotient of entry in magic square for $n=3$ with corresponding entry for $n=2$. Placing one square on top of another and preparing base square for $n=1$ with $Tri A+Tri B$ we obtain "magic cube" of Lie algebras. Exceptional symmetric spaces can be obtained as quotients of neighbour points in magic cube.

We can also replace given algebra $A$ by split version $\tilde A$ as I mentioned above. This way we can obtain non-compact versions of exceptional Lie algebras.

Future development

I would like to add what is still missing in this nice picture. It would be good to have focus on Lie group, not on Lie algebra. The geometry is hidden in the group. Lie algebra was created as algebraic tool to classify the groups.

It would be good to have uniform definition of exceptional symmetric spaces. For example Huang thesis contain definition of symmetric spaces as grassmanians.

It is not easy to define finite groups of Lie type for exceptional Lie groups. It would be good to have something also working for finite fields.

• When you say "all compact and non-compact exceptional Lie algebras", do you mean "all real forms"? Jul 6 '12 at 12:58
• There are three real forms of $F_4$ and the Barton & Sudbery paper constructs two of them. Jul 6 '12 at 13:04
• Bruce, I am only intrested in real Lie algebras. To obtain f4 with compact spin9 I have changed sign in last two A*B.
– user21230
Jul 30 '12 at 10:12