Determining Lie algebras from commutative diagrams of exact sequences. Suppose that we have the fol

Jazlyn Raymond

Jazlyn Raymond

Answered question

2022-05-14

Determining Lie algebras from commutative diagrams of exact sequences.
Suppose that we have the following commutative diagram of graded Lie algebras.
0 C n A n + 1 A n 0 0 D n B n + 1 B n 0
for all n Z + , where the both rows are split exact sequences and every vertical map is onto.
My question is, suppose that we know A n , C n , D n for all n Z + and B 1 , do we have enough information to uniquely determine up to isomorphism every B n (likely by induction)?

Answer & Explanation

Emmy Sparks

Emmy Sparks

Beginner2022-05-15Added 17 answers

It looks to me like it is true that B n is uniquely determined. Let K n be the kernel of the map A n B n and let E n be the kernel from C n to D n . Then we have an exact sequence 0 E n K n + 1 K n is known and K n is known inductively. That means that K n + 1 can be determined as the subspace of A n + 1 generated by all preimages of K n and images f D n . Therefore B n + 1 = A n + 1 / K n + 1 is uniquely determined.

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