Quantum algorithms bring great challenges to classical public key cryptosystems, which makes cryptosystems based on non-commutative algebraic systems hop topic. The braid groups, which are non-commutative, have attrac...Quantum algorithms bring great challenges to classical public key cryptosystems, which makes cryptosystems based on non-commutative algebraic systems hop topic. The braid groups, which are non-commutative, have attracted much attention as a new platform for constructing quantum attack-resistant cryptosystems. A ring signature scheme is proposed based on the difficulty of the root extraction problem over braid groups, which can resist existential forgery against the adaptively cho-sen-message attack under the random oracle model.展开更多
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...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 Pnfor 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.展开更多
Recently Turaev generalized the notion of a tensor category to that of a crossed group category.In[5]the authors constructed the representation category Rep(H) of a T-coalgebra H.In[2]the authors introduced the notion...Recently Turaev generalized the notion of a tensor category to that of a crossed group category.In[5]the authors constructed the representation category Rep(H) of a T-coalgebra H.In[2]the authors introduced the notions of a weak tensor category to characterize a weak bialgebra and a weak Hopf algebra.This paper is based on these ideas to naturally introduce the notions of a weak T-category and a weak braided T-category which are not under the usual way and prove that the categories of representations of a weak T-coalgebra and a weak braided T-coalgebra are a weak T-category and a weak braided T-category respectively.Furthermore we also discuss some properties of weak T-category.展开更多
Recent experiments with suspended graphene have indicated the crucial role of carrier mobility in the competition between Laughlin collective state and insulating state, probably of Wigner-crystal-type. Moreover, the ...Recent experiments with suspended graphene have indicated the crucial role of carrier mobility in the competition between Laughlin collective state and insulating state, probably of Wigner-crystal-type. Moreover, the fractional quantum Hall effect (FQHE) in graphene has been observed at a low carrier density where the interaction is reduced as a result of particles dilution. This suggests that the interaction may not be a sole factor in the triggering of FQHE as it was expected basing on the standard formulation of the composite fermion model. Here, the topological arguments are presented to explain the observed features of the FQHE in graphene and the triggering role of carrier mobility in formation of the Laughlin state.展开更多
The topology-based explanation of the origin of the fractional quantum Hall effect is summarized. The cyclotron braid subgroups crucial for this approach are introduced in order to identify the origin of Laughlin corr...The topology-based explanation of the origin of the fractional quantum Hall effect is summarized. The cyclotron braid subgroups crucial for this approach are introduced in order to identify the origin of Laughlin correlations in 2D Hall systems. The so-called composite fermions are explained in terms of the homotopy cyclotron braids. Some new concept for fractional Chern insulator states is formulated in terms of the homotopy condition applied to the Berry field flux quantization.展开更多
Let HLB be the category of generalized Long modules, that is, H-modules and B-comodules over Hopf algebras B and H. We describe a new Turaev braided group category over generalized Long module HLB （S（π）） where th...Let HLB be the category of generalized Long modules, that is, H-modules and B-comodules over Hopf algebras B and H. We describe a new Turaev braided group category over generalized Long module HLB （S（π）） where the opposite group S（π） of the semidirect product of the opposite group πopof a group π by π. As an application, we show that this is a Turaev braided group-category HLBfor a quasitriangular Turaev group-coalgebra H and a coquasitriangular Turaev group-algebra B.展开更多
Motivated by the work of Birman about the relationship between mapping class groups and braid groups,the authors discuss the relationship between the orbit braid group and the equivariant mapping class group on the cl...Motivated by the work of Birman about the relationship between mapping class groups and braid groups,the authors discuss the relationship between the orbit braid group and the equivariant mapping class group on the closed surface M with a free and proper group action in this paper.Their construction is based on the exact sequence given by the fibration F_(0)^(G) M→F(M/G,n).The conclusion is closely connected with the braid group of the quotient space.Comparing with the situation without the group action,there is a big difference when the quotient space is T^(2).展开更多
We unify and generalize several approaches to constructing braid group representa-tions from finite groups,using iterated twisted tensor products.We provide some general characterizations and classification of these r...We unify and generalize several approaches to constructing braid group representa-tions from finite groups,using iterated twisted tensor products.We provide some general characterizations and classification of these representations,focusing on the size of their images,which are typically finite groups.The well-studied Gaussian representations associated with metaplectic modular categories can be understood in this framework,and we give some new examples to illustrate their ubiquity.Our results suggest a relationship between the braiding on the G-gaugings of a pointed modular category C(A,Q)and that of C(A,Q)itself.展开更多
