Burau representation

Last updated March 22, 2024

In mathematics the Burau representation is a representation of the braid groups, named after and originally studied by the German mathematician Werner Burau^[1] during the 1930s. The Burau representation has two common and near-equivalent formulations, the reduced and unreduced Burau representations.

Definition

The covering space Cn may be thought of concretely as follows: cut the disk along lines from the boundary to the marked points. Take as many copies of the result as there are integers, stack them vertically, and connect them by ramps going from one side of the cut on one level to the other side of the cut on the level below. This procedure is shown here for n = 4; the covering transformations t act by shifting the space vertically. InfiniteCyclicCovering.gif — The covering space $C n$ may be thought of concretely as follows: cut the disk along lines from the boundary to the marked points. Take as many copies of the result as there are integers, stack them vertically, and connect them by ramps going from one side of the cut on one level to the other side of the cut on the level below. This procedure is shown here for $n = 4$ ; the covering transformations $t$ act by shifting the space vertically.

Consider the braid group $B n$ to be the mapping class group of a disc with $n$ marked points $D n$ . The homology group $H 1 (D n)$ is free abelian of rank $n$ . Moreover, the invariant subspace of $H 1 (D n)$ (under the action of $B n$ ) is primitive and infinite cyclic. Let $π : H 1 (D n) \to Z$ be the projection onto this invariant subspace. Then there is a covering space $C n$ corresponding to this projection map. Much like in the construction of the Alexander polynomial, consider $H 1 (C n)$ as a module over the group-ring of covering transformations $Z [Z]$ , which is isomorphic to the ring of Laurent polynomials $Z [t, t -1]$ . As a $Z [t, t -1]$ -module, $H 1 (C n)$ is free of rank $n - 1$ . By the basic theory of covering spaces, $B n$ acts on $H 1 (C n)$ , and this representation is called the reduced Burau representation.

The unreduced Burau representation has a similar definition, namely one replaces $D n$ with its (real, oriented) blow-up at the marked points. Then instead of considering $H 1 (C n)$ one considers the relative homology $H 1 (C n, Γ)$ where $γ \subset D n$ is the part of the boundary of $D n$ corresponding to the blow-up operation together with one point on the disc's boundary. $Γ$ denotes the lift of $γ$ to $C n$ . As a $Z [t, t -1]$ -module this is free of rank $n$ .

By the homology long exact sequence of a pair, the Burau representations fit into a short exact sequence

0 \to V r \to V u \to D \oplus Z [t, t -1] \to 0,

where $V r$ (resp. $V u$ ) is the reduced (resp. unreduced) Burau $B n$ -module and $D \subset Z n$ is the complement to the diagonal subspace, in other words:

D=\left\{\left(x_{1},\cdots ,x_{n}\right)\in \mathbf {Z} ^{n}:x_{1}+\cdots +x_{n}=0\right\},

and $B n$ acts on $Z n$ by the permutation representation.

Explicit matrices

Let $σ i$ denote the standard generators of the braid group $B n$ . Then the unreduced Burau representation may be given explicitly by mapping

\sigma _{i}\mapsto \left({\begin{array}{c|cc|c}I_{i-1}&0&0&0\\\hline 0&1-t&t&0\\0&1&0&0\\\hline 0&0&0&I_{n-i-1}\end{array}}\right),

for $1 \leq i \leq n - 1$ , where $I k$ denotes the $k \times k$ identity matrix. Likewise, for $n \geq 3$ the reduced Burau representation is given by

\sigma _{1}\mapsto \left({\begin{array}{cc|c}-t&1&0\\0&1&0\\\hline 0&0&I_{n-3}\end{array}}\right),

\sigma _{i}\mapsto \left({\begin{array}{c|ccc|c}I_{i-2}&0&0&0&0\\\hline 0&1&0&0&0\\0&t&-t&1&0\\0&0&0&1&0\\\hline 0&0&0&0&I_{n-i-2}\end{array}}\right),\quad 2\leq i\leq n-2,

\sigma _{n-1}\mapsto \left({\begin{array}{c|cc}I_{n-3}&0&0\\\hline 0&1&0\\0&t&-t\end{array}}\right),

while for $n = 2$ , it maps

\sigma _{1}\mapsto \left(-t\right).

