QUANTUM SUPERGROUPS AND LINK POLYNOMIALS

1993 ◽  
Vol 05 (03) ◽  
pp. 533-549 ◽  
Author(s):  
M. D. GOULD ◽  
I. TSOHANTJIS ◽  
A. J. BRACKEN
Keyword(s):  
Irreducible Representation ◽  
Quantum Groups ◽  
Braid Group ◽  
Usual Sense ◽  
Type I ◽  
Closed Formula ◽  
Matrix Representations ◽  
Quantum Supergroup ◽  
General Method

A general method for constructing invariants for quantum supergroups is applied to obtain a closed formula for link polynomials. For type I quantum supergroups, a realization of the braid group and corresponding link polynomial is determined, for each irreducible representation of the quantum supergroup in a certain class. Although these realizations are not matrix representations in the usual sense, nevertheless link polynomials are defined which are generalizations of those previously obtained from quantum groups. To illustrate the theory, link polynomials corresponding to the defining representations of the quantum supergroups Uq [gl(m|n)], Uq [C (m + 1)] are determined explicitly.

QUANTUM SUPERGROUPS, LINK POLYNOMIALS AND REPRESENTATION OF THE BRAID GENERATOR

1993 ◽  
Vol 05 (02) ◽  
pp. 345-361 ◽  
Author(s):  
J. R. LINKS ◽  
M. D. GOULD ◽  
R. B. ZHANG
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Symmetric Tensor ◽  
Irreducible Module ◽  
Closed Formula ◽  
Cyclic Module ◽  
Tensor Representation ◽  
Finite Dimensional ◽  
Quantum Supergroup ◽  
Rank 2

Unlike the quantum group case, it is shown that the braid generator σ is not always diagonalizable on V ⊗ V, V an irreducible module for a quantum supergroup. Nevertheless a generalization of the Reshetikhin form of the braid generator, obtained previously for quantum groups, is determined corresponding to every finite dimensional standard cyclic module V of a quantum supergroup. This result is applied to obtain a general closed formula for link polynomials arising from standard cyclic modules of a quantum supergroup belonging to a certain class. As explicit examples we determine link polynomials corresponding to the rank 2 symmetric tensor representation of Uq [gl(m|m)] and the defining representation of Uq [osp(2n|2n)].

scholarly journals Quantum double finite group algebras and link polynomials

1994 ◽  
Vol 49 (2) ◽  
pp. 177-204 ◽  
Author(s):  
I. Tsohantjis ◽  
M.D. Gould
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Symmetric Group ◽  
Braid Group ◽  
Unitary Representations ◽  
Group Algebras ◽  
Diagonal Form ◽  
Closed Formula ◽  
Quantum Double ◽  
Induced Representations

Unitary representations of the braid group and corresponding link polynomials are constructed corresponding to each irreducible representation of a quantum double finite group algebra. Moreover the diagonal form of the braid generator is derived from which a general closed formula is obtained for link polynomials. As an example, link polynomials corresponding to certain induced representations of the symmetric group and its subgroups are determined explicitly.

FINITE LINEAR QUOTIENTS OF ${\mathcal{B}}_3$ OF LOW DIMENSION

2010 ◽  
Vol 19 (05) ◽  
pp. 587-600 ◽  
Author(s):  
ERIC C. ROWELL ◽  
IMRE TUBA
Keyword(s):  
Group Theory ◽  
Irreducible Representation ◽  
Quantum Groups ◽  
Braid Group ◽  
Computational Group Theory ◽  
Finite Image ◽  
Low Dimension ◽  
Finite Linear ◽  
Linear Quotients

We study the problem of deciding whether or not the image of an irreducible representation of the braid group [Formula: see text] of degree ≤ 5 has finite image if we are only given the eigenvalues of a generator. We provide a partial algorithm that determines when the images are finite or infinite in all but finitely many cases, and use these results to study examples coming from quantum groups. Our technique uses two classification theorems and the computational group theory package GAP.

scholarly journals Curved Rickard complexes and link homologies

2020 ◽  
Vol 2020 (769) ◽  
pp. 87-119
Author(s):  
Sabin Cautis ◽  
Aaron D. Lauda ◽  
Joshua Sussan
Keyword(s):  
Spectral Sequence ◽  
Quantum Groups ◽  
Braid Group ◽  
Derived Categories ◽  
Duke Math ◽  
Khovanov Homology ◽  
Hilbert Schemes ◽  
Coherent Sheaves ◽  
Knot Homology ◽  
Original Motivation

AbstractRickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to obtain deformations of link homologies which generalize those of Batson–Seed [3] [J. Batson and C. Seed, A link-splitting spectral sequence in Khovanov homology, Duke Math. J. 164 2015, 5, 801–841] and Gorsky–Hogancamp [E. Gorsky and M. Hogancamp, Hilbert schemes and y-ification of Khovanov–Rozansky homology, preprint 2017] to arbitrary representations/partitions. Another is to relate the deformed homology defined algebro-geometrically in [S. Cautis and J. Kamnitzer, Knot homology via derived categories of coherent sheaves IV, colored links, Quantum Topol. 8 2017, 2, 381–411] to categorified quantum groups (this was the original motivation for this paper).

