scholarly journals Double Product Integrals and Enriquez Quantisation of Lie Bialgebras II: The Quantum Yang–Baxter Equation

2005 ◽  
Vol 72 (3) ◽  
pp. 211-224 ◽  
Author(s):  
R. L. Hudson ◽  
S. Pulmannová
Keyword(s):  
Baxter Equation ◽  
Lie Bialgebras ◽  
Double Product

scholarly journals LEFT-SYMMETRIC BIALGEBRAS AND AN ANALOGUE OF THE CLASSICAL YANG–BAXTER EQUATION

2008 ◽  
Vol 10 (02) ◽  
pp. 221-260 ◽  
Author(s):  
CHENGMING BAI
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Symmetric Solution ◽  
Baxter Equation ◽  
Symmetric Part ◽  
Lie Bialgebra ◽  
Lie Bialgebras ◽  
Poisson Lie Groups ◽  
Lagrangian Subalgebras

We introduce a notion of left-symmetric bialgebra which is an analogue of the notion of Lie bialgebra. We prove that a left-symmetric bialgebra is equivalent to a symplectic Lie algebra with a decomposition into a direct sum of the underlying vector spaces of two Lagrangian subalgebras. The latter is called a parakähler Lie algebra or a phase space of a Lie algebra in mathematical physics. We introduce and study coboundary left-symmetric bialgebras and our study leads to what we call "S-equation", which is an analogue of the classical Yang–Baxter equation. In a certain sense, the S-equation associated to a left-symmetric algebra reveals the left-symmetry of the products. We show that a symmetric solution of the S-equation gives a parakähler Lie algebra. We also show that such a solution corresponds to the symmetric part of a certain operator called "[Formula: see text]-operator", whereas a skew-symmetric solution of the classical Yang–Baxter equation corresponds to the skew-symmetric part of an [Formula: see text]-operator. Thus a method to construct symmetric solutions of the S-equation (hence parakähler Lie algebras) from [Formula: see text]-operators is provided. Moreover, by comparing left-symmetric bialgebras and Lie bialgebras, we observe that there is a clear analogue between them and, in particular, parakähler Lie groups correspond to Poisson–Lie groups in this sense.

Double product integrals and Enriquez quantization of Lie bialgebras I: The quasitriangular identities

2004 ◽  
Vol 45 (5) ◽  
pp. 2090-2105 ◽  
Author(s):  
R. L. Hudson ◽  
S. Pulmannová
Keyword(s):  
Lie Bialgebras ◽  
Double Product

scholarly journals Algebraic Geometry of Lie Bialgebras Defined by Solutions of the Classical Yang–Baxter Equation

2021 ◽  
Author(s):  
Raschid Abedin ◽  
Igor Burban
Keyword(s):  
Algebraic Geometry ◽  
Lie Algebra ◽  
Baxter Equation ◽  
Lie Bialgebra ◽  
Lie Bialgebras ◽  
Infinite Dimensional ◽  
Geometric Context ◽  
Trigonometric Solution ◽  
Manin Triples ◽  
Geometric Study

AbstractThis paper is devoted to algebro-geometric study of infinite dimensional Lie bialgebras, which arise from solutions of the classical Yang–Baxter equation. We regard trigonometric solutions of this equation as twists of the standard Lie bialgebra cobracket on an appropriate affine Lie algebra and work out the corresponding theory of Manin triples, putting it into an algebro-geometric context. As a consequence of this approach, we prove that any trigonometric solution of the classical Yang–Baxter equation arises from an appropriate algebro-geometric datum. The developed theory is illustrated by some concrete examples.

Solution of the deformed Schwarzschild metric by the Yang-Baxter equation

2019 ◽  
Vol 96 (4) ◽  
pp. 8-14
Author(s):  
A. Meirambay ◽  
◽  
K.K. Yerzhanov ◽  
Keyword(s):  
Baxter Equation ◽  
Schwarzschild Metric

scholarly journals Impact of exercise based cardiac rehabilitation program in patients with transcatheter aortic valve replacement

2021 ◽  
Vol 28 (Supplement_1) ◽  
Author(s):  
LE Estrada Martinez ◽  
JA Lara Vargas ◽  
JA Pineda Juarez ◽  
JD Morales Portano ◽  
JB Gomez Alvarez ◽  
...  
Keyword(s):  
Heart Rate ◽  
Cardiac Rehabilitation ◽  
Intervention Group ◽  
Rehabilitation Program ◽  
Control Group ◽  
Cardiac Rehabilitation Program ◽  
Double Product ◽  
Oxygen Pulse ◽  
Transcatheter Aortic Valve ◽  
Cardiopulmonary Performance

Abstract Funding Acknowledgements Type of funding sources: None. Introduction  Due to the increase in global prevalence of degenerative valve disease, aortic stenosis (AS) has played a preponderant role in the cardiovascular scenario, especially in patients undergoing transcatheter aortic valve replacement (TAVR). An alternative management for this patients are the cardiac rehabilitation programs (CRP); however, their effect has not been completely understood, both in exercise capacity and quality of life, but neither in the improvement of cardiopulmonary performance and other cardiovascular outcomes.  Purpose: To evaluate the effect of the CRP on exercise tolerance and cardiopulmonary performance in patients with AS undergoing TAVR.  Methods: A cohort study was conducted including 26 patients with AS undergoing TAVR and divided into an intervention group who performed a 4-week supervised training program in the Cardiac Rehabilitation Service and a control group to whom instructions and recommendations to performed unsupervised exercise at home were given. Demographic and clinical data (VO2Max, METS12, oxygen pulse, heart rate, double product, left ventricular ejection fraction, body mass index) were collected at baseline and after a 4-week follow-up. Results: 15 patients were included in the intervention group and 11 patients in the control group. There were no baseline significant differences between groups. After the intervention, significant differences were observed in the METS 12 final gain variable between the control and intervention group (4.55 vs 3.1 p = 0.01). Intergroup analysis showed significant differences (percentage changes) in the intervention group with an increase of METS12 (67.4%, p = 0.001), oxygen pulse (18.21%, p = 0.01), final METS (39.47% p = 0.001) and a decrease in VO2 recovery time (-12.5%, p = 0.05), in the ergometric performance index by heart rate (-38.17%, p = 0.001) and by double product (-38.1%, p = 0.001). Conclusions  A 4-week cardiac rehabilitation program is effective to improve exercise tolerance and cardiopulmonary response in patients with AS undergoing TAVR; improvement was statistically significant in METS12, oxygen pulse, VO2 recovery time, METS-load and ergometric performance index for heart rate and double product. METS12 final gain was statistically significant in intervention group in comparison with the control group. Abstract Figure. Control vs Intervention Group (METS12)

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