# Book: OddAndEvenOrthogonalLieAlgebras

Understand the Dynkin diagrams of the odd and even orthogonal Lie algebras.

• What does it mean that {$D_1$} is {$\mathfrak{so}_{2}$}?
• In what sense are {$B_2$} and {$C_2$} folded versions of {$A_3$}?
• In what sense is {$G_2$} a folded version of {$D_4$}?
• List out the Dynkin diagrams for odd and even orthogonal Lie algebras.

The index {$n$} for {$A_n$}, {$B_n$}, {$C_n$} and {$D_n$} refers to the number of nodes in the Dynkin diagram. This is related to the size of the matrices in the following way:

• {$A_n$}: {$\mathfrak{sl}_{n+1}$}
• {$B_n$}: {$\mathfrak{so}_{2n+1}$}
• {$C_n$}: {$\mathfrak{sp}_{2n}$}
• {$D_n$}: {$\mathfrak{so}_{2n}$}
• intuition of rotations in three-dimensions: {$A_1 \cong B_1 \cong C_1$}

Note that the special orthogonal group {$\mathfrak{so}_{2}$} consists simply of

reflections?

{$\mathfrak{sl}_{2} \cong \mathfrak{so}_{3} \cong \mathfrak{sp}_{2}$}

SO(2) is the circle, which can be understood in {$\mathbb{C}$} or in {$\mathbb{R}^2$}.

{$\mathfrak{so}_{2}$} is the Abelian Lie algebra given by

{$$\begin{pmatrix} 0 & \theta \\ - \theta & 0 \end{pmatrix}$$}

{$\textrm{Lie}(G) =\{X \in M(n;\mathbb{C}) | t \in \mathbb{R} \rightarrow e^{tX}\in G\}$}

{$X \in \textrm{Lie}(G) \Leftrightarrow e^{tX}\in G$}

{$$\begin{pmatrix} cos \theta & sin \theta \\ -sin \theta & cos \theta \end{pmatrix} \Leftrightarrow \begin{pmatrix} 0 & \theta \\ - \theta & 0 \end{pmatrix}$$}

{$$\begin{pmatrix} cos \alpha & sin \alpha \\ -sin \alpha & cos \alpha \end{pmatrix} \begin{pmatrix} cos \beta & sin \beta \\ -sin \beta & cos \beta \end{pmatrix} = \begin{pmatrix} cos \alpha cos \beta - sin \alpha sin \beta & cos \alpha sin \beta - sin \alpha cos \beta \\ - sin \alpha cos \beta + cos \alpha sin \beta & - sin \alpha sin \beta + cos \alpha cos \beta \end{pmatrix}$$}

{$$= \begin{pmatrix} cos (\alpha + \beta) & sin (\alpha + \beta) \\ -sin (\alpha + \beta) & cos (\alpha + \beta) \end{pmatrix}$$}

{$$\begin{pmatrix} 0 & \alpha \\ - \alpha & 0 \end{pmatrix} + \begin{pmatrix} 0 & \beta \\ - \beta & 0 \end{pmatrix} = \begin{pmatrix} 0 & \alpha + \beta \\ -\alpha - \beta & 0 \end{pmatrix}$$}

{$$(e^{i \theta}) \Leftrightarrow \begin{pmatrix} 0 & \theta \\ - \theta & 0 \end{pmatrix}$$}

Simple Lie algebras need to generate, under the bracket, all of themselves.

Parsiųstas iš http://www.ms.lt/sodas/Book/OddAndEvenOrthogonalLieAlgebras
Puslapis paskutinį kartą pakeistas 2019 birželio 17 d., 19:17