Analogues of quantum Schubert cell algebras in PBW-deformations of quantum groups

2016 ◽  
Vol 15 (10) ◽  
pp. 1650179 ◽  
Author(s):  
Yongjun Xu ◽  
Dingguo Wang ◽  
Jialei Chen
Keyword(s):  
Weyl Group ◽  
Quantum Groups ◽  
Braid Group ◽  
Group Actions ◽  
Necessary Condition ◽  
Quantum Algebras ◽  
Ore Extension ◽  
Schubert Cell ◽  
Sufficient And Necessary Condition

We focus on a class of filtered quantum algebras [Formula: see text] which are both coideal subalgebras of quantum groups and Poincaré–Birkhoff–Witt (PBW)-deformations of their negative parts. In [Y. Xu and S. Yang, PBW-deformations of quantum groups, J. Algebra 408 (2014) 222–249], Xu and Yang proved that braid group actions on [Formula: see text] introduced by Kolb and Pellegrini can be used to define root vectors and construct PBW bases for [Formula: see text]. In this present paper, for each element [Formula: see text] in the Weyl group of [Formula: see text] we first introduce a subspace [Formula: see text] and a subalgebra [Formula: see text] of [Formula: see text], where [Formula: see text] can be considered as an analogue of quantum Schubert cell algebra. Then a sufficient and necessary condition on [Formula: see text] is given for [Formula: see text]. Moreover, we prove that [Formula: see text] if and only if [Formula: see text] and [Formula: see text] can be generated by the same simple reflections. Finally, we characterize the algebra [Formula: see text] which can be obtained via an iterated Ore extension. Our results show that quantum groups and their PBW-deformations really have some different properties.

APPROXIMATELY INNER FLOWS ON SEPARABLE C*-ALGEBRAS

2002 ◽  
Vol 14 (07n08) ◽  
pp. 649-673 ◽  
Author(s):  
AKITAKA KISHIMOTO
Keyword(s):  
Irreducible Representation ◽  
Von Neumann Algebra ◽  
Automorphism Groups ◽  
Unitary Groups ◽  
Type I ◽  
Von Neumann ◽  
C Algebra ◽  
C Algebras ◽  
Inner Automorphism ◽  
Uniformly Continuous

We present two types of result for approximately inner one-parameter automorphism groups (referred to as AI flows hereafter) of separable C*-algebras. First, if there is an irreducible representation π of a separable C*-algebra A such that π(A) does not contain non-zero compact operators, then there is an AI flow α such that π is α-covariant and α is far from uniformly continuous in the sense that α induces a flow on π(A) which has full Connes spectrum. Second, if α is an AI flow on a separable C*-algebra A and π is an α-covariant irreducible representation, then we can choose a sequence (hn) of self-adjoint elements in A such that αt is the limit of inner flows Ad eithn and the sequence π(eithn) of one-parameter unitary groups (referred to as unitary flows hereafter) converges to a unitary flow which implements α in π. This latter result will be extended to cover the case of weakly inner type I representations. In passing we shall also show that if two representations of a separable simple C*-algebra on a separable Hilbert space generate the same von Neumann algebra of type I, then there is an approximately inner automorphism which sends one into the other up to equivalence.

THE REPRESENTATIONS AND ENDOMORPHISMS OF A SEPARABLE NUCLEAR C*-ALGEBRA

2003 ◽  
Vol 14 (03) ◽  
pp. 313-326 ◽  
Author(s):  
AKITAKA KISHIMOTO
Keyword(s):  
Hilbert Space ◽  
Irreducible Representation ◽  
Separable Hilbert Space ◽  
Type I ◽  
C Algebra

It is shown that if A is a separable, non-type I, nuclear simple C*-algebra and π is an irreducible representation of A, then for any representation ρ on a separable Hilbert space of A there is an endomorphism α of A such that ρ is unitarily equivalent to πα.

A general method for type I and type II g-C3N4/g-C3N4 metal-free isotype heterostructures with enhanced visible light photocatalysis

2015 ◽  
Vol 39 (6) ◽  
pp. 4737-4744 ◽  
Author(s):  
Fan Dong ◽  
Zilin Ni ◽  
Peidong Li ◽  
Zhongbiao Wu
Keyword(s):  
Visible Light ◽  
Band Alignment ◽  
Visible Light Photocatalysis ◽  
Type I ◽  
Type Ii ◽  
Metal Free ◽  
General Method

Composite precursors were used to construct type I and type II g-C3N4/g-C3N4 metal-free isotype heterostructures based on different band-alignment patterns.

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