\[
e \to K \to G \to G' \to e
\]

\[
e \to N_K(T) \to N_G(T) \to G' \to e
\]

\[
e \to K(\bar k) \to G(\bar k) \to G'(\bar k) \to e
\]

\[
H^i(k,G) \simeq H^i(\pi, G(\bar k))
\]

\[
e \to K \to G \to G' \to e
\]

\[
e \to N_K(\mathfrak{h}) \to N_G(\mathfrak{h}) \to G' \to e
\]

\[
e \to K \to G \to G' \to e
\]

\[
H^i(k,G) = H^i(k,G') = 0 \quad \text{pour } i > 0
\]

\[
H^2(k,K') = H^2(k,K')' = \{\ill{}\} \ \text{\uncertain{universellement}}.
\]

\[
e \to K'' \to \Pi(K) \to \Pi(K') \to e .
\]

\[
G = \left\lbrace
\begin{pmatrix} x & ay & & \\ y & x & & \\ & & u & v \\ & & w & z \end{pmatrix}
\;\middle|\;
\begin{vmatrix} x & ay \\ y & x \end{vmatrix}
= \begin{vmatrix} u & v \\ w & z \end{vmatrix} \neq 0
\right\rbrace .
\]

\[
\begin{pmatrix} x & ay & & \\ y & x & & \\ & & u & v \\ & & w & z \end{pmatrix}
\rightsquigarrow
\begin{pmatrix} u & v \\ w & z \end{pmatrix} \big/ (\text{centre})
\]

\[
Z = \left\lbrace
\begin{pmatrix} x & ay & & \\ y & x & & \\ & & u & 0 \\ & & 0 & u \end{pmatrix}
\;\middle|\;
x^2 - ay^2 = u^2 \neq 0
\right\rbrace
= \text{centre de } G = \text{radical de } G .
\]

\[
\mathfrak{g}/\mathfrak{L} \longrightarrow \mathrm{Grass}(\mathfrak{g})
\ (\ill{}) , \qquad \check{H}\ldots(\mathfrak{L}, \mathfrak{g}/\mathfrak{L})
\]

\[
H^1(B, \mathfrak{g}/\mathfrak{L}) = 0 \quad ??
\]

\[
H^1(\mathfrak{R}, \mathfrak{g}/\mathfrak{L})^{B/R} = 0
\]

\[
H^1(B', \mathfrak{g}/\mathfrak{L}) \longrightarrow
H^1(B, \mathfrak{g}/\mathfrak{L}) \longrightarrow
H^1(\,\cdot\,, \mathfrak{g}/\mathfrak{L})
\]

\[
(\mathfrak{g}/[\mathfrak{u},\mathfrak{u}])^T = 0, \qquad
\mathfrak{L}_1 = \mathfrak{L}, \qquad B = B', \qquad B_0 = B'_0
\]

\[
\text{i) } B = \mathrm{Norm}_G(\mathfrak{L}) \qquad
\text{ii) } \mathfrak{L} = \mathrm{Norm}_{\mathfrak{g}}(\mathfrak{L})
\]

\[
K \xrightarrow{\;u_n = n.\mathrm{id}_K\;} K .
\]

\[
\left\lbrace
\begin{array}{l}
\mathcal{U} = A\{X\} \quad \text{puissances divisées} \\
\Delta \gamma^n X = \displaystyle\sum_{p+q=n} \gamma^p X \otimes \gamma^q X
\end{array}
\right.
\]

\[
\Delta \exp X = \exp X \otimes \exp Y
\]

\[
\left\lbrace
\begin{array}{l}
\mathcal{U}(\underline{E}) = \mathcal{O}_S\{\check{\underline{E}}\} \quad
\text{puissances divisées sur } \check{\underline{E}} \\
\Delta \gamma^n X = \displaystyle\sum_{p+q=n} \gamma^p X \otimes \gamma^q Y
\end{array}
\right.
\]

\[
\langle \Delta X_n, s^p \otimes s^q \rangle = \langle X_n, s^{p+q} \rangle
= \delta_{n,p+q}
\qquad
\langle X_p . X_q, s^n \rangle = \langle X_p \otimes X_q, (\Delta s)^n \rangle
\]

\[
= \sum_{\alpha+\beta+\gamma = n} \frac{n!}{\alpha!\,\beta!\,\gamma!}\,
s^{\alpha+\gamma} \otimes s^{\beta+\gamma},
\qquad
\left\lbrace
\begin{array}{l}
\alpha + \gamma = p \\
\beta + \gamma = q
\end{array}
\right.
\]

\[
\left\lbrace
\begin{array}{ll}
0 & \text{si } n \notin [\sup(p,q), p+q] \\[4pt]
\dfrac{n!}{(n-p)!\,(n-q)!\,(p+q-n)!} & \text{sinon}
\end{array}
\right.
\]

