Quasi-free algebra

Key takeaways

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

Reference excerpt

In abstract algebra, a quasi-free algebra is an associative algebra that satisfies the lifting property similar to that of a formally smooth algebra in commutative algebra. The notion was introduced by Cuntz and Quillen for the applications to cyclic homology. A quasi-free algebra generalizes a free algebra, as well as the coordinate ring of a smooth affine complex curve. Because of the latter generalization, a quasi-free algebra can be thought of as signifying smoothness on a noncommutative space.

Definition Let A be an associative algebra over the complex numbers. Then A is said to be quasi-free if the following equivalent conditions are met:

Given a square-zero extension R → R / I {\displaystyle R\to R/I} , each homomorphism A → R / I {\displaystyle A\to R/I} lifts to A → R {\displaystyle A\to R} . The cohomological dimension of A with respect to Hochschild cohomology is at most one. Let ( Ω A , d ) {\displaystyle (\Omega A,d)} denotes the differential envelope of A; i.e., the universal differential-graded algebra generated by A. Then A is quasi-free if and only if Ω 1 A {\displaystyle \Omega ^{1}A} is projective as a bimodule over A. There is also a characterization in terms of a connection. Given an A-bimodule E, a right connection on E is a linear map

∇ r : E → E ⊗ A Ω 1 A {\displaystyle \nabla _{r}:E\to E\otimes _{A}\Omega ^{1}A}

that satisfies ∇ r ( a s ) = a ∇ r ( s ) {\displaystyle \nabla _{r}(as)=a\nabla _{r}(s)} and ∇ r ( s a ) = ∇ r ( s ) a + s ⊗ d a {\displaystyle \nabla _{r}(sa)=\nabla _{r}(s)a+s\otimes da} . A left connection is defined in the similar way. Then A is quasi-free if and only if Ω 1 A {\displaystyle \Omega ^{1}A} admits a right connection.

Properties and examples One of basic properties of a quasi-free algebra is that the algebra is left and right hereditary (i.e., a submodule of a projective left or right module is projective or equivalently the left or right global dimension is at most one). This puts a strong restriction for algebras to be quasi-free. For example, a hereditary (commutative) integral domain is precisely a Dedekind domain. In particular, a polynomial ring over a field is quasi-free if and only if the number of variables is at most one. An analog of the tubular neighborhood theorem, called the formal tubular neighborhood theorem, holds for quasi-free algebras.

References

Bibliography Cuntz, Joachim (June 2013). "Quillen's work on the foundations of cyclic cohomology". Journal of K-Theory. 11 (3): 559–574. arXiv:1202.5958. doi:10.1017/is012011006jkt201. ISSN 1865-2433. Cuntz, Joachim; Quillen, Daniel (1995). "Algebra Extensions and Nonsingularity". Journal of the American Mathematical Society. 8 (2): 251–289. doi:10.2307/2152819. ISSN 0894-0347. JSTOR 2152819. Kontsevich, Maxim; Rosenberg, Alexander L. (2000). "Noncommutative Smooth Spaces". The Gelfand Mathematical Seminars, 1996–1999. Birkhäuser. pp. 85–108. arXiv:math/9812158. doi:10.1007/978-1-4612-1340-6_5. ISBN 978-1-4612-7102-4. Maxim Kontsevich, Alexander Rosenberg, Noncommutative spaces, preprint MPI-2004-35 Vale, R. (2009). "notes on quasi-free algebras" (PDF).

Further reading https://ncatlab.org/nlab/show/quasi-free+algebra

Worked examples

Example 1 — a first encounter with Quasi-free algebra

Start with the simplest possible case. Write down what Quasi-free algebra 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 Quasi-free algebra 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 Quasi-free algebra 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 Quasi-free algebra

In research: Quasi-free algebra 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 Quasi-free algebra 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: Quasi-free algebra is common in secondary-school and first-year university syllabi. It links to neighbouring topics Abstract algebra, Abstract algebra stubs, so understanding it makes those chapters shorter.
In everyday life: Look for Quasi-free algebra 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 Quasi-free algebra in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Quasi-free algebra 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 Quasi-free algebra out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Quasi-free algebra in simple terms?

In abstract algebra, a quasi-free algebra is an associative algebra that satisfies the lifting property similar to that of a formally smooth algebra in commutative algebra. The notion was introduced by Cuntz and Quillen for the applications to cyclic homology.

Why does Quasi-free algebra 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 Quasi-free algebra?

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 Quasi-free algebra.