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Bott periodicity, Lie group embeddings, Lie theory
Understand Lie group embeddings in terms of Dynkin diagrams
Dynkin diagrams
Dynkin diagrams for classical Lie algebras
I want to consider how the chain of embeddings of Lie algebras can be understood in terms of their Dynkin diagrams.
Note that the Dynkin diagrams are related to the Lie algebras over {$\mathbb{C}$}. Whereas the unitary group {$\frak{u}$}(n)$} of skew-Hermitian matrices is a Lie algebra over {$\mathbb{R}$}. For {$i$} times a skew-Hermitian matrix is Hermitian.
- {$A_n=\frak{sl}$}{$(n+1)$} The maximal compact subgroup is {$SU(n+1)$}.
- {$C_n=\frak{sp}$}{$(2n)$} The maximal compact subgroup is {$Sp(2n)$}.
- {$D_n=\frak{o}$}{$(2n)$} The maximal compact subgroup is {$SO(2n)$}.
| Lie group | nodes | diagram | zero |
| {$U(2r)$} | 2r - 1 | {$\circ - \circ - \circ - \circ$} | dual |
| {$U(r)\times U(r)$} | 2r - 2 | {$\circ - \circ - \circ - \circ \;\;\;\; \circ - \circ - \circ - \circ$} | dual |
| {$U(r)$} | r - 1 | {$\circ - \circ - \circ - \circ$} | dual |
| Lie group | nodes | diagram | zero |
| {$O(32r)$} | 8r | {$\circ - \circ - \circ \langle \begin{matrix}\circ \\ \circ \\ \end{matrix} $} | folding |
| {$U(16r)$} | 8r - 1 | {$\circ - \circ - \circ - \circ$} | dual |
| {$Sp(8r)$} | 4r | {$\circ - \circ - \circ -- \circ$} | fusing |
| {$Sp(4r)\times Sp(4r)$} | 4r | {$\circ - \circ - \circ -- \circ \;\;\;\; \circ - \circ - \circ -- \circ$} | fusing disconnected |
| {$Sp(4r)$} | 2r | {$\circ - \circ - \circ -- \circ$} | fusing |
| {$U(4r)$} | 4r-1 | {$\circ - \circ - \circ - \circ$} | dual |
| {$O(4r)$} | 2r | {$\circ - \circ - \circ \langle \begin{matrix}\circ \\ \circ \\ \end{matrix} $} | folding |
| {$O(2r)\times O(2r)$} | 2r | {$\circ - \circ - \circ \langle \begin{matrix}\circ \\ \circ \\ \end{matrix} $} | folding disconnected |
| {$O(2r)$} | r | {$\circ - \circ - \circ \langle \begin{matrix}\circ \\ \circ \\ \end{matrix} $} | folding |