\[
\alpha + \beta + 2\gamma = p + q, \qquad
\left\lbrace
\begin{array}{l}
\gamma = p + q - n \\
\alpha = n - q \\
\beta = n - p
\end{array}
\right.
\]

\[
\sup(p,q) \leq n \leq p + q
\]

\[
\boxed{
\begin{array}{l}
\Delta X_n = \displaystyle\sum_{p+q=n} X_p \otimes X_q \\[10pt]
X_p X_q = \displaystyle\sum_{\sup(p,q) \leq n \leq p+q}
\frac{n!}{(n-p)!\,(n-q)!\,(p+q-n)!}\, X_n
\end{array}}
\]

\[
\delta X_n = \sum_{\substack{p+q=n \\ p,q>0}} X_p \otimes X_q
= X_n \otimes X_1 + X_{n-1} \otimes X_2 + \cdots + X_2 \otimes X_{n-1}
+ X_1 \otimes X_n
\]

\[
X_1^2 = 2X_2 + X_1, \qquad
\left\lbrace
\begin{array}{l}
2X_2 = X_1^2 - X_1 \\
\delta X_2 = X_1 \otimes X_1
\end{array}
\right.
\qquad ? \quad 2\delta X_2 = \delta(X_1^2 - X_1)
\]

\[
2X_1 \otimes X_1 = \delta(X_1^2 - X_1)
\]

\[
\delta(X_1) = X_1 \otimes 1 + 1 \otimes X_1, \qquad
\delta(X_1^2) = X_1^2 \otimes 1 + 1 \otimes X_1^2 + 2 X_1 \otimes X_1
\]

\[
A \longleftarrow A \otimes A \longleftarrow A, \qquad
G \xrightarrow{\ \text{diag.}\ } G \times G \xrightarrow{\ \text{mult.}\ } G
\]

\[
X_n(t^m) = \binom{m}{n} t^m \qquad
\langle X_n, t^m \rangle = \binom{m}{n}
\]

\[
X_n X_{n'}(t^m) = \binom{m}{n}\binom{m}{n'}
\]

\[
\frac{m(m-1)\cdots(m-n+1)}{n!}\;\frac{m(m-1)\cdots(m-n'+1)}{n'!}
\qquad \text{\uncertain{nul} si}
\]

\[
t = 1 + s, \quad s = t - 1
\]

\[
\langle X_n, s^m \rangle = \langle X_n, (t-1)^m \rangle
\]

\[
= \binom{m}{n} - \binom{m}{1}\binom{m-1}{n} + \binom{m}{2}\binom{m-2}{n}
+ \cdots + (-1)^k \binom{m}{k}\binom{m-k}{n} + \cdots
\]

\[
\langle X_1, (t-1)^m \rangle = t\,m\,(t-1)^{m-1}\big|_{t=1}
= \left\lbrace
\begin{array}{ll}
0 & \text{si } m \neq 1 \\
1 & \text{si } m = 1
\end{array}
\right.
\]

\[
\langle X_2, (t-1)^m \rangle = \Big\langle \frac{X_1^2}{2} - \frac{X_1}{2} \Big\rangle
\]

\[
\frac{m(m-1)}{2!} - \frac{m}{\ \ }
\]

\[
X_2(t-1) = X_2(t) = \binom{1}{2}
\qquad X_n = \binom{X}{n}
\]

\[
X_n(t^m) = \left\lbrace
\begin{array}{ll}
0 & \text{si } m < n \\
t^m & \text{si } m = n \\
\ldots &
\end{array}
\right. \ (m \geq 0)
\qquad
X_n((t-1)^m) = \left\lbrace
\begin{array}{ll}
0 & \text{si } 0 \leq m < n \\
t^n & \text{si } m = n \\
0 & \text{si } \ldots
\end{array}
\right.
\]

\[
(t-1)^r \qquad \langle X_n, s^m \rangle \qquad
\langle X_n, s^m \rangle = \delta_{nm}
\]

\begin{align*}
&\mathrm{Aff}_A(G) \quad \text{son algèbre affine} \\
&\mathcal{U}_A(G) \quad \text{son hyperalgèbre}
\end{align*}

\[
\langle f, X \rangle = \langle X, f \rangle
\]

\[
\left\lbrace
\begin{array}{l}
\langle Y, X * f \rangle = \langle Y * X, f \rangle = \langle f, Y * X \rangle
= \langle f * Y, X \rangle \\
\langle \varepsilon, X * f \rangle = \langle \varepsilon, f * X \rangle
= \langle X, f \rangle
\end{array}
\right.
\]

\[
\langle g, f \lrcorner X \rangle = \langle gf, X \rangle
\qquad \text{en particulier } \langle f, X \rangle = \varepsilon(f \lrcorner X)
\]

