Infinite Braid Group Research Articles

Yang–Baxter endomorphisms

Every unitary solution of the Yang-Baxter equation (R-matrix) in dimension $d$ can be viewed as a unitary element of the Cuntz algebra ${\mathcal O}_d$ and as such defines an endomorphism of ${\mathcal O}_d$. These Yang-Baxter endomorphisms restrict and extend to endomorphisms of several other $C^*$- and von Neumann algebras and furthermore define a II$_1$ factor associated with an extremal character of the infinite braid group. This paper is devoted to a detailed study of such Yang-Baxter endomorphisms. Among the topics discussed are characterizations of Yang-Baxter endomorphisms and the relative commutants of the various subfactors they induce, an endomorphism perspective on algebraic operations on R-matrices such as tensor products and cabling powers, and properties of characters of the infinite braid group defined by R-matrices. In particular, it is proven that the partial trace of an R-matrix is an invariant for its character by a commuting square argument. Yang-Baxter endomorphisms also supply information on R-matrices themselves, for example it is shown that the left and right partial traces of an R-matrix coincide and are normal, and that the spectrum of an R-matrix can not be concentrated in a small disc. Upper and lower bounds on the minimal and Jones indices of Yang-Baxter endomorphisms are derived, and a full characterization of R-matrices defining ergodic endomorphisms is given. As examples, so-called simple R-matrices are discussed in any dimension $d$, and the set of all Yang-Baxter endomorphisms in $d=2$ is completely analyzed.

Conjugation-invariant norms on the commutator subgroup of the infinite braid group

In this paper, we give another proof of the result of Brandenbursky and Kȩdra which says that the commutator subgroup of the infinite braid group admits stably unbounded norms. Moreover, we observe that the norms which we construct are equivalent to the bi-invariant word norm studied by Brandenbursky and Kȩdra.

The dilogarithmic central extension of the Ptolemy–Thompson group via the Kashaev quantization

Quantization of universal Teichmüller space provides projective representations of the Ptolemy–Thompson group, which is isomorphic to the Thompson group T. This yields certain central extensions of T by Z, called dilogarithmic central extensions. We compute a presentation of the dilogarithmic central extension TˆKash of T resulting from the Kashaev quantization, and show that it corresponds to 6 times the Euler class in H2(T;Z). Meanwhile, the braided Ptolemy–Thompson groups T⁎, T♯ of Funar–Kapoudjian are extensions of T by the infinite braid group B∞, and by abelianizing the kernel B∞ one constructs central extensions Tab⁎, Tab♯ of T by Z, which are of topological nature. We show TˆKash≅Tab♯. Our result is analogous to that of Funar and Sergiescu, who computed a presentation of another dilogarithmic central extension TˆCF of T resulting from the Chekhov–Fock(–Goncharov) quantization and thus showed that it corresponds to 12 times the Euler class and that TˆCF≅Tab⁎. In addition, we suggest a natural relationship between the two quantizations in the level of projective representations.

Concordance group and stable commutator length in braid groups

We define a quasihomomorphism from braid groups to the concordance group of knots and examine its properties and consequences of its existence. In particular, we provide a relation between the stable four ball genus in the concordance group and the stable commutator length in braid groups, and produce examples of infinite families of concordance classes of knots with uniformly bounded four ball genus. We also provide applications to the geometry of the infinite braid group. In particular, we show that its commutator subgroup admits a stably unbounded conjugation invariant norm. This answers an open problem posed by Burago, Ivanov and Polterovich.

Subfactors from braided C∗tensor categories

We extend subfactor constructions originally defined for unitary braid representations to the setting of braided C -tensor categories. The categorical approach is then used to compute the principal graph of these subfactors. We also determine the dual principal graph for several important cases. Here invertibility of the so-called S-matrix of a subcategory and certain related group actions play an important role. It was noted by Vaughan Jones that his examples of subfactors gave rise to unitary braid representations. By this we mean representations of the infinite braid group B1 defined by infinitely many generators 1; 2;::: which satisfy the familiar braid relations. Subsequently, unitary braid representations were used by A. Ocneanu and by H. Wenzl to construct new examples of subfactors; here the subfactor is given by the subgroup B2;1 generated by 2; 3;::: . This construction was denoted as the one-sided subfactor construction by J. Erlijman, as opposed to her multisided subfactors. Here, for a given integer s > 1, the s-sided subfactor is obtained as a suitable inductive limit of the embeddings of the quotients of B s D Bn

STRANGE QUESTIONS ABOUT BRAIDS

The infinite braid group B∞ admits a left self-distributive structure. In particular, it includes a free monogenerated left self-distributive system, and, therefore, it inherits all properties of the latter object. Here we discuss how such algebraic properties translate into the language of braids. We state new results about braids and propose a list of open questions.

On braid words and irreflexivity

A left-distributive algebra is a set B equipped with a binary operation (here written as concatenation) such that a(bc) = (ab)(ac) for all a, b, c ∈ B. The free left-distributive algebra on n letters is denoted An, and we write A for A1 If P,Q ∈ B write P <L Q iff one can write P as a strict prefix of Q, i.e., Q = ((PQ1) . . .)Qk for some Q1, . . . , Qk, k ≥ 1. Then a proof that <L is irreflexive (that is, that P 6= ((PQ1) . . .)Qk for all P,Q1, . . . , Qk ∈ A) on A was found by R. Laver ([Lav 92]), under large cardinal assumptions, as part of a theorem that A is isomorphic to a certain algebra of elementary embeddings from set theory. It was also proved in [Lav 92] that <L linearly orders A, the part that for all P,Q ∈ A at least one of P <L Q,P = Q,Q <L P holds being proved independently and by a different method by P. Dehornoy ([Deh 89a, Deh 89b]). The linear ordering of A gives left cancellation, the solvability of the word problem, and other consequences. Left open was whether irreflexivity, and hence the linear ordering, can be proved in ZFC. Recently, Dehornoy ([Deh 92]) has found such a proof, involving an extension of the infinite braid group but without invoking axioms extending ZFC. The purpose of this note is to prove the result without the additional machinery of this extended group, and at shorter length.