What is a quantum vector space? September 24, 2010

Posted by Sarah in Uncategorized.
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I went to the Friday grad seminar — this one by Hyun Kyu on the quantum Teichmuller space. (Here’s his paper, which I haven’t read as of now.) I thought it might be helpful to learn some background about this whole “quantum” business.

One way of thinking about the ordinary plane is to consider it the algebra freely generated by the elements x and y subject to the commutation relationship yx = xy. Now, what if we alter this description to instead have yx = qxy? This defines something known as the quantum plane. Here, q is an element of the ground field. Obviously, except when q = 1, this is a non-commutative algebra. For any pair of integers i and j, we have
$y^i x^i = q^{ij} x^i y^j.$
We can define quantum versions of lots of things — for example, $SL_q(2, \mathbb{C})$ is the group of 2-by-2 matrices with determinant 1, satisfying the relations

ab = 2ba, bc = cb, cd = q dc, ac = q ca, bd = q db, and ad-da = (q – 1/q)bc.

The “quantum” here is in the mathematical sense of a non-commutative deformation of a commutative algebra.
More on the subject: “What is a Quantum Group?”