Denseness and Zariski denseness of Jones braid representations

Abstract

Using various tools from representation theory and group theory, but without using hard classification theorems such as the classification of finite simple groups, we show that the Jones representations of braid groups are dense in the (complex) Zariski topology when the parameter t is not a root of unity. As first established by Freedman, Larsen and Wang, we obtain the same result when t is a nonlattice root of unity, other than one initial case when t has order 10. We also compute the real Zariski closure of these representations (meaning, the closure in Zariski closure of the real Weil restriction). When such a representation is indiscrete in the analytic topology, then its analytic closure is the same as its real Zariski closure. In this article we will study representations of braid groups associated with the Jones polynomial J.L;t/. Our question is to determine the closures of these representations, which are then Lie groups. Freedman, Larsen and Wang [4] computed these closures in the important case where tD exp.2 i=r/ is a principal root of unity. In this case, the (reduced) braid representations are unitary, and the braids can be interpreted as quantum circuits. Freedman, Larsen and Wang established that the Jones representations are eventually dense if rD 5 or r 7. This has the important corollary that these representations are universal for quantum computation. In this article t will usually be a complex number which is not a root of unity. Although the Jones polynomial is not directly a model of quantum computation for these values of t , the closure of the braid group representation is still interesting for related questions in complexity theory; see Ahargonov, Arad, Eban and Landau [1]. Also, we will say more about the Zariski closure of the braid group action in the target group GL.N;C/, rather than the closure in the usual topology. Switching to the Zariski topology simplifies the question, and yet in many cases it does not change the question very much. To be precise, we will consider two Zariski topologies on GL.N;C/ and SL.N;C/. The usual Zariski topology is generated by polynomials in the complex matrix entries, and