\[
\mathrm{Aff}_B(G) = \mathrm{Aff}_A(G) \otimes_A B \supset \mathrm{Aff}_A(G)
\]

\[
\mathcal{U}_B(G) = \mathcal{U}_A(G) \otimes_A B \supset \mathcal{U}_A(G)
\]

\[
\langle g, f \lrcorner X \rangle = \langle gf, X \rangle
= \langle f, g \lrcorner X \rangle \in \langle f, \mathcal{U}_A(G) \rangle
\subset A \ ].
\]

\[
\mathfrak{g} = L_S(G)
\]

\[
G \simeq \underline{\mathrm{Aut}}^0_S(\mathfrak{g})
\]

\[
(3) \quad H^2(\mathfrak{g}_0, \mathfrak{g}_0) = 0
\]

\[
H^*(G, \mathfrak{g}) \longrightarrow H^*(\mathfrak{g}, \mathfrak{g})
\]

\[
H^*(G, \mathfrak{g}) = H^*(\mathfrak{g}, \mathfrak{g}) = 0 \quad ??
\]

\[
X \quad Y \quad H
\qquad
\left\lbrace
\begin{array}{l}
[X, H] = \lambda(H) X \\
{[Y, H]} = -\lambda(H) Y \\
{[X, Y]} = H
\end{array}
\right.
\qquad
T \longrightarrow T'
\]

\[
\left\lbrace
\begin{array}{l}
[H, X] = 0 \\
{[H, Y]} = 0 \\
{[X, Y]} = H
\end{array}
\right.
\qquad \mathcal{U}
\]

\[
\left\lbrace
\begin{array}{l}
[H', X] = X \\
{[H', Y]} = -Y \\
{[X, Y]} = 0
\end{array}
\right.
\]

\[
H^1(S, \Pi_1) \longrightarrow H^1(S, \Gamma)
= \text{\uncertain{ens.}\ des classes d'\uncertain{isom.}}
\]

\[
e \longrightarrow \Pi_1^{\rho} \longrightarrow \Gamma_1^{\rho}
\longrightarrow E_1^{\rho} \longrightarrow e
\qquad \Gamma_1^{\rho} \simeq E_1^{\rho} \cdot \Pi_1^{\rho}
\]

\[
H^0(S, E_1^{\rho}) \longrightarrow H^1(S, \Pi_1^{\rho}) \longrightarrow
H^1(S, \Gamma_1^{\rho}) \longrightarrow H^1(S, E^{\rho})
\]

\[
H^1(S, \Gamma_1^{\rho}) \simeq H^1(S, \Gamma)
\]

\[
e \longrightarrow (\Pi_1^{\rho})^{\xi} \longrightarrow
(\Gamma_1^{\rho})^{\xi} \longrightarrow (E_1^{\rho})^{\xi} \longrightarrow e
\]

\[
e \longrightarrow \mathrm{Ad}(G) \longrightarrow \mathrm{Aut}(G)
\longrightarrow \mathrm{Ext}(G) \longrightarrow e
\]

\[
(iii') \Longleftrightarrow (i) \Longrightarrow (i') \Longleftrightarrow (ii)
\qquad (iii) \Longleftrightarrow (i'')
\]

\[
(ax - by)^2 + (cx + dy)^2 = x^2 + y^2
\qquad
\left\lbrace
\begin{array}{l}
a^2 + c^2 = 1 \\
b^2 + d^2 = 1 \\
2(-ab + cd) = 0
\end{array}
\right.
\]

\[
e \to SO(2m) \to O(2m) \to \mathbb{Z}/2\mathbb{Z} \to e
\]

\[
e \to SO(2m+1) \to O(2m+1) \to \mu_2 \to e
\]

\begin{gather*}
(x, y) \longmapsto (\lambda x + u(y),\ y) \\
(\lambda(x) + u(y))^2 + Q_0(y) \\
\lambda^2 x^2 + u(y)^2 = x^2 \\
u(y)^2 = 0 \\
\lambda^2 = 1 \\
u(y)^2 = \\
\underline{\lambda^2 = 1} \qquad \mu_2 \\
(\mu_2\ \ \alpha_2\ \ \alpha_2\ \ \ldots\ \ \alpha_2)
\end{gather*}

\begin{gather*}
(\xi, y) \longmapsto (\lambda \xi + u(y),\ y) \\
\lambda^2 = 1 \qquad u(y)^2 = 0 \\
\lambda' \lambda \xi + \lambda' u(y) + u'(y) \\
(\lambda', u')(\lambda, u) = (\lambda' \lambda,\ \lambda' u + u') \\
\mu_2 .
\end{gather*}

\[
f(x \wedge y, x \wedge y) =
\begin{vmatrix}
f(x, x) & f(x, y) \\
f(y, x) & f(y, y)
\end{vmatrix}
= Q(x, x) Q(y, y) - f(x, y)^2
\]

