Braid Group Representations from Twisted Tensor Products of Algebras

Abstract

We unify and generalize several approaches to constructing braid group representations from finite groups, using iterated twisted tensor products. We provide some general characterizations and classification of these representations, focusing on the size of their images, which are typically finite groups. The well-studied Gaussian representations associated with metaplectic modular categories can be understood in this framework, and we give some new examples to illustrate their ubiquity. Our results suggest a relationship between the braiding on the G-gaugings of a pointed modular category ${\mathcal {C}}(A,Q)$ and that of ${\mathcal {C}}(A,Q)$ itself.

Appendix: Computations for \(G=S_3\) and \({\mathcal {A}}_n(Q_8)\)

In what follows we provide some details classifying solutions to the ${\mathcal {A}}(S_3,\tau )$ and ${\mathcal {A}}(Q_8)$-YBE.

1.1 Symmetric Group $S_3$

We let u, v be the generators for $S_3$ with $u^2=v^3=1$ and $uvu=v^2$. For example, we could take $u=(1\;2)$ and $v=(1\;2\;3).$ By the theory above, we initialize with the following Magma code to find conditions on $a,b,c,d,e\in {\mathbb {C}}$ so that $r=1+au+bv+cv^2+duv+euv^2$ is an ${\mathcal {A}}(S_3,\tau )$-YBO.

The ideal of solutions is generated by the coefficients of the monomials in $u_i,v_j$. We enforce invertibility of r by assuming the determinant of the image of r under the faithful $S_3$ representation on ${\mathbb {C}}^3$ is non-zero. The output of the Gröbner basis is the following set of polynomials:

$$\begin{aligned}&\{c, b, e(a^2+d^2+e^2+1), ad+ae+de, a^3+a^2e+2ae^2+de^2+e^3+a+e,\\&\quad -a^2e+ae^2+d^3+2de^2+d\}. \end{aligned}$$

Notice that $c=b=0$, in all cases. If $e=0$ then $ad=0$, and $a^3+a=d^3+d=0$, which are degenerate solutions of the form $1+xu$ that can be obtained from ${\mathbb {Z}}_2$ (see [15]).

If $e\ne 0$, we find that e is a free parameter, and the following code shows that we may normalize to get $r^4=1$. There is a 1-parameter family of solutions for (a, d, e). Moreover, one sees that if we require a unitary solution each of a, d, e should be pure imaginary, and consequently the equation $a^2+d^2+e^2+1=0$ implies that $(a/\mathrm {i},d/\mathrm {i},e/\mathrm {i})$ is a point on the unit sphere. Geometrically, this is the intersection of the unit sphere with the surface given by $xy+xz+yz=0$.

1.2 Quaterionic Algebra ${\mathcal {A}}_n(Q_8)$

For the case of the algebra ${\mathcal {A}}_n(Q_8)$, we use Magma to classify ${\mathcal {A}}(Q_8)$-YBOs. The following is the final code, where the last polynomial relations are the coefficients obtained from an initial run of the normal form command on an initial run (i.e., without the last set of relations). One finds that the non-trivial solutions for (a, b, c) are all $\pm 1$, so that if we want unitary solutions, the inverse of R1 is of the form given as R1i since $u^*=u^{-1}=-u$, etc. We conclude that all unitary solutions are equivalent to the choice $(a,b,c)=(1,1,1)$.