Bimodule Connections for Relative Line Modules over the Irreducible Quantum Flag Manifolds

IF 0.9 3区 物理与天体物理 Q2 MATHEMATICS
A. Carotenuto, R. O. Buachalla
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引用次数: 2

Abstract

It was recently shown (by the second author and D\'{i}az Garc\'{i}a, Krutov, Somberg, and Strung) that every relative line module over an irreducible quantum flag manifold $\mathcal{O}_q(G/L_S)$ admits a unique $\mathcal{O}_q(G)$-covariant connection with respect to the Heckenberger-Kolb differential calculus $\Omega^1_q(G/L_S)$. In this paper we show that these connections are bimodule connections with an invertible associated bimodule map. This is proved by applying general results of Beggs and Majid, on principal connections for quantum principal bundles, to the quantum principal bundle presentation of the Heckenberger-Kolb calculi recently constructed by the authors and D\'{i}az Garc\'{i}a. Explicit presentations of the associated bimodule maps are given first in terms of generalised quantum determinants, then in terms of the FRT presentation of the algebra $\mathcal{O}_q(G)$, and finally in terms of Takeuchi's categorical equivalence for relative Hopf modules.
不可约量子旗流形上相对线模的双模连接
最近(由第二作者和D\ \ {i}az Garc\ \ {i}a, Krutov, Somberg, and string)证明了不可约量子标志流形$\mathcal{O}_q(G/L_S)$上的每个相对线模对于Heckenberger-Kolb微分$\Omega^1_q(G/L_S)$具有唯一的$\mathcal{O}_q(G)$-协变连接。在本文中,我们证明了这些连接是具有可逆关联双模映射的双模连接。将Beggs和Majid关于量子主束的主连接的一般结果应用于作者和D\ \ i}az Garc\ \ i}a最近构造的Heckenberger-Kolb演算的量子主束表示,证明了这一点。首先用广义量子行列式给出相关双模映射的显式表示,然后用代数$\mathcal{O}_q(G)$的FRT表示,最后用相对Hopf模的Takeuchi的范畴等价给出相关双模映射的显式表示。
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来源期刊
CiteScore
1.80
自引率
0.00%
发文量
87
审稿时长
4-8 weeks
期刊介绍: Scope Geometrical methods in mathematical physics Lie theory and differential equations Classical and quantum integrable systems Algebraic methods in dynamical systems and chaos Exactly and quasi-exactly solvable models Lie groups and algebras, representation theory Orthogonal polynomials and special functions Integrable probability and stochastic processes Quantum algebras, quantum groups and their representations Symplectic, Poisson and noncommutative geometry Algebraic geometry and its applications Quantum field theories and string/gauge theories Statistical physics and condensed matter physics Quantum gravity and cosmology.
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