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About what Anton said at the end about deformations of a group. Suppose that I take the group $G=GL_{2}(\mathbb{C})$ for concretness. Let $m_0$ be the standard multiplication. Then I want to consider a deformation of the form $m:(G \times \epsilon \mathfrak{g}) \times (G \times \epsilon \mathfrak{g}) \to G \times \epsilon \mathfrak{g}$ where $m(g_1, g_2) = m_{0}(g_1,g_2) + \epsilon m_1 (g_1,g_2)$. When you write out the associativity condition $m\circ (m \times 1) = m \circ (1 \times m)$ it seems that you find that $(g_1,g_2) \mapsto (m_{1}(g_{1},g_{2}))(g_{1}g_{2})^{-1}$ is a group cohomology cocycle for G acting on $\mathfrak{g}$ by the adjoint representation. Now one has to identify $H^{2}(G,Ad)$ with $H^{2}(BG,Ad)$ (taking care of the topology somehow).

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About what you Anton said at the end about deformations of a group. Suppose that I take the group $G=GL_{2}(\mathbb{C})$ for concretness. Let $m_0$ be the standard multiplication. Then I want to consider a deformation of the form $m:(G \times \epsilon \mathfrak{g}) \times (G \times \epsilon \mathfrak{g}) \to G \times \epsilon \mathfrak{g}$ where $m(g_1, g_2) = m_{0}(g_1,g_2) + \epsilon m_1 (g_1,g_2)$. When you write out the associativity condition $m\circ (m \times 1) = m \circ (1 \times m)$ it seems that you find that $(g_1,g_2) \mapsto (m_{1}(g_{1},g_{2}))(g_{1}g_{2})^{-1}$ is a group cohomology cocycle for G acting on $\mathfrak{g}$ by the adjoint representation. Now one has to identify $H^{2}(G,Ad)$ with $H^{2}(BG,Ad)$ (taking care of the topology somehow).

About what you said at the end about deformations of a group. Suppose that I take the group $G=GL_{2}(\mathbb{C})$ for concretness. Let $m_0$ be the standard multiplication. Then I want to consider a deformation of the form $m:(G \times \epsilon \mathfrak{g}) \times (G \times \epsilon \mathfrak{g}) \to G \times \epsilon \mathfrak{g}$ where $m(g_1, g_2) = m_{0}(g_1,g_2) + \epsilon m_1 (g_1,g_2). g_1,g_2)$. When you write out the associativity condition $m\circ (m \times 1) = m \circ (1 \times m)$ it seems that you find that $(g_1,g_2) \mapsto m_{1}(g_{1},g_{2})g_{1}g_{2}$ (m_{1}(g_{1},g_{2}))(g_{1}g_{2})^{-1}$is a group cohomology cocycle for G acting on$\mathfrak{g}$by the adjoint representation. Now one has to identify$H^{2}(G,Ad)$with$H^{2}(BG,Ad)\$ (taking care of the topology somehow).