5 Lie Algebras and Lie Algebra Representations

In this chapter we will introduce Lie algebras and Lie algebra representations, which provide a tractable linear construction that captures much of the behavior of Lie groups and Lie group representations. We have so far seen the case of $U(1)$ , for which the Lie algebra is trivial, and a little bit about the $SU(2)$ case, where the first non-trivial Lie algebra appears. Chapters 6 and 8 will provide details showing how the general theory works out for the basic examples of $SU(2),SO(3)$ and their representations. The very general nature of the material in this chapter may make it hard to understand until one has some experience with examples that only appear in later chapters. The reader is thus advised that it may be a good idea to first skim the material of this chapter, returning for a deeper understanding and better insight into these structures after first seeing them in action later on in more concrete contexts.

For a group $G$ we have defined unitary representations $(\pi ,V)$ for finite dimensional vector spaces $V$ of complex dimension $n$ as homomorphisms

$$\pi :G\to U(n)$$

Recall that in the case of $G=U(1)$ (see the proof of theorem 2.3) we could use the homomorphism property of $\pi$ to determine $\pi$ in terms of its derivative at the identity. This turns out to be a general phenomenon for Lie groups $G$: we can study their representations by considering the derivative of $\pi$ at the identity, which we will call ${\pi }^{\prime }$. Because of the homomorphism property, knowing ${\pi }^{\prime }$ is often sufficient to characterize the representation $\pi$ it comes from. ${\pi }^{\prime }$ is a linear map from the tangent space to $G$ at the identity to the tangent space of $U(n)$ at the identity. The tangent space to $G$ at the identity will carry some extra structure coming from the group multiplication, and this vector space with this structure will be called the Lie algebra of $G$. The linear map ${\pi }^{\prime }$ will be an example of a Lie algebra representation.

The subject of differential geometry gives many equivalent ways of defining the tangent space at a point of manifolds like $G$, but we do not want to enter here into the subject of differential geometry in general. One of the standard definitions of the tangent space is as the space of tangent vectors, with tangent vectors defined as the possible velocity vectors of parametrized curves $g(t)$ in the group $G$.

More advanced treatments of Lie group theory develop this point of view (see for example [99]) which applies to arbitrary Lie groups, whether or not they are groups of matrices. In our case though, since we are interested in specific groups that are usually explicitly given as groups of matrices, in such cases we can give a more concrete definition, using the exponential map on matrices. For a more detailed exposition of this subject, using the same concrete definition of the Lie algebra in terms of matrices, see for instance [42] or the abbreviated on-line version [40].

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正文：英文 · 原著转录

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原书 PDF · 印刷页 47、48

来源版本：2025-10-20

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Concept index

• Lie algebra
• adjoint representation
• Lie bracket
• Lie algebra representation
• Lie algebra representation
• abstract Lie algebra
• complexification