Let M and N be topological spaces,let G be a group,and letτ:G×M→M be a proper free action of G.In this paper,we define a Borsuk-Ulam-type property for homotopy classes of maps from M to N with respect to the pa...Let M and N be topological spaces,let G be a group,and letτ:G×M→M be a proper free action of G.In this paper,we define a Borsuk-Ulam-type property for homotopy classes of maps from M to N with respect to the pair(G,τ)that generalises the classical antipodal Borsuk-Ulam theorem of maps from the n-sphere S^(n) to R^(n).In the cases where M is a finite pathwise-connected CWcomplex,G is a finite,non-trivial Abelian group,τis a proper free cellular action,and N is either R^(2) or a compact surface without boundary different from S^(2) and RP^(2),we give an algebraic criterion involving braid groups to decide whether a free homotopy class β∈[M,N]has the Borsuk-Ulam property.As an application of this criterion,we consider the case where M is a compact surface without boundary equipped with a free actionτof the finite cyclic group Zn.In terms of the orient ability of the orbit space Mof M by the actionτ,the value of n modulo 4 and a certain algebraic condition involving the first homology group of M,we are able to determine if the single homotopy class of maps from M to R^(2) possesses the Borsuk-Ulam property with respect to(Z_(n),τ).Finally,we give some examples of surfaces on which the symmetric group acts,and for these cases,we obtain some partial results regarding the Borsuk-Ulam property for maps whose target is R^(2).展开更多
Let X be a topological space.In this survey the authors consider several types of configuration spaces,namely,the classical(usual)configuration spaces F_n(X)and D_n(X),the orbit configuration spaces F_n^G(X)and F_n^G(...Let X be a topological space.In this survey the authors consider several types of configuration spaces,namely,the classical(usual)configuration spaces F_n(X)and D_n(X),the orbit configuration spaces F_n^G(X)and F_n^G(X)/S_nwith respect to a free action of a group G on X,and the graph configuration spaces F_n~Γ(X)and F_n~Γ(X)/H,whereΓis a graph and H is a suitable subgroup of the symmetric group S_n.The ordered configuration spaces F_n(X),F_n^G(X),F_n~Γ(X)are all subsets of the n-fold Cartesian product ∏_1~nX of X with itself,and satisfy F_n^G(X)?F_n(X)?F_n~Γ(X)?∏_1~nX.If A denotes one of these configuration spaces,the authors analyse the difference between A and ∏_1~nXfrom a topological and homotopical point of view.The principal results known in the literature concern the usual configuration spaces.The authors are particularly interested in the homomorphism on the level of the homotopy groups of the spaces induced by the inclusionι:A-→∏_1~nX,the homotopy type of the homotopy fibre I_ιof the mapιvia certain constructions on various spaces that depend on X,and the long exact sequence in homotopy of the fibration involving I_ιand arising from the inclusionι.In this respect,if X is either a surface without boundary,in particular if X is the 2-sphere or the real projective plane,or a space whose universal covering is contractible,or an orbit space S^k/Gof the k-dimensional sphere by a free action of a Lie group G,the authors present recent results obtained by themselves for the first case,and in collaboration with Golasi′nski for the second and third cases.The authors also briefly indicate some older results relative to the homotopy of these spaces that are related to the problems of interest.In order to motivate various questions,for the remaining types of configuration spaces,a few of their basic properties are described and proved.A list of open questions and problems is given at the end of the paper.展开更多
Both in Majid's double-bosonization theory and in Rosso's quantum shuffle theory, the rankinductive and type-crossing construction for U_q(g)'s is still a remaining open question. In this paper, working in...Both in Majid's double-bosonization theory and in Rosso's quantum shuffle theory, the rankinductive and type-crossing construction for U_q(g)'s is still a remaining open question. In this paper, working in Majid's framework, based on the generalized double-bosonization theorem we proved before, we further describe explicitly the type-crossing construction of U_q(g)'s for(BCD)_n series directly from type An-1via adding a pair of dual braided groups determined by a pair of(R, R′)-matrices of type A derived from the respective suitably chosen representations. Combining with our results of the first three papers of this series, this solves Majid's conjecture, i.e., any quantum group U_q(g) associated to a simple Lie algebra g can be grown out of U_q(sl_2)recursively by a series of suitably chosen double-bosonization procedures.展开更多
基金Supported by the National Natural Science Foundation of China (No. 10501053)
文摘Quantum algorithms bring great challenges to classical public key cryptosystems, which makes cryptosystems based on non-commutative algebraic systems hop topic. The braid groups, which are non-commutative, have attracted much attention as a new platform for constructing quantum attack-resistant cryptosystems. A ring signature scheme is proposed based on the difficulty of the root extraction problem over braid groups, which can resist existential forgery against the adaptively cho-sen-message attack under the random oracle model.