\[
Q(x \wedge y) = Q(x, x) Q(y, y) - f(x, y)^2
\]

\begin{gather*}
e_1 e_2 = e_1^2 = 0 \\
(x e_1 + y e_2)(x' e_1 + y' e_2) = y y' \\
(e_1 + e_2)(e_1 + e_2) \qquad e_1 + e_2 - \ldots + e_n \qquad
\Sigma x_i^2 \\
e_1\ \ \underbrace{e_2\ e_3 - \ldots}\ \ e_1 - e_2 \\
e_1,\ e_1 - e_2,\ e_1 - e_3,\ \ldots,\ e_1 - e_n \\
e'_i e'_j = 0 \quad i, j \neq 1 \\
a_{11} x_1 x'_1 + \sum_{1 \leq i < j} a_{ij} (x_i x'_j + x_j x'_i)
\qquad (e_1 - e_i)(e_1 - e_j) = 1 \\
a_{11} x_1 x'_1
\end{gather*}

\[
0 \longrightarrow \Sigma^2(\check E) \longrightarrow \mathrm{Symm}^2(E)
\longrightarrow \Sigma^2(\check E) \longrightarrow \mathrm{Symm}^2(E)
\]

\[
\textstyle\sum a_{ij} x_i x_j \leadsto \sum a_{ij} x_i \otimes x_j
\]

\[
0 \to \mathrm{Alt}^2(\check E) \to \Sigma^2(\check E) \to
\mathrm{Symm}^2(\check E) \to \mathrm{Alt}^2(\check E) \to 0
\]

\[
0 \to \check E^{(2)} \to \mathrm{Symm}^2(\check E) \to \Sigma^2(\check E)
\to \check E^{(2)} \to 0
\]

\[
\boxed{\mathrm{Im} = \mathrm{Ker} = \mathrm{Alt}^2(\check E)
\qquad \Sigma^2(\check E) \rightleftarrows \mathrm{Symm}^2(\check E)
\qquad \mathrm{Im} = \mathrm{Noyau} = \check E^{(2)}}
\]

\[
x_1^2 + \cdots + x_n^2 + \struck{x_1 x_2}\ y_1 y_2 + y_3 y_4 + \cdots +
y_{2m-1} y_{2m}
\]

\[
\coprod_{n, m \geq 0} E^{n,m}(S) = \bar E(S)
\]

\[
E(S) \xrightarrow{\ \varepsilon_1 \times \varepsilon_2\ } \mathbb{Z}
\times \mathbb{Z}
\]

\[
D(A) = A^{*}/A^{*2} = H^1(A, \mathbb{Z}/2\mathbb{Z})
\]

\[
\varphi : \mathbb{Z}(D) \longrightarrow Q(A)
\]

\[
a + b = a' + b' \quad , \quad a.b = a'.b'
\]

\[
B = H^2(A, \mathbb{Z}/2\mathbb{Z})
\]

\[
e_{a+u+v} - e_{a+u} - e_{a+v} + e_a
\]

\[
W^1 : D \longrightarrow H^1
\]

\[
\mathrm{grad}\, Q(A) \to H^{\cdot} .
\]

\[
H^{\cdot}_{1} = H^{\cdot}(A, \mathbb{Z}/2\mathbb{Z}) ,
\]

\[
W^i : Q(A) \to H^i(A, \mathbb{Z}/2\mathbb{Z}) .
\]

\[
\lambda_t(\varphi_a) = 1 + \varphi_a t ,
\]

\[
H^{\cdot}(S_{\mathrm{top}}, \mathbb{Z}/2\mathbb{Z}) \to H^{\cdot}(A,
\mathbb{Z}/2\mathbb{Z}) ,
\]

\[
\left.
\begin{array}{l}
F(A)/F_1(A) \simeq \mathbb{Z} \\
F_1(A)/F_2(A) \simeq A^{*}/A^{*2} \\
F_2(A)/F_3(A) \subset {}_2\mathrm{Br}(A)
\end{array}
\right\rbrace
\]

\[
\left\lbrace
\begin{array}{l}
F_1(A)/F_2(A) \simeq H^1(A, \mathbb{Z}/2\mathbb{Z}) \\
F_2(A)/F_3(A) \simeq H^2(A, \mathbb{Z}/2\mathbb{Z})
\end{array}
\right.
\]

\[
\mathrm{Pic}(A)/2\,\mathrm{Pic}(A) \subset H^2(A, \mu_2)
\]

\[
H^1(K, \mathbb{Z}/2\mathbb{Z}) \to \sum_p H^1(K_p, \mathbb{Z}/2\mathbb{Z})
\qquad
H^1(K, \mathbb{G}_m) \to
\]