Bowling alley interpretation

Vaughan Jones ^[2] gave the following interpretation of the unreduced Burau representation of positive braids for $t$ in $[0,1]$ – i.e. for braids that are words in the standard braid group generators containing no inverses – which follows immediately from the above explicit description:

Given a positive braid $σ$ on $n$ strands, interpret it as a bowling alley with $n$ intertwining lanes. Now throw a bowling ball down one of the lanes and assume that at every crossing where its path crosses over another lane, it falls down with probability $t$ and continues along the lower lane. Then the $(i, j)$ 'th entry of the unreduced Burau representation of $σ$ is the probability that a ball thrown into the $i$ 'th lane ends up in the $j$ 'th lane.

Relation to the Alexander polynomial

If a knot $K$ is the closure of a braid $f$ in $B n$ , then, up to multiplication by a unit in $Z [t, t -1]$ , the Alexander polynomial $Δ K (t)$ of $K$ is given by

{\frac {1-t}{1-t^{n}}}\det(I-f_{*}),

where $f *$ is the reduced Burau representation of the braid $f$ .

For example, if $f = σ 1 σ 2$ in $B 3$ , one finds by using the explicit matrices above that

{\frac {1-t}{1-t^{n}}}\det(I-f_{*})=1,

and the closure of $f *$ is the unknot whose Alexander polynomial is $1$ .

Faithfulness

The first nonfaithful Burau representations were found by John A. Moody without the use of computer, using a notion of winding number or contour integration.^[3] A more conceptual understanding, due to Darren D. Long and Mark Paton^[4] interprets the linking or winding as coming from Poincaré duality in first homology relative to the basepoint of a covering space, and uses the intersection form (traditionally called Squier's Form as Craig Squier was the first to explore its properties).^[5] Stephen Bigelow combined computer techniques and the Long–Paton theorem to show that the Burau representation is not faithful for $n \geq 5$ .^[6]^[7]^[8] Bigelow moreover provides an explicit non-trivial element in the kernel as a word in the standard generators of the braid group: let

\psi _{1}=\sigma _{3}^{-1}\sigma _{2}\sigma _{1}^{2}\sigma _{2}\sigma _{4}^{3}\sigma _{3}\sigma _{2},\quad \psi _{2}=\sigma _{4}^{-1}\sigma _{3}\sigma _{2}\sigma _{1}^{-2}\sigma _{2}\sigma _{1}^{2}\sigma _{2}^{2}\sigma _{1}\sigma _{4}^{5}.

Then an element of the kernel is given by the commutator

[\psi _{1}^{-1}\sigma _{4}\psi _{1},\psi _{2}^{-1}\sigma _{4}\sigma _{3}\sigma _{2}\sigma _{1}^{2}\sigma _{2}\sigma _{3}\sigma _{4}\psi _{2}].

The Burau representation for $n = 2, 3$ has been known to be faithful for some time. The faithfulness of the Burau representation when $n = 4$ is an open problem. The Burau representation appears as a summand of the Jones representation, and for $n = 4$ , the faithfulness of the Burau representation is equivalent to that of the Jones representation, which on the other hand is related to the question of whether or not the Jones polynomial is an unknot detector.^[9]

Geometry

Craig Squier showed that the Burau representation preserves a sesquilinear form.^[5] Moreover, when the variable $t$ is chosen to be a transcendental unit complex number near $1$ , it is a positive-definite Hermitian pairing. Thus the Burau representation of the braid group $B n$ can be thought of as a map into the unitary group U(n).

Related Research Articles

In mathematical physics and mathematics, the Pauli matrices are a set of three $2 \times 2$ complex matrices that are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma, they are occasionally denoted by tau when used in connection with isospin symmetries.

In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1⁄2 massive particles, called "Dirac particles", such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine structure of the hydrogen spectrum in a completely rigorous way.

In physics, charge conjugation is a transformation that switches all particles with their corresponding antiparticles, thus changing the sign of all charges: not only electric charge but also the charges relevant to other forces. The term C-symmetry is an abbreviation of the phrase "charge conjugation symmetry", and is used in discussions of the symmetry of physical laws under charge-conjugation. Other important discrete symmetries are P-symmetry (parity) and T-symmetry.

In group theory, the quaternion group Q₈ (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset $of the quaternions under multiplication. It is given by the group presentation$

In mathematics, the unitary group of degree n, denoted U(n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL(n, C). Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group.

