Abstract
This chapter deals with general theories that do not depend on the types of Lie groups and Lie algebras. As generalizations, it addresses projective representations of Lie groups and Lie algebras by combining the contents of Chap. 2. Then, it introduces the Fourier transform for Lie groups including the case of projective representations. It also prepares several concepts for Chap. 6. Also, this chapter introduces complex Lie groups and complex Lie algebras, which are helpful for real Lie groups and real Lie algebras.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In physics, we identify elements of a real Lie algebra and a Hermitian matrix as a physical quantity. However, in our definition, the real Lie algebra of \(\mathop {\mathrm{SU}}\nolimits (d) \) consists of skew-Hermitian matrices. That is, in physics, the products of \(\sqrt{-1}\) and the elements of Lie algebra defined here are regarded as elements of a Lie algebra.
- 2.
When \(\mathop {{\mathfrak {g}}}\nolimits \) is not semi simple, the center \(C(\tilde{G}) \) might be a continuous group. In particular, when \(\mathop {{\mathfrak {g}}}\nolimits \) is a nilpotent Lie algebra, the center \(C(\tilde{G}) \) is a continuous group, and hence, the Lie algebra of the Lie group \(\tilde{G}/C(\tilde{G}) \) is different from the Lie algebra \(\mathop {{\mathfrak {g}}}\nolimits \).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2017 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Hayashi, M. (2017). Foundation of Representation Theory of Lie Group and Lie Algebra. In: Group Representation for Quantum Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-44906-7_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-44906-7_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-44904-3
Online ISBN: 978-3-319-44906-7
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)