In this paper we describe connections among extraspecial 2-groups, unitary representations of the braid group and multi-qubit braiding quantum gates. We first construct new representations of extraspecial 2-groups. Extending the latter by the symmetric group, we construct new unitary braid representations, which are solutions to generalized Yang-Baxter equations and use them to realize new brai...

We shed some light on the problem of determining the orbits of the braid group action on semiorthonormal bases of Mukai lattices as considered in [7] and [8]. We show that there is an algebraic (and in particular algorithmic) equivalence between this problem and the Hurwitz problem for integer matrix groups finitely generated by involutions. In particular we consider the case of K0(P n) n > 4 w...

The goal of this paper is to define new deformations of group algebras of Coxeter groups. Recall that a Coxeter group W is generated by elements si, i ∈ I modulo two kinds of relations – the involutivity relations si = 1 and the relations (sisj) mij = 1, where 2 ≤ mij = mji ≤ ∞; in presence of the involutivity relations these are equivalent to the braid relations sisjsi... = sjsisj... (mij fact...

We give a new infinite family of group homomorphisms from the braid group Bk to the symmetric group Smk for all k and m ≥ 2. Most known permutation representations of braids are included in this family. We prove that the homomorphisms in this family are non-cyclic and transitive. For any divisor l of m, 1 ≤ l < m, we prove in particular that if m l is odd then there are 1+ m l non-conjugate hom...

W. Magnus’ representations of submonoids E ≤ End(F ) of the endomorphisms of a free group F of finite rank are generalised by identifying them with the first homology group of F with particular coefficient modules. By considering a suitable free resolution of the integers over the semidirect product of free groups, a class of representations of the braid group can be obtained on higher homology...

1a. Generalities. This paper investigates the connection between symplectic geometry and those parts of representation theory which revolve around the notion of categorification. The existence of such a connection, in an abstract sense, follows from simple general ideas. The difficult thing is to make it explicit. On the symplectic side, the tools needed for a systematic study of this question ...

We find stability conditions ([D2], [Br]) on some derived categories of differential graded modules over a graded algebra studied in [RZ], [KS]. This category arises in both derived Fukaya categories and derived categories of coherent sheaves. This gives the first examples of stability conditions on the A-model side of mirror symmetry, where the triangulated category is not naturally the derive...

In this paper, we describe the braid group, first mentioned in [2]. We give applications to cryptography mentioned in [1] and [8]. We then describe an apparently new method for shortening braid descriptions which could be useful for transmitting braids over a limited bandwidth channel.

In this paper, we give a Gröbner-Shirshov basis of the braid group Bn+1 in the Artin–Garside generators. As results, we obtain a new algorithm for getting the Garside normal form, and a new proof that the braid semigroup B + n + 1 is the subsemigroup in Bn+1.

In the paper, we describe the teleportation from the viewpoints of the braid group and Temperley–Lieb algebra. We propose the virtual braid teleportation which exploits the teleportation swapping and identifies unitary braid representations with universal quantum gates, and further suggest the braid teleportation which is explained in terms of the crossed measurement and the state model of knot...