基金supported by the National Natural Science Foundation of China(No.11971144)the State Contract of the Sobolev Institute of Mathematics+2 种基金SB RAS(No.I.1.5,FWNF-2022-0009)the High-level Scientific Research Foundation of Hebei Provincethe Start-up Research Fund from Yanqi Lake Beijing Institute of Mathematical Sciences and Applications。
文摘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 Pnfor 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.
基金Supported by the Natural Science Foundation of Shandong Province(ZR2012AL02)
文摘Recently Turaev generalized the notion of a tensor category to that of a crossed group category.In[5]the authors constructed the representation category Rep(H) of a T-coalgebra H.In[2]the authors introduced the notions of a weak tensor category to characterize a weak bialgebra and a weak Hopf algebra.This paper is based on these ideas to naturally introduce the notions of a weak T-category and a weak braided T-category which are not under the usual way and prove that the categories of representations of a weak T-coalgebra and a weak braided T-coalgebra are a weak T-category and a weak braided T-category respectively.Furthermore we also discuss some properties of weak T-category.
基金Supported by Polish National Center of Science Project no:DEC-2011/02/A/ST3/001.
文摘Recent experiments with suspended graphene have indicated the crucial role of carrier mobility in the competition between Laughlin collective state and insulating state, probably of Wigner-crystal-type. Moreover, the fractional quantum Hall effect (FQHE) in graphene has been observed at a low carrier density where the interaction is reduced as a result of particles dilution. This suggests that the interaction may not be a sole factor in the triggering of FQHE as it was expected basing on the standard formulation of the composite fermion model. Here, the topological arguments are presented to explain the observed features of the FQHE in graphene and the triggering role of carrier mobility in formation of the Laughlin state.
基金The support from the NCN Project UMO-2011/02/A/ST3/00116 is acknowledged.
文摘The topology-based explanation of the origin of the fractional quantum Hall effect is summarized. The cyclotron braid subgroups crucial for this approach are introduced in order to identify the origin of Laughlin correlations in 2D Hall systems. The so-called composite fermions are explained in terms of the homotopy cyclotron braids. Some new concept for fractional Chern insulator states is formulated in terms of the homotopy condition applied to the Berry field flux quantization.
基金The NSF (11101128) of Chinathe NSF (102300410049) of Henan Provincethe NSF (BK2012736) of Jiangsu Province
文摘Let HLB be the category of generalized Long modules, that is, H-modules and B-comodules over Hopf algebras B and H. We describe a new Turaev braided group category over generalized Long module HLB （S（π）） where the opposite group S（π） of the semidirect product of the opposite group πopof a group π by π. As an application, we show that this is a Turaev braided group-category HLBfor a quasitriangular Turaev group-coalgebra H and a coquasitriangular Turaev group-algebra B.
基金supported by the National Natural Science Foundation of China(No.11971112)。
文摘Motivated by the work of Birman about the relationship between mapping class groups and braid groups,the authors discuss the relationship between the orbit braid group and the equivariant mapping class group on the closed surface M with a free and proper group action in this paper.Their construction is based on the exact sequence given by the fibration F_(0)^(G) M→F(M/G,n).The conclusion is closely connected with the braid group of the quotient space.Comparing with the situation without the group action,there is a big difference when the quotient space is T^(2).
文摘We unify and generalize several approaches to constructing braid group representa-tions from finite groups,using iterated twisted tensor products.We provide some general characterizations and classification of these representations,focusing on the size of their images,which are typically finite groups.The well-studied Gaussian representations associated with metaplectic modular categories can be understood in this framework,and we give some new examples to illustrate their ubiquity.Our results suggest a relationship between the braiding on the G-gaugings of a pointed modular category C(A,Q)and that of C(A,Q)itself.
