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Let $\mu:\mathbb{R}^n \times \mathbb{R}^n \longrightarrow \mathbb{R}^n$ be a alternating bilinear map, i.e. $\mu(X,Y)=-\mu(Y,X)$ (anticommutativity) and so, let $\mathfrak{a}=(\mathbb{R}^n,\mu)$ be a skew-symmetric algebra (this one is not necessarily a Lie algebra).

Questions:

  1. Is there a "famous" example of a skew-symmetric algebra that is not a Lie algebra?

  2. We assume that for any $X \in \mathfrak{a}$, $\mu(X,\cdot):\mathbb{R}^n \longrightarrow \mathbb{R}^n$ is a nilpotent linear transformation. Is it true that $\mathfrak{a}$ is a nilpotent algebra? (recall that $\mathfrak{a}$ is not necessarily a Lie algebra)

Thank you for your help.

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If you don't assume associativity, then the multiplication of imaginary octonions gives you a famous example of 1. –  José Figueroa-O'Farrill Jun 3 '11 at 12:04

1 Answer 1

  1. To rephrase José Figueroa-O'Farrill's comment above - a 7-dimensional simple exceptional Malcev algebra (which is a quotient of octonions under the commutator by 1-dimensional center). But, really, I find the question is formulated badly: just skew-commutativity is a very mild restriction to say something meaningful about an algebra in general, so it's almost the same as to ask for "famous" examples of a (nonassociative) algebra.

  2. No, this is not true. I was able to construct counterexamples on computer, as finite-dimensional quotients of free algebras in respective "Engel varieties", but all they are large and cumbersome. Shorther examples can be found in the following paper by Koreshkov and Kharitonov, which, apparently, was published twice:

    About nilpotency of Engel algebras, Formal Power Series and Algebraic Combinatorics (ed. D. Krob et al.), Springer, 2000, 461-467; ZBL: 0983.17003 [available on google books and amazon]

    Nilpotency of the Engel algebras, Russ. Math. (Izv. VUZ) 45 (2001), No. 11, 15-18; ZBL: 1103.17300

They prove also that this is true for algebras of dimension $\le 4$. It is also easy to see (and is recorded, for example, in: V.T. Filippov, Binary Lie algebras satisfying the third Engel condition, Siber. Math. J. 49 (2008), N4, 744-748; DOI: 10.1007/s11202-008-0071-3) that the second Engel condition implies nilpotency of degree $4$.

On the other hand, Engel(-like) theorems were established for many particular ("famous"?) classes of anticommutative algebras considered in the literature: binary-Lie, Malcev, and some other generalizations of Lie algebras, and it is an open question, as far as I know, for Sagle algebras.

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For some reason I like your "which, apparently, was published twice" remark a lot. –  Vladimir Dotsenko Jun 11 '11 at 10:56
    
@Vladimir Dotsenko: No offense or sarcasm meant, really. Just a mere (bibliographical) fact. I like this paper and have used it on some other occasion. –  Pasha Zusmanovich Jun 11 '11 at 11:11
    
Oh I did understand that. It's the option of a tongue-in-cheek interpretation that makes it lovely. –  Vladimir Dotsenko Jun 11 '11 at 22:26

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