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In this article the authors prove theorem on Lifting for the set of virtual pure braid groups. This theorem says that if they know presentation of virtual pure braid group V P 4 , then they can find presentation of V P n for arbitrary n > 4. Using this theorem they find the set of generators and defining relations for simplicial group T * which was defined in [Bardakov, V. G. and Wu, J., On virtual cabling and structure of 4-strand virtual pure braid group, J. Knot Theory and Ram. , 29 (10), 2020, 1–32]. They find a decomposition of the Artin pure braid group P n in semi-direct product of free groups in the cabled generators.

We first construct an action of the extended double affine braid group$$\\mathcal {\\ddot{B}}$$B ¨ on the quantum toroidal algebra$$U_{q}(\\mathfrak {g}_{\\textrm{tor}})$$U q ( g tor ) in untwisted and twisted types. As a crucial step in the proof, we obtain a finite Drinfeld new style presentation for a broad class of quantum affinizations. In the simply laced cases, using our action and certain involutions of$$\\mathcal {\\ddot{B}}$$B ¨ we produce automorphisms and anti-involutions of$$U_{q}(\\mathfrak {g}_{\\textrm{tor}})$$U q ( g tor ) which exchange the horizontal and vertical subalgebras. Moreover, they switch the central elements C and$$k_{0}^{a_{0}}\\dots k_{n}^{a_{n}}$$k 0 a 0 ⋯ k n a n up to inverse. This can be viewed as the analogue, for these quantum toroidal algebras, of the duality for double affine braid groups used by Cherednik to realise the difference Fourier transform in his celebrated proof of the Macdonald evaluation conjectures. Our work generalises existing results in type A due to Miki which have been instrumental in the study of the structure and representation theory of$$U_{q}(\\mathfrak {sl}_{n+1,\\textrm{tor}})$$U q ( sl n + 1 , tor ) .

We investigate the rational cohomology of the quotient of (generalized) braid groups by the commutator subgroup of the pure braid groups. We provide a combinatorial description of it using isomorphism classes of certain families of graphs. We establish Poincaré dualities for them and prove a stabilization property for the infinite series of reflection groups.

Cactus groups J n are currently attracting considerable interest from diverse mathematical communities. This work explores their relations to right-angled Coxeter groups and, in particular, twin groups T w n and Mostovoy’s Gauss diagram groups D n , which are better understood. Concretely, we construct an injective group 1-cocycle from J n to D n and show that T w n (and its k -leaf generalizations) inject into J n . As a corollary, we solve the word problem for cactus groups, determine their torsion (which is only even) and center (which is trivial), and answer the same questions for pure cactus groups, P J n . In addition, we yield a 1-relator presentation of the first non-abelian pure cactus group P J 4 . Our tools come mainly from combinatorial group theory.

The most general construction of double affine Artin groups (DAAG) and Hecke algebras (DAHA) associates such objects to pairs of compatible reductive group data. We show that DAAG/DAHA It turns out that the structural intricacies of DAAG/DAHA are captured by the underlying We first give a new Coxeter-type presentation for adjoint DAAG as quotients of the Coxeter braid groups associated to certain crystallographic diagrams that we call double affine Coxeter diagrams. As a consequence we show that the rank two Artin groups of type We show further that the above rank two Artin group action descends to an outer action of the congruence subgroup

Given a representation φ : B n → G n of the braid group B n , n ≥ 2 into a group G n , we are considering the problem of whether it is possible to extend this representation to a representation Φ : S M n → A n , where S M n is the singular braid monoid and A n is an associative algebra, in which the group of units contains G n . We also investigate the possibility of extending the representation Φ : S M n → A n to a representation Φ ~ : S B n → A n of the singular braid group S B n . On the other hand, given two linear representations φ 1 , φ 2 : H → G L m ( k ) of a group H into a general linear group over a field k , we define the defect of one of these representations with respect to the other. Furthermore, we construct a linear representation of S B n which is an extension of the Lawrence–Krammer–Bigelow representation (LKBR) and compute the defect of this extension with respect to the exterior product of two extensions of the Burau representation. Finally, we discuss how to derive an invariant of classical links from the Lawrence–Krammer–Bigelow representation.

Representations of braid group Bn on n≥2 strands by automorphisms of a free group of rank n go back to Artin. In 1991, Kauffman introduced a theory of virtual braids, virtual knots, and links. The virtual braid group VBn on n≥2 strands is an extension of the classical braid group Bn by the symmetric group Sn. In this paper, we consider flat virtual braid groups FVBn on n≥2 strands and construct a family of representations of FVBn by automorphisms of free groups of rank 2n. It has been established that these representations do not preserve the forbidden relations between classical and virtual generators. We investigated some algebraic properties of the constructed representations. In particular, we established conditions of faithfulness in case n=2 and proved that the kernel contains a free group of rank two for n≥3.

In this paper, we describe how to explicitly construct infinitely many finite simple groups as characteristic quotients of the rank 2 free group F_2 . This shows that a “baby” version of the Wiegold conjecture [in: Geometry, Rigidity, and Group Actions (2011), 609–643] fails for F_2 and provides counterexamples to two conjectures in the theory of noncongruence subgroups of SL_2(Z) by Chen [Math. Ann. 371 (2018), 41–126]. Our main result explicitly produces, for every prime power q 7 , the groups SL_3(F_q) and SU_3(F_q) as characteristic quotients of  F_2 . Our strategy is to study specializations of the Burau representation for the braid group B_4 , exploiting an exceptional relationship between F_2 and B_4 first observed by Dyer, Formanek, and Grossman [Arch. Math. (Basel) 38 (1982), 404–409]. Weisfeiler’s strong approximation theorem guarantees that our specializations are surjective for infinitely many primes, but they are not effective. To make our result effective, we give another proof of surjectivity via a careful analysis of the maximal subgroup structures of SL_3(F_q) and SU_3(F_q) . These examples are minimal in the sense that no finite simple group of the form PSL_2(F_q) appears as a characteristic quotient of F_2 .

We develop a method based on the Burau matrix to detect conditions on the linking numbers of braid strands. Our main application is to iterated exchanged braids. Unless the braid permutation fixes both braid edge strands, we establish under some fairly generic conditions on the linking numbers a ‘subsymmetry’ property; in particular at most two such braids can be mutually conjugate. As an addition, we prove that the Burau kernel is contained in the commutator subgroup of the pure braid group. We discuss also some properties of the Burau image.

We propose a definition by generators and relations of the rank$n-2$Askey-Wilson algebra$\\mathfrak{aw}(n)$for any integer$n$ , generalising the known presentation for the usual case$n=3$ . The generators are indexed by connected subsets of$\\{1,\\dots,n\\}$and the simple and rather small set of defining relations is directly inspired from the known case of$n=3$ . Our first main result is to prove the existence of automorphisms of$\\mathfrak{aw}(n)$satisfying the relations of the braid group on$n+1$strands. We also show the existence of coproduct maps relating the algebras for different values of$n$ . An immediate consequence of our approach is that the Askey-Wilson algebra defined here surjects onto the algebra generated by the intermediate Casimir elements in the$n$ -fold tensor product of the quantum group${\\rm U}_q(\\mathfrak{sl}_2)$or, equivalently, onto the Kauffman bracket skein algebra of the$(n+1)$ -punctured sphere. We also obtain a family of central elements of the Askey-Wilson algebras which are shown, as a direct by-product of our construction, to be sent to$0$in the realisation in the$n$ -fold tensor product of${\\rm U}_q(\\mathfrak{sl}_2)$ , thereby producing a large number of relations for the algebra generated by the intermediate Casimir elements.