Lie group–Lie algebra correspondence

Key takeaways

Lie group–Lie algebra correspondence belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Lie group–Lie algebra correspondence to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Lie group–Lie algebra correspondence from memory before moving on to harder problems.

Reference excerpt

In mathematics, Lie group–Lie algebra correspondence allows one to correspond a Lie group to a Lie algebra or vice versa, and study the conditions for such a relationship. Lie groups that are isomorphic to each other have Lie algebras that are isomorphic to each other, but the converse is not necessarily true. One obvious counterexample is R n {\displaystyle \mathbb {R} ^{n}} and T n {\displaystyle \mathbb {T} ^{n}} (see real coordinate space and the circle group respectively) which are non-isomorphic to each other as Lie groups but their Lie algebras are isomorphic to each other, being R n {\displaystyle \mathbb {R} ^{n}} with a trivial bracket. However, for simply connected Lie groups, the Lie group-Lie algebra correspondence is one-to-one. In this article, a Lie group refers to a real Lie group. For the complex and p-adic cases, see complex Lie group and p-adic Lie group. In this article, manifolds (in particular Lie groups) are assumed to be second countable; in particular, they have at most countably many connected components.

Basics

The Lie algebra of a Lie group There are various ways one can understand the construction of the Lie algebra of a Lie group G. One approach uses left-invariant vector fields. A vector field X on G is said to be invariant under left translations if, for any g, h in G,

( d L g ) h ( X h ) = X g h {\displaystyle (dL_{g})_{h}(X_{h})=X_{gh}}

where L g : G → G {\displaystyle L_{g}:G\to G} is defined by L g ( x ) = g x {\displaystyle L_{g}(x)=gx} and ( d L g ) h : T h G → T g h G {\displaystyle (dL_{g})_{h}:T_{h}G\to T_{gh}G} is the differential of L g {\displaystyle L_{g}} between tangent spaces. Let Lie ⁡ ( G ) {\displaystyle \operatorname {Lie} (G)} be the set of all left-translation-invariant vector fields on G. It is a real vector space. Moreover, it is closed under the Lie bracket of vector fields; i.e., [ X , Y ] {\displaystyle [X,Y]} is a left-translation-invariant vector field if X and Y are. Thus, Lie ⁡ ( G ) {\displaystyle \operatorname {Lie} (G)} is a Lie subalgebra of the Lie algebra of all vector fields on G and is called the Lie algebra of G. One can understand this more concretely by identifying the space of left-invariant vector fields with the tangent space at the identity, as follows: Given a left-invariant vector field, one can take its value at the identity, and given a tangent vector at the identity, one can extend it to a left-invariant vector field. This correspondence is one-to-one in both directions, so is bijective. Thus, the Lie algebra can be thought of as the tangent space at the identity and the bracket of X and Y in T e G {\displaystyle T_{e}G} can be computed by extending them to left-invariant vector fields, taking the bracket of the vector fields, and then evaluating the result at the identity. There is also another incarnation of Lie ⁡ ( G ) {\displaystyle \operatorname {Lie} (G)} as the Lie algebra of primitive elements of the Hopf algebra of distributions on G with support at the identity element; for this, see Related constructions below.

Matrix Lie groups Suppose G is a closed subgroup of GL(n;C), and thus a Lie group, by the closed subgroups theorem. Then the Lie algebra of G may be computed as

Lie ⁡ ( G ) = { X ∈ M ( n ; C ) ∣ e t X ∈ G for all t ∈ R } . {\displaystyle \operatorname {Lie} (G)=\left\{X\in M(n;\mathbb {C} )\mid e^{tX}\in G{\text{ for all }}t\in \mathbb {R} \right\}.}

For example, one can use the criterion to establish the correspondence for classical compact groups (cf. the table in "compact Lie groups" below.)

Homomorphisms If f : G → H {\displaystyle f:G\to H} is a Lie group homomorphism, then its differential at the identity element

d f = d f e : Lie ⁡ ( G ) → Lie ⁡ ( H ) {\displaystyle df=df_{e}:\operatorname {Lie} (G)\to \operatorname {Lie} (H)}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Lie group–Lie algebra correspondence

Start with the simplest possible case. Write down what Lie group–Lie algebra correspondence claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Lie group–Lie algebra correspondence before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Lie group–Lie algebra correspondence ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Lie group–Lie algebra correspondence

In research: Lie group–Lie algebra correspondence appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Lie group–Lie algebra correspondence in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Lie group–Lie algebra correspondence is common in secondary-school and first-year university syllabi. It links to neighbouring topics Differential geometry, Lie algebras, Lie groups, so understanding it makes those chapters shorter.
In everyday life: Look for Lie group–Lie algebra correspondence outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Lie group–Lie algebra correspondence in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Lie group–Lie algebra correspondence means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Lie group–Lie algebra correspondence out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Lie group–Lie algebra correspondence in simple terms?

In mathematics, Lie group–Lie algebra correspondence allows one to correspond a Lie group to a Lie algebra or vice versa, and study the conditions for such a relationship. Lie groups that are isomorphic to each other have Lie algebras that are isomorphic to each other, but the converse is not neces…

Why does Lie group–Lie algebra correspondence matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Lie group–Lie algebra correspondence?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Lie group–Lie algebra correspondence.