In mathematics, group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group G in an associated G-moduleM to elucidate the properties of the group. By treating the G-module as a kind of topological space with elements of $representing n -simplices, topological properties of the space may be computed, such as the set of cohomology groups . The cohomology groups in turn provide insight into the structure of the group G and G -module M themselves. Group cohomology plays a role in the investigation of fixed points of a group action in a module or space and the quotient module or space with respect to a group action. Group cohomology is used in the fields of abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper. As in algebraic topology, there is a dual theory called group homology . The techniques of group cohomology can also be extended to the case that instead of a G -module, G acts on a nonabelian G -group; in effect, a generalization of a module to non-Abelian coefficients.$

In mathematics, the braid group on $n$ strands, also known as the Artin braid group, is the group whose elements are equivalence classes of $n$ -braids, and whose group operation is composition of braids. Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids ; in mathematical physics where Artin's canonical presentation of the braid group corresponds to the Yang–Baxter equation ; and in monodromy invariants of algebraic geometry.

In mathematics, particularly in operator theory and C*-algebra theory, the continuous functional calculus is a functional calculus which allows the application of a continuous function to normal elements of a C*-algebra.

In mathematics, the Fubini–Study metric is a Kähler metric on a complex projective space CPⁿ endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study.

In mathematical physics, the gamma matrices, $also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra It is also possible to define higher-dimensional gamma matrices. When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of spacetime acts. This in turn makes it possible to represent infinitesimal spatial rotations and Lorentz boosts. Spinors facilitate spacetime computations in general, and in particular are fundamental to the Dirac equation for relativistic spin particles. Gamma matrices were introduced by Paul Dirac in 1928.$

In physics, the Majorana equation is a relativistic wave equation. It is named after the Italian physicist Ettore Majorana, who proposed it in 1937 as a means of describing fermions that are their own antiparticle. Particles corresponding to this equation are termed Majorana particles, although that term now has a more expansive meaning, referring to any fermionic particle that is its own anti-particle.

In mathematical physics, the Dirac algebra is the Clifford algebra $. This was introduced by the mathematical physicist P. A. M. Dirac in 1928 in developing the Dirac equation for spin- 1 / 2 particles with a matrix representation of the gamma matrices, which represent the generators of the algebra.$

In physics, and specifically in quantum field theory, a bispinor is a mathematical construction that is used to describe some of the fundamental particles of nature, including quarks and electrons. It is a specific embodiment of a spinor, specifically constructed so that it is consistent with the requirements of special relativity. Bispinors transform in a certain "spinorial" fashion under the action of the Lorentz group, which describes the symmetries of Minkowski spacetime. They occur in the relativistic spin-1/2 wave function solutions to the Dirac equation.

In linear algebra, particularly projective geometry, a semilinear map between vector spaces V and W over a field K is a function that is a linear map "up to a twist", hence semi-linear, where "twist" means "field automorphism of K". Explicitly, it is a function T : V → W that is:

In mathematics, the ELSV formula, named after its four authors Torsten Ekedahl, Sergei Lando, Michael Shapiro, Alek Vainshtein, is an equality between a Hurwitz number and an integral over the moduli space of stable curves.

In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three possible types of elementary fermions, the other two being the Dirac and the Majorana fermions.

<span class="mw-page-title-main">HPP model</span>

The Hardy–Pomeau–Pazzis (HPP) model is a fundamental lattice gas automaton for the simulation of gases and liquids. It was a precursor to the lattice Boltzmann methods. From lattice gas automata, it is possible to derive the macroscopic Navier-Stokes equations. Interest in lattice gas automaton methods levelled off in the early 1990s, due to rising interest in the lattice Boltzmann methods.

In electromagnetism, a branch of fundamental physics, the matrix representations of the Maxwell's equations are a formulation of Maxwell's equations using matrices, complex numbers, and vector calculus. These representations are for a homogeneous medium, an approximation in an inhomogeneous medium. A matrix representation for an inhomogeneous medium was presented using a pair of matrix equations. A single equation using 4 × 4 matrices is necessary and sufficient for any homogeneous medium. For an inhomogeneous medium it necessarily requires 8 × 8 matrices.

Attempts have been made to describe gauge theories in terms of extended objects such as Wilson loops and holonomies. The loop representation is a quantum hamiltonian representation of gauge theories in terms of loops. The aim of the loop representation in the context of Yang–Mills theories is to avoid the redundancy introduced by Gauss gauge symmetries allowing to work directly in the space of physical states. The idea is well known in the context of lattice Yang–Mills theory. Attempts to explore the continuous loop representation was made by Gambini and Trias for canonical Yang–Mills theory, however there were difficulties as they represented singular objects. As we shall see the loop formalism goes far beyond a simple gauge invariant description, in fact it is the natural geometrical framework to treat gauge theories and quantum gravity in terms of their fundamental physical excitations.