基金supported by the CNPq project n°140836the Capes/COFECUB project n°12693/13-8+2 种基金supported by the Capes/INCTMat project n°88887.136371/2017-00-465591/2014-0partially supported by the Projeto Temático FAPESP,grant n°2016/24707-4:Topologia AlgébricaGeométrica e Diferencial。
文摘Let M and N be topological spaces,let G be a group,and letτ:G×M→M be a proper free action of G.In this paper,we define a Borsuk-Ulam-type property for homotopy classes of maps from M to N with respect to the pair(G,τ)that generalises the classical antipodal Borsuk-Ulam theorem of maps from the n-sphere S^(n) to R^(n).In the cases where M is a finite pathwise-connected CWcomplex,G is a finite,non-trivial Abelian group,τis a proper free cellular action,and N is either R^(2) or a compact surface without boundary different from S^(2) and RP^(2),we give an algebraic criterion involving braid groups to decide whether a free homotopy class β∈[M,N]has the Borsuk-Ulam property.As an application of this criterion,we consider the case where M is a compact surface without boundary equipped with a free actionτof the finite cyclic group Zn.In terms of the orient ability of the orbit space Mof M by the actionτ,the value of n modulo 4 and a certain algebraic condition involving the first homology group of M,we are able to determine if the single homotopy class of maps from M to R^(2) possesses the Borsuk-Ulam property with respect to(Z_(n),τ).Finally,we give some examples of surfaces on which the symmetric group acts,and for these cases,we obtain some partial results regarding the Borsuk-Ulam property for maps whose target is R^(2).
基金supported by the CNRS/FAPESP programme no226555(France)and n^(o) 2014/50131-7(Brazil)FAPESP–Fundacao de Amparo a Pesquisa do Estado de Sao Paulo,Projeto Tematico Topologia Algebrica,Geometrica 2012/24454-8(Brazil)for partial supportthe Institute for Mathematical Sciences,National University of Singapore
文摘Let X be a topological space.In this survey the authors consider several types of configuration spaces,namely,the classical(usual)configuration spaces F_n(X)and D_n(X),the orbit configuration spaces F_n^G(X)and F_n^G(X)/S_nwith respect to a free action of a group G on X,and the graph configuration spaces F_n~Γ(X)and F_n~Γ(X)/H,whereΓis a graph and H is a suitable subgroup of the symmetric group S_n.The ordered configuration spaces F_n(X),F_n^G(X),F_n~Γ(X)are all subsets of the n-fold Cartesian product ∏_1~nX of X with itself,and satisfy F_n^G(X)?F_n(X)?F_n~Γ(X)?∏_1~nX.If A denotes one of these configuration spaces,the authors analyse the difference between A and ∏_1~nXfrom a topological and homotopical point of view.The principal results known in the literature concern the usual configuration spaces.The authors are particularly interested in the homomorphism on the level of the homotopy groups of the spaces induced by the inclusionι:A-→∏_1~nX,the homotopy type of the homotopy fibre I_ιof the mapιvia certain constructions on various spaces that depend on X,and the long exact sequence in homotopy of the fibration involving I_ιand arising from the inclusionι.In this respect,if X is either a surface without boundary,in particular if X is the 2-sphere or the real projective plane,or a space whose universal covering is contractible,or an orbit space S^k/Gof the k-dimensional sphere by a free action of a Lie group G,the authors present recent results obtained by themselves for the first case,and in collaboration with Golasi′nski for the second and third cases.The authors also briefly indicate some older results relative to the homotopy of these spaces that are related to the problems of interest.In order to motivate various questions,for the remaining types of configuration spaces,a few of their basic properties are described and proved.A list of open questions and problems is given at the end of the paper.
基金supported by National Natural Science Foundation of China(Grant No.11271131)
文摘Both in Majid's double-bosonization theory and in Rosso's quantum shuffle theory, the rankinductive and type-crossing construction for U_q(g)'s is still a remaining open question. In this paper, working in Majid's framework, based on the generalized double-bosonization theorem we proved before, we further describe explicitly the type-crossing construction of U_q(g)'s for(BCD)_n series directly from type An-1via adding a pair of dual braided groups determined by a pair of(R, R′)-matrices of type A derived from the respective suitably chosen representations. Combining with our results of the first three papers of this series, this solves Majid's conjecture, i.e., any quantum group U_q(g) associated to a simple Lie algebra g can be grown out of U_q(sl_2)recursively by a series of suitably chosen double-bosonization procedures.
