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What are the most important examples of non-power associative algebras? Why are they important, and what are their applications?

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This is not the best way of taking advantange of MO. The FAQ explains that questions which are of the form «tell me about X» are quite suboptimal. –  Mariano Suárez-Alvarez Nov 4 '11 at 16:47
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I personally have not encountered any examples. –  Bruce Westbury Nov 4 '11 at 17:01
    
Small remark that the algebras obtained by repeated Cayley-Dickson doubling (starting from $\mathbb{R}$, say) are all power-associative. –  Todd Trimble Nov 11 '11 at 18:02

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A tensor product of composition algebras, e.g. ${\mathbb O}\otimes{\mathbb O}$, is usually not power associative. One can use them to construct exceptional Lie algebras: see for example Tensor products of composition algebras, Albert forms and some exceptional Lie algebras by B.N.Allison, Trans. Am. Math. Soc. Vol. 306, 1988.

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Well... the free algebra on one generator $x$ is not power-associative.

It has a basis given by $1$, $x$, $xx$, $(xx)x$, $x(xx)$, $((xx)x)x$, $(x(xx))x$, $(xx)(xx)$, $x((xx)x)$, $x(x(xx)$, $(((xx)x)x)x$, etc.

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To be pedantic, you would probably like to say "free magmatic algebra". (A free algebra on the given set of generators depends on the operad over which it is an algebra.) –  Vladimir Dotsenko Nov 17 '11 at 14:42

The algebras known variously as left-symmetric algebras (or right-symmetric algebras), Vinberg algebras, or pre-Lie algebras are in general not power associative. These algebras occur in at least two apparently unrelated contexts. One is the study of flat affine structures, the other is the study of the algebraic structure of renormalization.

Let $[x, y, z] = (xy)z - x(yz)$ be the associator. An algebra is left-symmetric if \begin{align} (LSA)\qquad [x, y, z] = [y, x, z] \end{align} for all $x$, $y$, and $z$ (the opposite algebra of a left-symmetric algebra is right-symmetric, meaning it satisfies the identity $[x, y, z] = [x, z, y]$).

These certainly do not have to be power associative. A simple example, taken from E. Kleinfeld's paper "Assosymmetric rings" is the following. The algebra is spanned by $x_{1}$, $x_{2}$, and $x_{3}$, with the non-zero products $x_{1}^{2} = x_{2}$ and $x_{2}x_{1} = x_{3}$. The only non-zero associator of the form $[x_{i}, x_{j}, x_{k}]$ is $[x_{1}, x_{1}, x_{1}] = x_{3}$. This shows that the algebra is left symmetric but not power associative. Here's a more interesting example. Consider the algebra with basis $x_{1}, x_{2}, x_{3}, \dots$ and product $x_{i}x_{j} = (j+1)x_{i+j}$. Then $[x_{i}, x_{j}, x_{k}] = -k(k+1)x_{i+j+k} = [x_{j}, x_{i}, x_{k}]$, so the algebra is left symmetric. Since $[x_{i}, x_{i}, x_{i}] = -i(i+1)x_{3i}$, this algebra is not power associative. The associated Lie algebra is the Lie algebra of polynomial vector fields on the line. A concrete realization is given by $x_{i} = z^{i+1}\partial_{z}$. The left symmetric multiplication is that given by the standard (flat) covariant derivative.

Left symmetric algebras were introduced by E.B. Vinberg (hence the terminology "Vinberg algebras") and used by him to give a sort of classification of homogeneous convex cones (the title of the English translation is "The Theory of convex homogeneous cones"). In a related context, a left invariant flat (torsion free) connection on a Lie group gives its Lie algebra a structure of a left symmetric algebra. (If $\nabla$ is an affine connection, then the product on vector fields defined by $xy = \nabla_{x}y$ satisfies $[x, y] = xy - yx$ if $\nabla$ is torsion-free, and satisfies (LSA) if $\nabla$ is moreover flat).

Roughly contemporaneously, M. Gerstenhaber defined a product on the space of Hochschild cochains of an associative algebra and showed that makes the cochains into what he called a (graded) pre-Lie algebra. This alternative terminology reflects that for any left symmetric algebra the commutator $[x, y] = xy - yx$ satisfies the Jacobi identity, so is a Lie bracket (the condition (LSA) means that the left regular representation of the algebra is a Lie algebra homomorphism). In the context in which Gerstenhaber was working, the algebras are graded, meaning the rule of signs has to applied to all brackets, associators, etc.

There is a more sophisticated story involving operads and rooted trees and related to the work of Connes and Kreimer on the Hopf algebra of rooted trees, but I don't know the details well enough to recount it here. The key thing is that there is a product on rooted trees which makes them into a pre-Lie algebra. See the paper of Chapoton and Livernet ("Pre-Lie algebras and the rooted trees operad") and a survey by D. Manchon ("A short survey on pre-Lie algebras") for this point of view and references. For the more classical material related to cones and affine structures, in addition to Vinberg's article, some of the standard references are D. Segal's "Complete left-symmetric algebras" and J. Helmstetter's "Radical d'une algebra symetrique gauche". There are surveys by D. Burde and P. Cartier ("Vinberg algebras").

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Yes, somehow the relationship between pre-Lie algebras and the Connes--Kreimer story is possible precisely because of lack of power-associativity: this makes the Lie algebra associated to the free pre-Lie algebra on one generator interesting (in particular, non-abelian). –  Vladimir Dotsenko Nov 17 '11 at 14:44

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