We introduce skein theoretic techniques to compute the Yang-Baxter (YB) homology and cohomology groups of the R-matrix corresponding to the Jones polynomial. More specifically, we show that the YB ...operator R for Jones, normalized for homology, admits a skein decomposition R=I+βα, where α:V⊗2→k is a “cup” pairing map and β:k→V⊗2 is a “cap” copairing map, and differentials in the chain complex associated to R can be decomposed into horizontal tensor concatenations of cups and caps. We apply our skein theoretic approach to determine the second and third YB homology groups, confirming a conjecture of Przytycki and Wang. Further, we compute the cohomology groups of R, and provide computations in higher dimensions that yield some annihilations of submodules.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
We construct ternary self-distributive (TSD) objects from compositions of binary Lie algebras, 3-Lie algebras and, in particular, ternary Nambu-Lie algebras. We show that the structures obtained ...satisfy an invertibility property resembling that of racks. We prove that these structures give rise to Yang-Baxter operators in the tensor product of the base vector space and, upon defining suitable twisting isomorphisms, we obtain representations of the infinite (framed) braid group. We consider examples for low-dimensional Lie algebras, where the ternary bracket is defined by composition of the binary ones, along with simple 3-Lie algebras. We show that the Yang-Baxter operators obtained are not gauge equivalent to the transposition operator, and we consider the problem of deforming the operators to obtain new solutions to the Yang-Baxter equation. We discuss the applications of this deformation procedure to the construction of (framed) link invariants.
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Bożejko and Speicher associated a finite von Neumann algebra MT to a self-adjoint operator T on a complex Hilbert space of the form $\mathcal {H}\otimes \mathcal {H}$ which satisfies the Yang–Baxter ...relation and $ \left\| T \right\| < 1$. We show that if dim$(\mathcal {H})$ ⩾ 2, then MT is a factor when T admits an eigenvector of some special form.
This is a paper in a series to study vertex algebra-like structures arising from various algebras including quantum affine algebras and Yangians. In this paper, we develop a theory of what we call ...(weak) quantum vertex
F
(
(
t
)
)
-algebras with
F
a field of characteristic zero and
t a formal variable, and we give a conceptual construction of (weak) quantum vertex
F
(
(
t
)
)
-algebras and their modules. As an application, we associate weak quantum vertex
F
(
(
t
)
)
-algebras to quantum affine algebras, providing a solution to a problem posed by Frenkel and Jing. We also explicitly construct an example of quantum vertex
F
(
(
t
)
)
-algebras from a certain quantum
βγ-system.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Kauffman and Lomonaco (New J Phys 4:73.1–73.18,
2002
.
arXiv:quant-ph/0401090
, New J Phys 6:134.1–134.40,
2004
) explored the idea of understanding quantum entanglement (the non-local correlation of ...certain properties of particles) topologically by viewing unitary entangling operators as braiding operators. In Alagic et al. (Yang–Baxter operators need quantum entanglement to distinguish knots,
2015
.
arXiv:1507.05979v1
), it is shown that entanglement is a necessary condition for forming non-trivial invariants of knots from braid closures via solutions to the Yang–Baxter equation. We show that the arguments used by Alagic et al. (
2015
) generalize to essentially the same results for quantum invariant state summation models of knots. In one case (the unoriented swap case) we give an example of a Yang–Baxter operator, and associated quantum invariant, that can detect the Hopf link. Again this is analogous to the results of Alagic et al. (
2015
). We also give a class of
R
matrices that are entangling and are weak invariants of classical knots and links yet strong invariants of virtual knots and links. We also give an example of an
SU
(2) representation of the three-strand braid group that models the Jones polynomial for closures of three-strand braids. This invariant is a quantum model for the Jones polynomial restricted to three-strand braids, and it does not involve quantum entanglement. These relationships between topological braiding and quantum entanglement can be used as a framework for future work in understanding the properties of entangling gates in topological quantum computing. The paper ends with a discussion of the Aravind hypothesis about the direct relationship of knots and quantum entanglement and the
E
R
=
E
P
R
hypothesis about the relationship of quantum entanglement with the connectivity of space. We describe how, given a background space and a quantum tensor network, to construct a new topological space that welds the network and the background space together. This construction embodies the principle that quantum entanglement and topological connectivity are intimately related.
In this paper we introduce the notions of weak Yang–Baxter operator and weak braided Hopf algebra. We prove that it is possible to obtain examples of these notions working with Yetter–Drinfeld ...modules associated to a weak Hopf algebra
H with invertible antipode. Finally, we complete the study of the structure of weak Hopf algebras with a projection obtaining a categorical equivalence between the category of weak Hopf algebra projections associated to
H and the category of Hopf algebras in the non-strict braided monoidal category of left–left Yetter–Drinfeld modules over
H.
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In his study of quantum groups, Drinfeld suggested considering set-theoretic solutions of the Yang-Baxter equation as a discrete analogon. As a typical example, every conjugacy class in a group or, ...more generally, every rack Q c_Q \colon x \otimes y \mapsto y \otimes x^y within the space of Yang-Baxter operators over some complete ring. Infinitesimally these deformations are classified by Yang-Baxter cohomology. We show that the Yang-Baxter cochain complex of c_Q, including the modular case which had previously been left in suspense, by establishing that every deformation of c_Q interact; if all elements of Q collapses to its diagonal part, which we identify with rack cohomology. This establishes a strong relationship between the classical deformation theory following Gerstenhaber and the more recent cohomology theory of racks, both of which have numerous applications in knot theory.
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In this paper we give an introduction to the theory of weak braided Hopf algebras proposed as a braided version of weak Hopf algebras. Also, we obtain the fundamental theorem of Hopf modules for ...these algebraic structures.
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