# $q$-Deformed Quillen–Suslin Theorem for the Quantum Vector Spaces?

Define n-quantum vector space to be the algebra $${\mathbb C}_q^n := \mathbb{C}\left< x_i \mid i =1, \ldots, N\right>/\left<x_i x_j = q x_j x_i \mid i<j\right>.$$ For $q=1$, we get the usual polynomial ring in $n$-variables, and so, Serre's conjecture (Quillen–Suslin theorem) tells that every finitely generated projective module over ${\mathbb C}_1^n$ is free. How does this work for $q \neq 1$? Is there a $q$-deformed Quillen–Suslin theorem? The not a root of unity case is the most interesting to me.

• Have you checked out Lam's book "Serre's problem on projective modules" (the 2006 edition)? Chapter VIII (New developments since 1977) contains subsections on non-commutative and quantum versions. I do not have my copy handy, so I can not check right now if your question is answered there... – Matthias Wendt Sep 22 '14 at 20:54
• Thanks, but I checked the book and it gives a result only when $N=2$. – Juan Corrida Sep 22 '14 at 21:13
• See also this remark on PhysicsOverflow to the same questions: physicsoverflow.org/24083/… – Dilaton Sep 30 '14 at 6:39

F.g. projective modules of sufficiently HIGH RANK are free, at least at two cases: (1) $q_{ij}$ are roots of unity (2) $q_{ij}$ are at general position.