Heat Kernel on Lie Groups and Maximally Symmetric Spaces

The heat kernel for the spin-tensor Laplacians on Lie groups and maximally symmetric spaces is studied in this monograph, which introduces new ideas, methods, and tools. It provides explicit results for the heat kernel on spheres and hyperbolic spaces and is a valuable resource for researchers and graduate students in mathematics and mathematical and theoretical physics.

Format: Paperback / softback
Length: 190 pages
Publication date: 26 April 2023
Publisher: Birkhauser Verlag AG

This comprehensive monograph delves into the study of the heat kernel for the spin-tensor Laplacians on Lie groups and maximally symmetric spaces. It presents a wealth of innovative ideas, methods, and tools developed by the author, along with a comprehensive list of all known exact results for the heat kernel on spheres and hyperbolic spaces, both in explicit form and through derivations. Part I provides a detailed exploration of the geometry of simple Lie groups and maximally symmetric spaces, while Part II focuses on the computation of the heat kernel for scalar, spinor, and generic Laplacians on spheres and hyperbolic spaces across various dimensions. This text serves as an invaluable resource for researchers and graduate students engaged in diverse fields of mathematics, including global analysis, spectral geometry, stochastic processes, and financial mathematics, as well as in areas of mathematical and theoretical physics, such as quantum field theory, quantum gravity, string theory, and statistical physics.

Introduction:
The heat kernel is a fundamental object in mathematical physics, playing a crucial role in understanding various physical phenomena. In this monograph, we focus on the study of the heat kernel for the spin-tensor Laplacians on Lie groups and maximally symmetric spaces. These spaces are of interest in various areas of mathematics and physics, including global analysis, spectral geometry, stochastic processes, and financial mathematics.

Original Ideas and Methods:
The author of this monograph has developed a range of original ideas and methods that contribute to the understanding of the heat kernel for spin-tensor Laplacians. These ideas and methods are based on a deep understanding of the geometry and dynamics of these spaces. The monograph provides a detailed exposition of these ideas and methods, making them accessible to a wide audience.

Exact Results:
One of the key contributions of this monograph is the presentation of a comprehensive list of all known exact results for the heat kernel on spheres and hyperbolic spaces. These results are derived using a variety of techniques, including analytical and numerical methods. The monograph also provides detailed proofs of these results, making it a valuable resource for researchers and graduate students in the field.

Geometry of Simple Lie Groups and Maximally Symmetric Spaces:
Part I of the monograph delves into the geometry of simple Lie groups and maximally symmetric spaces in detail. The author discusses the basic concepts and properties of these spaces, including Lie groups, Lie algebras, and maximally symmetric spaces. The monograph also provides an introduction to the theory of heat kernels and their applications.

Calculation of the Heat Kernel:
Part II of the monograph focuses on the calculation of the heat kernel for scalar, spinor, and generic Laplacians on spheres and hyperbolic spaces in various dimensions. The author discusses the methods and techniques used to compute the heat kernel, including the use of differential operators, Fourier transforms, and integral transforms. The monograph also provides examples and applications of the heat kernel in various fields.

Applications:
The heat kernel has numerous applications in various fields of mathematics and physics. In particular, it plays a crucial role in understanding the behavior of quantum field theories, quantum gravity, and string theory. The monograph discusses the applications of the heat kernel in these areas, providing insights into the structure and dynamics of these theories.

Conclusion:
This monograph is a valuable resource for researchers and graduate students working in various areas of mathematics and theoretical physics. It provides a comprehensive overview of the heat kernel for spin-tensor Laplacians on Lie groups and maximally symmetric spaces, along with a wealth of original ideas, methods, and tools. The book also presents a comprehensive list of all known exact results for the heat kernel on spheres and hyperbolic spaces, making it an essential reference for anyone interested in this field.

Weight: 367g
Dimension: 240 x 168 (mm)
ISBN-13: 9783031274503
Edition number: 1st ed. 2023