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Reading @Qiaochu Yuan's blog the other day, I came across a new term: Poisson algebra. I had never heard of it before and wondered what other algebras are out there that I'm not aware of. Neither Wikipedia nor Google are of much help in this regard, since one can't regex google and Wikipedia has a rather paltry list of algebras.

Here are some algebras I'm familiar with:

  • Boolean algebra
  • Algebra of sets
  • various group algebras
  • Heyting algebra
  • Flag algebra
  • $\sigma$-algebras (OK, not really the same use of the word)
  • Exterior algebra

What are some more algebras that I'm missing out on?

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1  
Term algebras in mathematical logic. Lots of special cases of associative algebras: offhand I can recall seeing Banach, Clifford, Hopf, incidence, and Lie algebras. –  Brian M. Scott Dec 4 '11 at 19:23
    
Here's a WP list of algebras. –  Mitch Dec 4 '11 at 20:21
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What is the purpose of constructing this list? –  Mariano Suárez-Alvarez Dec 4 '11 at 21:03
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(I mean: I can probably name 50 «algebras»... What good would it be for the OP or anyone else?) –  Mariano Suárez-Alvarez Dec 4 '11 at 21:10
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en.wikipedia.org/w/… –  sdcvvc Dec 6 '11 at 20:33

3 Answers 3

You forgot the von Neumann algebras and $C^*$-algebras, which are very important in non-commutative geometry, among other things. The basic idea is that you can understand certain topological spaces $T$ by looking at their (commutative) $C^*$-algebra $C^0(T,\mathbb{C})$. Non commutative geometry is a generalization of sorts of this correspondence to non-commutative algebras.

The classification of von Neumann algebras is a difficult and recent topic (see Connes, "Noncommutative Geometry"), and it is of course completely hopeless to classify all algebras...

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I agree with the last one. I can invent a new algebra right now, it forms a group with the elements S, T, E, V and E, and the addition operation is binary and takes any two and gives you some other one (defined by a table), and I'll call it Steve algebra. –  Dhaivat Pandya Dec 4 '11 at 19:23
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«see Connes, "Non commutative geometry"» is funny in the same way as «it's en Euler's œuvre somewhere». –  Mariano Suárez-Alvarez Dec 4 '11 at 21:05
    
@MarianoSuárez-Alvarez have you seen that video of J.P. Serre, "how to write bad mathematics" ? =) –  Glougloubarbaki Dec 6 '11 at 19:19

A very important class of algebras: Lie algebras. An easy example is that of $\mathbb{R}^3$ paired with the cross product. Or take any associative algebra $\mathcal{A}$ and replace the multiplication operator with the commutator: $[a,b]=ab-ba$. In some sense all Lie algebra arise in this way (see: Universal Enveloping Algebra).

Not totally unrelated are Jordan algebras. To get examples take an associative algebra and replace the multiplication operator with $1/2$ the anti-commutator: $a \circ b = \frac{1}{2}(ab+ba)$.

A non-classical collection of algebras which in some sense unifies associative commutative algebras, Lie algebra, and Jordan algebras is Vertex Algebras.

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Relational Algebra and Process Algebra. Although they're both narrow in problem scope, they're pretty important for Data Management and Simulation "Management" (translating model to actual simulation).

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