You are currently browsing the tag archive for the ‘strong law of small numbers’ tag.

For sake of concreteness we will work here over the complex numbers ${{\bf C}}$, although most of this discussion is valid for arbitrary algebraically closed fields (but some care needs to be taken in characteristic ${2}$, as always, particularly when defining the orthogonal and symplectic groups). Then one has the following four infinite families of classical Lie groups for ${n \geq 1}$:

1. (Type ${A_n}$) The special linear group ${SL_{n+1}({\bf C})}$ of volume-preserving linear maps ${T: {\bf C}^{n+1} \rightarrow {\bf C}^{n+1}}$.
2. (Type ${B_n}$) The special orthogonal group ${SO_{2n+1}({\bf C})}$ of (orientation preserving) linear maps ${T: {\bf C}^{2n+1} \rightarrow {\bf C}^{2n+1}}$ preserving a non-degenerate symmetric form ${\langle, \rangle: {\bf C}^{2n+1} \times {\bf C}^{2n+1} \rightarrow {\bf C}}$, such as the standard symmetric form

$\displaystyle \langle (z_1,\ldots,z_{2n+1}), (w_1,\ldots,w_{2n+1}) \rangle := z_1 w_1 + \ldots + z_{2n+1} w_{2n+1}.$

(this is the complexification of the more familiar real special orthogonal group ${SO_{2n+1}({\bf R})}$).

3. (Type ${C_n}$) The symplectic group ${Sp_{2n}({\bf C})}$ of linear maps ${T: {\bf C}^{2n} \rightarrow {\bf C}^{2n}}$ preserving a non-degenerate antisymmetric form ${\omega: {\bf C}^{2n} \times {\bf C}^{2n} \rightarrow {\bf C}}$, such as the standard symplectic form

$\displaystyle \omega((z_1,\ldots,z_{2n}), (w_1,\ldots,w_{2n})) := \sum_{j=1}^n z_j w_{n+j} - z_{n+j} w_j.$

4. (Type ${D_n}$) The special orthogonal group ${SO_{2n}({\bf C})}$ of (orientation preserving) linear maps ${{\bf C}^{2n} \rightarrow {\bf C}^{2n}}$ preserving a non-degenerate symmetric form ${\langle,\rangle: {\bf C}^{2n} \times {\bf C}^{2n} \rightarrow {\bf C}}$ (such as the standard symmetric form).

For this post I will abuse notation somewhat and identify ${A_n}$ with ${SL_{n+1}({\bf C})}$, ${B_n}$ with ${SO_{2n+1}({\bf C})}$, etc., although it is more accurate to say that ${SL_{n+1}({\bf C})}$ is a Lie group of type ${A_n}$, etc., as there are other forms of the Lie algebras associated to ${A_n, B_n, C_n, D_n}$ over various fields. Over a non-algebraically closed field, such as ${{\bf R}}$, the list of Lie groups associated with a given type can in fact get quite complicated; see for instance this list. One can also view the double covers ${Spin_{2n+1}({\bf C})}$ and ${Spin_{2n}({\bf C})}$ of ${SO_{2n+1}({\bf C})}$, ${SO_{2n}({\bf C})}$ (i.e. the spin groups) as being of type ${B_n, D_n}$ respectively; however, I find the spin groups less intuitive to work with than the orthogonal groups and will therefore focus more on the orthogonal model.

The reason for this subscripting is that each of the classical groups ${A_n, B_n, C_n, D_n}$ has rank ${n}$, i.e. the dimension of any maximal connected abelian subgroup of simultaneously diagonalisable elements (also known as a Cartan subgroup) is ${n}$. For instance:

1. (Type ${A_n}$) In ${SL_{n+1}({\bf C})}$, one Cartan subgroup is the diagonal matrices in ${SL_{n+1}({\bf C})}$, which has dimension ${n}$.
2. (Type ${B_n}$) In ${SO_{2n+1}({\bf C})}$, all Cartan subgroups are isomorphic to ${SO_2({\bf C})^n \times SO_1({\bf C})}$, which has dimension ${n}$.
3. (Type ${C_n}$) In ${Sp_{2n}({\bf C})}$, all Cartan subgroups are isomorphic to ${SO_2({\bf C})^n \leq Sp_2({\bf C})^n \leq Sp_{2n}({\bf C})}$, which has dimension ${n}$.
4. (Type ${D_n}$) in ${SO_{2n}({\bf C})}$, all Cartan subgroups are isomorphic to ${SO_2({\bf C})^n}$, which has dimension ${n}$.

(This same convention also underlies the notation for the exceptional simple Lie groups ${G_2, F_4, E_6, E_7, E_8}$, which we will not discuss further here.)

With two exceptions, the classical Lie groups ${A_n,B_n,C_n,D_n}$ are all simple, i.e. their Lie algebras are non-abelian and not expressible as the direct sum of smaller Lie algebras. The two exceptions are ${D_1 = SO_2({\bf C})}$, which is abelian (isomorphic to ${{\bf C}^\times}$, in fact) and thus not considered simple, and ${D_2 = SO_4({\bf C})}$, which turns out to “essentially” split as ${A_1 \times A_1 = SL_2({\bf C}) \times SL_2({\bf C})}$, in the sense that the former group is double covered by the latter (and in particular, there is an isogeny from the latter to the former, and the Lie algebras are isomorphic).

The adjoint action of a Cartan subgroup of a Lie group ${G}$ on the Lie algebra ${{\mathfrak g}}$ splits that algebra into weight spaces; in the case of a simple Lie group, the associated weights are organised by a Dynkin diagram. The Dynkin diagrams for ${A_n, B_n, C_n, D_n}$ are of course well known, and can be found for instance here.

For small ${n}$, some of these Dynkin diagrams are isomorphic; this is a classic instance of the tongue-in-cheek strong law of small numbers, though in this case “strong law of small diagrams” would be more appropriate. These accidental isomorphisms then give rise to the exceptional isomorphisms between Lie algebras (and thence to exceptional isogenies between Lie groups). Excluding those isomorphisms involving the exceptional Lie algebras ${E_n}$ for ${n=3,4,5}$, these isomorphisms are

1. ${A_1 = B_1 = C_1}$;
2. ${B_2 = C_2}$;
3. ${D_2 = A_1 \times A_1}$;
4. ${D_3 = A_3}$.

There is also a pair of exceptional isomorphisms from (the ${Spin_8}$ form of) ${D_4}$ to itself, a phenomenon known as triality.

These isomorphisms are most easily seen via algebraic and combinatorial tools, such as an inspection of the Dynkin diagrams (see e.g. this Wikipedia image). However, the isomorphisms listed above can also be seen by more “geometric” means, using the basic representations of the classical Lie groups on their natural vector spaces (${{\bf C}^{n+1}, {\bf C}^{2n+1}, {\bf C}^{2n}, {\bf C}^{2n}}$ for ${A_n, B_n, C_n, D_n}$ respectively) and combinations thereof (such as exterior powers). (However, I don’t know of a simple way to interpret triality geometrically; the descriptions I have seen tend to involve some algebraic manipulation of the octonions or of a Clifford algebra, in a manner that tended to obscure the geometry somewhat.) These isomorphisms are quite standard (I found them, for instance, in this book of Procesi), but it was instructive for me to work through them (as I have only recently needed to start studying algebraic group theory in earnest), and I am recording them here in case anyone else is interested.