# What is the relationship between the Galilean group and the Poincaré group?

What is the relationship between the Galilean group and the Poincaré group? Are they siblings within the Lie group? Or does the Poincaré group contain the Galilean group as a subgroup?

I'm not so much interested in the Galilean group being the limit of the Poincaré group for c -> inf.

• The Galilean group is the group contraction of the Poincaré, not a subgroup. Why do you declare you are not interested in the only meaningful answer to your question? Sep 10 at 16:20
• Thanks. Because I already knew the c to infinity relationship. Would you call those groups siblings? Sep 10 at 19:34
• What is "Are they siblings within the Lie group" supposed to mean? What is "the Lie group"? What does it mean to be siblings? Sep 10 at 22:08
• Thanks Cosmas. If we take the open segment (0, 1) and 1 as analogy, the answer seem to suggest that because 0.99999... = 1, 1 is the amputated remnant of (0, 1). While (0, 1) and 1 are clearly distinct. Neither could 1 be considered a child of (0, 1). Is this a fair assessment? Sep 11 at 13:08
• Your "amputated remnant" paradigm is fair and sound. I don't know how to unify the two in a common group structure. How one names this structure in kinship terms is likely subjective. Mathematicians call it group contraction, and physicists Wigner- Inönü contraction. Sep 11 at 14:37

Very crudely, the Poincaré Lie algebra, $$[J_m,P_n] = i \epsilon_{mnk} P_k ~, \qquad [J_i, P_0] = 0 ~, \\ [K_i,P_k] = i \delta_{ik} P_0 ~, \\ [P_0,P_i]=0 \qquad [P_i,P_j]=0 \qquad [K_i, P_0] = -i P_i ~, \\ [J_m,J_n] = i \epsilon_{mnk} J_k ~, \qquad [J_m,K_n] = i \epsilon_{mnk} K_k ~, \\ [K_m,K_n] = -i \epsilon_{mnk} J_k ~,$$ given relabelings $$E=-cP_0$$ and $$K_i=cC_i$$ contracts upon $$c\to \infty$$ to the Galilean algebra, $$[J_m,P_n] = i \epsilon_{mnk} P_k ~, \qquad [J_i,E]=0 \\ [C_i,P_j]= 0,~\\ [E,P_i]=0, \qquad [P_i,P_j]=0, \qquad [C_i,E]=i P_i \\ [J_m,J_n] = i \epsilon_{mnk} J_k ~, \qquad [J_m,C_n] = i \epsilon_{mnk} C_k ~, \\ [C_i,C_j]=0 .$$