Ilia Smilga: Some tables of real simple Lie groups and of their real finite-dimensional representations
Homepage Page d'accueil
Here are some tables classifying simple Lie groups and their representations. These are drafts, originally intended for the author's use only; caveat lector! There are lots of scaffolding, erasures, obscure comments etc. On the other hand, the author did double-check the tables for factual mistakes. If you would like to use those tables, feel completely free to contact me for any clarifications.
I apologize for the messiness of these tables (and, for that matter, of the explicative page you are reading now). Making everything clean would take a lot of time; since some of these tables do not seem to exist in the literature, I felt that making them public was more urgent than putting them into a presentable shape.
Inside these tables, everything is written in French. Most such comments should be either irrelevant or guessable for an English speaker ; however, for those who do not speak French, here is some useful vocabulary:
- "pair" = "even"
- "impair" = "odd"
- "premier" = "prime"
- "racine" = "root"
- "poids" = "weight"
- "aucun, aucune" = "none"
A table of small-rank real simple Lie algebras (those with complex rank <= 6, and additionally E7 and E8), ordered first by complex then by real rank. For every such algebra, we give its Vogan diagram (but see warning below), its common names (in the presence of exceptionnal isomorphisms, there may be several of them) and its restricted root system. Also listed are exceptionnal isomorphisms of so(4, C) and of its real forms.
WARNING: WATCH OUT FOR MISLEADING NOTATION! Some Vogan diagrams are drawn with two painted roots. As such, these are NOT correct Vogan diagrams for the corresponding algebras. To obtain a correct Vogan diagram, choose *one* of those two roots (no matter which one: there is no obvious choice; this is why I included both!), paint it and leave all the other roots unpainted.
A table of fundamental groups of small-rank real simple Lie groups with trivial center. An explanation of the notations: g0 is some real simple Lie algebra (given with its Vogan diagram); g is its complexification; k0 is the maximal compact subalgebra of g0; Int(g) is the connected centerless Lie group with Lie algebra g; the map i: g0 -> g is the canonical inclusion; and the map i# is the morphism of fundamental groups induced by i.
A table of all connected complex simple Lie groups (taking into account all possible covers of the centerless version).
A table of all connected compact simple Lie groups (taking into account all possible covers of the centerless version).
A table of all connected simple Lie groups that are linear (not "algebraic" as claimed in the title; technically e.g. the algebraic group SO(p, q) is not connected, and its identity component is not algebraic) (taking into account all real forms of the complex version and all possible covers of the centerless version). Note that for a given simple Lie algebra, the corresponding simply-connected group is not always linear (for example PSL(2, R) is linear, but its universal cover is not; the only linear cover of PSL(2, R) is SL(2, R)). The largest linear cover (or, equivalently, the largest cover having a complexification) of the centerless version is sometimes called the "Zariski-simply-connected" version. For every Zariski-simply-connected group, we give its fundamental group (in ordinary topology). This shows "how many groups are missing" to the right of the picture.
Some tables listing the small-dimensional (complex) irreducible representations of A2, B2, G2, A3 (described as D3), B3 and C3. For every representation, we give a geometric picture of all its weights with multiplicities. The dimension of the representation (sometimes explicitly given) is the sum of all weight multiplicities. Also, for every real form of the algebra being represented, we give a picture of all the restricted weights of the corresponding representation, given with multiplicities; this is just a projection of the picture of all weights onto a suitable linear subspace. (We leave out the compact form, as the restricted weights for compact groups live in a 0-dimensional space).
At the beginning of a table, we sometimes give the lattice of all possible highest weights, colored blue or red according to whether they also belong or not to the root lattice.
There is also an older version for B2, without the restricted weights; I'm not sure if it is of any use.
A concise table allowing determination of the real, complex or quaternionic type of any irreducible representation of any simple real, non complex Lie algebra (not necessarily compact). Here is how to use this table:
A special note for so*(2n) when n is even: care must be taken when establishing the correspondence between nodes of the Dynkin diagram and simple roots, as this particular real form breaks the symmetry between the two "horns" of the Dn diagram. Among the two horns, the node that is painted in the last column of this table corresponds to the node that is painted in the Vogan diagram.
- a representation is of complex type iff its highest weight is not invariant by the involution given in the "complex" column;
- otherwise, a representation is of quaternionic type iff the sum of the coordinates of its highest weight corresponding to roots that are painted in the "quaternionic" column is odd;
- otherwise the representation is real.
For a complex Lie algebra g, any real representation is the tensor product of a representation of g and of a representation of the complex conjugate of g; hence its highest weight is given by a pair of elements of the weight lattice. The representation is of real type if these two elements are equal, of complex type otherwise.
A summary table of all small-dimensional real representations of simple real Lie groups with small complex rank (namely 1, 2 or 3). All such groups are listed, grouped as follows:
For every group, the weight lattice is drawn. At every node of the weight lattice, we write the real dimension of the representation with this highest weight. Numbers are written in ink for faithful representations, in pencil otherwise. They are underlined 0, 1 or 2 times to indicate that the representation is respectively of real, complex or quaternionic type. Finally, "blobs" are drawn around sets of representations having the same highest restricted weight.
- first by complex rank. Here by "complex rank", we really mean "rank of the complexification"; so complex groups are listed with twice the rank they should have.
- then by the type of the root system of their complexification.
- then by their Lie algebra, represented as a Vogan diagram.
- then all the connected linear groups with the given Lie algebra are listed.