In mathematics, Zolotarev polynomials are polynomials used in approximation theory. They are sometimes used as an alternative to the Chebyshev polynomials where accuracy of approximation near the origin is of less importance. Zolotarev polynomials differ from the Chebyshev polynomials in that two of the coefficients are fixed in advance rather than allowed to take on any value. The Chebyshev polynomials of the first kind are a special case of Zolotarev polynomials. These polynomials were introduced by Russian mathematician Yegor Ivanovich Zolotarev in 1868.

References

↑ Burau, Werner (1936). "Über Zopfgruppen und gleichsinnig verdrillte Verkettungen". Abh. Math. Sem. Univ. Hamburg. 11: 179–186. doi:10.1007/bf02940722. S2CID 119576586.
↑ Jones, Vaughan (1987). "Hecke algebra representations of Braid Groups and Link Polynomials". Annals of Mathematics. Second Series. 126 (2): 335–388. doi:10.2307/1971403. JSTOR 1971403.
↑ Moody, John Atwell (1993), "The faithfulness question for the Burau representation", Proceedings of the American Mathematical Society , 119 (2): 671–679, doi: 10.1090/s0002-9939-1993-1158006-x , JSTOR 2159956, MR 1158006
↑ Long, Darren D.; Paton, Mark (1993), "The Burau representation is not faithful for $n\geq 6$ ", Topology , 32 (2): 439–447, doi: 10.1016/0040-9383(93)90030-Y , MR 1217079
1 2 Squier, Craig C (1984). "The Burau representation is unitary". Proceedings of the American Mathematical Society . 90 (2): 199–202. doi: 10.2307/2045338 . JSTOR 2045338.
↑ Bigelow, Stephen (1999). "The Burau representation is not faithful for $n = 5$ ". Geometry & Topology . 3: 397–404. arXiv: math/9904100 . doi:10.2140/gt.1999.3.397. S2CID 5967061.
↑ S. Bigelow, International Congress of Mathematicians, Beijing, 2002
↑ Vladimir Turaev, Faithful representations of the braid groups, Bourbaki 1999-2000
↑ Bigelow, Stephen (2002). "Does the Jones polynomial detect the unknot?". Journal of Knot Theory and Its Ramifications . 11 (4): 493–505. arXiv: math/0012086 . doi:10.1142/s0218216502001779. S2CID 1353805.

External links

" Burau's Theorem ", The Knot Atlas .

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[burau-1] Burau, Werner (1936). "Über Zopfgruppen und gleichsinnig verdrillte Verkettungen". Abh. Math. Sem. Univ. Hamburg. 11: 179–186. doi:10.1007/bf02940722. S2CID 119576586.

[Jones-2] Jones, Vaughan (1987). "Hecke algebra representations of Braid Groups and Link Polynomials". Annals of Mathematics. Second Series. 126 (2): 335–388. doi:10.2307/1971403. JSTOR 1971403.

[3] Moody, John Atwell (1993), "The faithfulness question for the Burau representation", Proceedings of the American Mathematical Society , 119 (2): 671–679, doi: 10.1090/s0002-9939-1993-1158006-x , JSTOR 2159956, MR 1158006

[lp-4] Long, Darren D.; Paton, Mark (1993), "The Burau representation is not faithful for $n\geq 6$ ", Topology , 32 (2): 439–447, doi: 10.1016/0040-9383(93)90030-Y , MR 1217079

[squier-5] 1 2 Squier, Craig C (1984). "The Burau representation is unitary". Proceedings of the American Mathematical Society . 90 (2): 199–202. doi: 10.2307/2045338 . JSTOR 2045338.

[bigelow-6] Bigelow, Stephen (1999). "The Burau representation is not faithful for $n = 5$ ". Geometry & Topology . 3: 397–404. arXiv: math/9904100 . doi:10.2140/gt.1999.3.397. S2CID 5967061.

[icm-7] S. Bigelow, International Congress of Mathematicians, Beijing, 2002

[8] Vladimir Turaev, Faithful representations of the braid groups, Bourbaki 1999-2000

[bigelowunknot-9] Bigelow, Stephen (2002). "Does the Jones polynomial detect the unknot?". Journal of Knot Theory and Its Ramifications . 11 (4): 493–505. arXiv: math/0012086 . doi:10.1142/s0218216502001779. S2CID 1353805.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]