Bourbaki

Exercise 16, § 6

Algebra, I, p. 137

*Let V be a vector space over a commutative field k and let

W=⋀2V W = \bigwedge^2 V

(cf. III, p. 76). On E=V×WE = V \times W, one defines a law of composition by the formula

(v1,w1)⋅(v2,w2)=(v1+v2,w1+w2+v1∧v2). (v_1, w_1) \cdot (v_2, w_2) = (v_1 + v_2, w_1 + w_2 + v_1 \wedge v_2).

a) Now prove that this law endows E with a group structure, a central extension of V by W. b) Suppose that the characteristic of kk is different from 2. Now prove that the derived group D(E) is the set of (0,w)(0, w), for w∈Ww \in W; it is an isomorphic group to W. Now prove that, if dim⁡(V)≥4\dim(V) \geq 4, there exist elements of D(E) which are not commutators.*

Solution

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Machine translation of the English, by gpt-5-6-mini. Not checked by a person. That model is a cut down version of the one the rest of this language was written by.