Related papers: Multi-switches and representations of braid groups
A notion of an algebroid - a generalization of a Lie algebroid structure is introduced. We show that many objects of the differential calculus on a manifold M associated with the canonical Lie algebroid structure on T^M can be obtained in…
We introduce a visual representation of qubits to assist in explaining quantum computing to a broad audience. The representation follows from physical devices that we developed to explain superposition, entanglement, measurement, phases,…
This paper defines a theory of cobordism for virtual knots and studies this theory for standard and rotational virtual knots and links. Non-trivial examples of virtual slice knots are given. Determinations of the four-ball genus of positive…
Using the theory of representations of the symmetric group, we propose an algorithm to compute the invariant ring of a permutation group. Our approach have the goal to reduce the amount of linear algebra computations and exploit a thinner…
We introduce a generalization of representations of quivers that contains also representations of posets, vectorspace problems and other matrix problems. Many examples, some of which are given in the paper, show that the language of marked…
Motivated by appearance of multisemigroups in the study of additive $2$-categories, we define and investigate the notion of a multisemigroup with multiplicities. This notion seems to be better suitable for applications in higher…
A network representation is useful for describing the structure of a large variety of complex systems. However, most real and engineered systems have multiple subsystems and layers of connectivity, and the data produced by such systems is…
For $n \geq 2$ we describe an $O(l^3n)$-time algorithm that determines if a length $l$ virtual braid word in the standard presentation of the virtual braid group ${\mathcal VB}_n$ represents the trivial virtual braid.
This is a systematic introduction for physicists to the theory of algebras and groups with braid statistics, as developed over the last three years by the author. There are braided lines, braided planes, braided matrices and braided groups…
Multicrossings, which have previously been defined for classical knots and links, are extended to virtual knots and links. In particular, petal diagrams are shown to exist for all virtual knots.
A virtual link can be understood as a link in a trivial I-bundle over an orientable compact surface with genus. A twisted virtual link is a link in a trivial I-bundle over a not-necessarily orientable compact surface. A twisted virtual…
The purpose of this paper is to make the theory of vertex algebras trivial. We do this by setting up some categorical machinery so that vertex algebras are just ``singular commutative rings'' in a certain category. This makes it easy to…
We investigate the braid group representations arising from categories of representations of twisted quantum doubles of finite groups. For these categories, we show that the resulting braid group representations always factor through finite…
We clarify some aspects of quantum group gauge theory and its recent generalisations (by T. Brzezinski and the author) to braided group gauge theory and coalgebra gauge theory. We outline the diagrammatic version of the braided case. We…
The notion of mixed representations of quivers can be derived from ordinary quiver representations by considering the dual action of groups on "vertex" vector spaces together with the usual action. A generating system for the algebra of…
In this paper we state the basics for a signal processing framework on quiver representations. A quiver is a directed graph and a quiver representation is an assignment of vector spaces to the nodes of the graph and of linear maps between…
We consider several classes of knotted objects, namely usual, virtual and welded pure braids and string links, and two equivalence relations on those objects, induced by either self-crossing changes or self-virtualizations. We provide a…
Motivated by the notion of the multi-virtual braid group introduced by L. Kauffman and by the study of extensions of the well-known twin group T_n, n >= 2, we introduce a new group called the multi-virtual twin group M_kVT_n, where k >= 1…
Let $p\ge 3$ be a prime. A generalised multi-edge spinal group is a subgroup of the automorphism group of a regular $p$-adic rooted tree T that is generated by one rooted automorphism and $p$ families of directed automorphisms, each family…
We propose a variational scheme to represent composite quantum systems using multiple parameterized functions of varying accuracies on both classical and quantum hardware. The approach follows the variational principle over the entire…