Locally compact quantum group

Key takeaways

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

Reference excerpt

In mathematics and theoretical physics, a locally compact quantum group is a C*-algebraic approach toward quantum groups that generalizes the Kac algebra, compact-quantum-group and Hopf-algebra approaches. Earlier attempts at a unifying definition of quantum groups using, for example, multiplicative unitaries have enjoyed some success but have also encountered several technical problems. One of the main features distinguishing this new approach from its predecessors is the axiomatic existence of left and right invariant weights. This gives a noncommutative analogue of left and right Haar measures on a locally compact Hausdorff group.

Definitions Before we can even begin to properly define a locally compact quantum group, we first need to define a number of preliminary concepts and also state a few theorems. Definition (weight). Let A {\displaystyle A} be a C*-algebra, and let A ≥ 0 {\displaystyle A_{\geq 0}} denote the set of positive elements of A {\displaystyle A} . A weight on A {\displaystyle A} is a function ϕ : A ≥ 0 → [ 0 , ∞ ] {\displaystyle \phi :A_{\geq 0}\to [0,\infty ]} such that

ϕ ( a 1 + a 2 ) = ϕ ( a 1 ) + ϕ ( a 2 ) {\displaystyle \phi (a_{1}+a_{2})=\phi (a_{1})+\phi (a_{2})} for all a 1 , a 2 ∈ A ≥ 0 {\displaystyle a_{1},a_{2}\in A_{\geq 0}} , and

ϕ ( r ⋅ a ) = r ⋅ ϕ ( a ) {\displaystyle \phi (r\cdot a)=r\cdot \phi (a)} for all r ∈ [ 0 , ∞ ) {\displaystyle r\in [0,\infty )} and a ∈ A ≥ 0 {\displaystyle a\in A_{\geq 0}} . Some notation for weights. Let ϕ {\displaystyle \phi } be a weight on a C*-algebra A {\displaystyle A} . We use the following notation:

M ϕ + := { a ∈ A ≥ 0 ∣ ϕ ( a ) < ∞ } {\displaystyle {\mathcal {M}}_{\phi }^{+}:=\{a\in A_{\geq 0}\mid \phi (a)<\infty \}} , which is called the set of all positive ϕ {\displaystyle \phi } -integrable elements of A {\displaystyle A} .

N ϕ := { a ∈ A ∣ ϕ ( a ∗ a ) < ∞ } {\displaystyle {\mathcal {N}}_{\phi }:=\{a\in A\mid \phi (a^{*}a)<\infty \}} , which is called the set of all ϕ {\displaystyle \phi } -square-integrable elements of A {\displaystyle A} .

M ϕ := Span M ϕ + = Span N ϕ ∗ N ϕ {\displaystyle {\mathcal {M}}_{\phi }:={\text{Span}}~{\mathcal {M}}_{\phi }^{+}={\text{Span}}~{\mathcal {N}}_{\phi }^{*}{\mathcal {N}}_{\phi }} , which is called the set of all ϕ {\displaystyle \phi } -integrable elements of A {\displaystyle A} . Types of weights. Let ϕ {\displaystyle \phi } be a weight on a C*-algebra A {\displaystyle A} .

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Locally compact quantum group

Start with the simplest possible case. Write down what Locally compact quantum group claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, 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 Locally compact quantum group 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 Locally compact quantum group 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 Locally compact quantum group

In research: Locally compact quantum group appears in physics 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 Locally compact quantum group 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: Locally compact quantum group is common in secondary-school and first-year university syllabi. It links to neighbouring topics C*-algebras, Functional analysis, Harmonic analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Locally compact quantum group 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 Locally compact quantum group in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Locally compact quantum group 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 Locally compact quantum group out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Locally compact quantum group in simple terms?

In mathematics and theoretical physics, a locally compact quantum group is a C*-algebraic approach toward quantum groups that generalizes the Kac algebra, compact-quantum-group and Hopf-algebra approaches. Earlier attempts at a unifying definition of quantum groups using, for example, multiplicativ…

Why does Locally compact quantum group matter?

Because it connects several physics 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 Locally compact quantum group?

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 Locally compact quantum group.