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

Question

Define n-quantum vector space to be the algebra

$${\mathbb C}_q^n := \mathbb{C}< x_i ~ | ~ i =1, \ldots, N>/<x_i x_j = q x_j x_i ~ | ~ i< j>.$$ 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.

This post imported from StackExchange MathOverflow at 2014-09-29 17:32 (UTC), posted by SE-user User1298

Comments

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...

This post imported from StackExchange MathOverflow at 2014-09-29 17:32 (UTC), posted by SE-user Matthias Wendt

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Thanks, but I checked the book and it gives a result only when $N=2$.

This post imported from StackExchange MathOverflow at 2014-09-29 17:32 (UTC), posted by SE-user User1298

Answer 1

These algebras are discussed by Odesskii in his  article on Elliptic Algebras. He states that there is no general result known for $N>3$. This might be a good starting point. (When $N=3$, this is space of vacua for the generic Leigh-Strassler deformation of $\mathcal{N}=4$ supersymmetric Yang-Mills theory.)