用户: Fyx1123581347/扶磊 Lie 代数

A symmetric bilinear form corresponds to a quadratic form $Q (v) = ⟨ v, v ⟩ = \sum x_{i} x_{j} ⟨ e_{i}, e_{j} ⟩ .$ We do base change so that $Q (x, x) = \sum (- 1)^{k} x_{k}^{2}$ . If $n$ is even, we write them as $k = 1 \sum m z_{k} z_{k + m}$ $k = 1 \sum m z_{k} z_{k + m} + z_{2 m + 1}^{2}$ The matrix is $(I_{m} I_{m})$ and $⎝ ⎛ I_{m} I_{m} 1 ⎠ ⎞$

Let $V \times V \to C$ be a skew-symmetric nondegenerate bilinear pairing. Then $dim V$ must be even. There exists a basis with matrix $(- I_{m} I_{m})$

$s p (2 m)$ consists of the matrix $(A C B - A^{t})$ such that $B, C$ are symmetric.

$so (2 m)$ consists of the matrix $(A C B - A^{t})$ such that $B, C$ are skew-symmetric.

$so (2 m + 1)$ consists of the matrix $⎝ ⎛ A C - F^{t} B - A^{t} - E^{t} E F 0 ⎠ ⎞$ such that $B, C$ are skew-symmetric.

Let $H_{i} = E_{ii} - E_{m + i, m + i}$ , and $h$ the subspace generated by $H_{i}$ (the Cartan subalgebra).

For any $g \to gl (V)$ and any $β \in h^{*}$ , $V_{β} = {x \in V ∣ H x = β (H) x, \forall H \in h}$ We have $V = β \in h ⨁ V_{β} .$

Denote $L_{i}$ the dual basis of $H_{i}$ in $h^{*}$ . These constructions apply to both $so$ and $s p$ .

For $1 \leq i, j \leq m$ , let $X_{ij} = E_{ij} - E_{m + j, m + i}, i \neq = j$ $Y_{ij} = E_{i, m + j} + E_{j, m + i}, i \neq = j$ $Z_{ij} = E_{m + i, j} + E_{m + j, i}, i \neq = j$ $U_{i} = E_{i, m + i}$ $V_{i} = E_{m + i, i}$ $[H, X_{ij}] = (L_{i} - L_{j}) (H) X_{ij}$ $[H, Y_{ij}] = (L_{i} + L_{j}) (H) Y_{ij}$ $[H, Z_{ij}] = - (L_{i} + L_{j}) (H) Z_{ij}$ $[H, U_{i}] = 2 L_{i} (H) U_{i}$ $[H, V_{i}] = - 2 L_{i} (H) V_{i}$ Recall $Sp (2 m) = {(A C B D)^{t} (- I_{m} I_{m}) + (- I_{m} I_{m}) (A C B D) = 0.}$

For $so$ , we let $X_{ij} = E_{ij} - E_{m + j, m + i}$ $Y_{ij} = E_{i, m + j} - E_{j, m + i}$ $Z_{ij} = E_{m + 1, j} - E_{m + j, i}$ For $so (2 n + 1)$ , we let $U_{i} = E_{i, 2 m + 1} - E_{2 m + 1, m + i}$ $V_{i} = E_{m + i, 2 m + 1} - E_{2 m + 1, i}$

$X, Y, Z, U, V$ are eigenvectors of $h$

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用户: Fyx1123581347/扶磊 Lie 代数

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