Comparing Lusztig's algebras and Hall algebras at v=−1

There are two abstract approaches to (positive parts of) quantized enveloping algebras of Kac–Moody algebras: Lusztig's axiomatic approach leading to non-degenerate objects in Green categories, and Ringel's approach via Hall algebras. Generically and for certain values of the quantum parameter these two approaches produce isomorphic objects, as shown by Green. This note studies a case where these objects turn out to be different. For v=−1, the algebra arising from Lusztig's approach is shown to be super-commutative for any datum (I,⋅), whereas Hall algebras are super-commutative only in a trivial case.