Return to Question

Let us consider the Poincaré algebra, characterized by the following commutators: \begin{align} [H,P_i]&=0\\ [H,K_i]&=P_i\\ [P_i,P_j]&=0\\ [K_i,K_j]&=-\epsilon_{ijk}K_k\\ [P_i,K_j]&=\delta_{ij}H\\ [J_i,J_j]&=\epsilon_{ijk}J_k\\ [J_i,K_j]&=\epsilon_{ijk}K_k\\ [J_i,P_j]&=\epsilon_{ijk}P_k\\ [J_i,H]&=0 \end{align} How could I know - using the algebra only - if the subgroup, generated by the $K_i$ generators, is compact or not? Is there a criteria for establishing compactness?

My understanding of a compact group is related to the notion of bounded and connected sets. For example the Lorentz group has four disconnected pieces, so it's a non-compact group.

Let us consider the Poincaré algebra, characterized by the following commutators: \begin{align} [H,P_i]&=0\\ [H,K_i]&=P_i\\ [P_i,P_j]&=0\\ [K_i,K_j]&=-\epsilon_{ijk}J_k\\ [P_i,K_j]&=\delta_{ij}H\\ [J_i,J_j]&=\epsilon_{ijk}J_k\\ [J_i,K_j]&=\epsilon_{ijk}K_k\\ [J_i,P_j]&=\epsilon_{ijk}P_k\\ [J_i,H]&=0 \end{align} How could I know - using the algebra only - if the subgroup, generated by the $K_i$ generators, is compact or not? Is there a criteria for establishing compactness?

My understanding of a compact group is related to the notion of bounded and connected sets. For example the Lorentz group has four disconnected pieces, so it's a non-compact group.

Let us consider the Poincaré algebra, characterized by the following commutators: \begin{align} [H,P_i]&=0\\ [H,K_i]&=P_i\\ [P_i,P_j]&=0\\ [K_i,K_j]&=-\epsilon_{ijk}K_k\\ [P_i,K_j]&=\delta_{ij}H\\ [J_i,J_j]&=\epsilon_{ijk}J_k\\ [J_i,K_j]&=\epsilon_{ijk}K_k\\ [J_i,P_j]&=\epsilon_{ijk}P_k\\ [J_i,H]&=0 \end{align} How could I know - using the algebra only - if the subgroup, generated by the $K_i$ generators, is compact or not? Is there a criteria for establishing compactness?

My understanding of a compact group is related to the notion of bounded and connected sets. For example the Lorentz group has four disconnected pieces, so it's a non-compact group.

Let us consider the Poincaré algebra, characterized by the following commutators: \begin{align} [H,P_i]&=0\\ [H,K_i]&=P_i\\ [P_i,P_j]&=0\\ [K_i,K_j]&=-\epsilon_{ijk}K_k\\ [P_i,K_j]&=\delta_{ij}H\\ [J_i,J_j]&=\epsilon_{ijk}J_k\\ [J_i,K_j]&=\epsilon_{ijk}K_k\\ [J_i,P_j]&=\epsilon_{ijk}P_k\\ [J_i,H]&=0 \end{align} How could I know - using the algebra only - if the subgroup, generated by the $K_i$ generators, is compact or not? Is there a criteria for establishing compactness?

My understanding of a compact group is related to the notion of bounded and connected sets. For example the Lorentz group has four disconnected pieces, so it's a non-compact group.

Let us consider the Poincaré algebra, characterized by the following commutators: \begin{align} [H,P_i]&=0\\ [H,K_i]&=P_i\\ [P_i,P_j]&=0\\ [K_i,K_j]&=-\epsilon_{ijk}P_k\\ [P_i,K_j]&=\delta_{ij}H\\ [J_i,J_j]&=\epsilon_{ijk}J_k\\ [J_i,K_j]&=\epsilon_{ijk}K_k\\ [J_i,P_j]&=\epsilon_{ijk}P_k\\ [J_i,H]&=0 \end{align} How could I know - using the algebra only - if the subgroup, generated by the $K_i$ generators, is compact or not? Is there a criteria for establishing compactness?

My understanding of a compact group is related to the notion of bounded and connected sets. For example the Lorentz group has four disconnected pieces, so it's a non-compact group.

Let us consider the Poincaré algebra, characterized by the following commutators: \begin{align} [H,P_i]&=0\\ [H,K_i]&=P_i\\ [P_i,P_j]&=0\\ [K_i,K_j]&=-\epsilon_{ijk}K_k\\ [P_i,K_j]&=\delta_{ij}H\\ [J_i,J_j]&=\epsilon_{ijk}J_k\\ [J_i,K_j]&=\epsilon_{ijk}K_k\\ [J_i,P_j]&=\epsilon_{ijk}P_k\\ [J_i,H]&=0 \end{align} How could I know - using the algebra only - if the subgroup, generated by the $K_i$ generators, is compact or not? Is there a criteria for establishing compactness?

My understanding of a compact group is related to the notion of bounded and connected sets. For example the Lorentz group has four disconnected pieces, so it's a non-compact group.