# vector spaces: sum of the maximum abelian ideal and attached map cores in a nilpotent Lie algebra

Leave $$mathbb {K}$$ be a field with $$char ( mathbb {K}) neq 2$$ Y $$L$$ a nilpotent lie algebra of finite dimension on $$mathbb {K}$$.

Suppose that $$dim L = 5 + dim Z (L)$$ Y $$L = span {x, y, z, u, v } oplus Z (L)$$ where $$dim A = 2 + dim Z (L)$$ Y $$A = span {u, v } oplus Z (L)$$ It is the ultimate abelian ideal.

Define $$H_ {a + x}: = A + ker (ad_ {a + x})$$ for all $$a in A$$. So $$x + A in H_ {a + x} / A$$ for all $$a en A setminus Z (L)$$. Suppose we have $$dim (H_x / A) = dim (H_ {a + x} / A) = 2$$ for all $$a en A setminus Z (L)$$.

Can we show that it exists $$a neq b en A setminus Z (L)$$ such that $$H_x subseteq H_ {a + x}$$ or $$H_ {a + x} subseteq H_x$$ or $$H_ {a + x} subseteq H_ {b + x}$$?

Now we can prove the statement when $$mathbb {K} = mathbb {F} _q$$ It is a finite field as follows. Note that $$L / A = span {x + A, y + A, z + A }$$ Y $$H_x / A = span {x + A, alpha_0y + beta_0z + A }$$ for some $$(0, 0) neq ( alpha_0, beta_0) in mathbb {F} _q ^ 2$$ and for each $$a en A setminus Z (L), H_ {a + x} / A = span {x + A, alpha_ay + beta_az + A }$$ for some $$(0, 0) neq ( alpha_a, beta_a) in mathbb {F} _q ^ 2$$. As $$dim (A / Z (L)) = 2$$, $$| {H_ {a + x} | a en A setminus Z (L) } cup {H_x } | leq q ^ 2$$. Keep in mind that there is at most $$q ^ 2 – 1$$ form subspaces $$span {x + A, alpha and + beta z + A }$$ where $$(0, 0) neq ( alpha, beta) in mathbb {F} _q ^ 2$$. Therefore, we have the result of the fourth theorem of isomorphism.

Is it still true when $$mathbb {K}$$ It's infinite?

In case we need to use them, we can also assume that

1. $$H_ {a + x} cap H_ {b + x} neq A$$ for all $$a, b in A$$
2. $$(A, L) subseteq Z (L)$$
3. $$(L, L) = (A, L)$$.

But I feel we don't need these conditions.