Quasi-Lie algebra

In mathematics, a quasi-Lie algebra in abstract algebra is just like a Lie algebra, but with the usual axiom

[math]\displaystyle{ [x,x]=0 }[/math]

replaced by

[math]\displaystyle{ [x,y]=-[y,x] }[/math] (anti-symmetry).

In characteristic other than 2, these are equivalent (in the presence of bilinearity), so this distinction doesn't arise when considering real or complex Lie algebras. It can however become important, when considering Lie algebras over the integers.

In a quasi-Lie algebra,

[math]\displaystyle{ 2[x,x]=0. }[/math]

Therefore, the bracket of any element with itself is 2-torsion, if it does not actually vanish.

See also

• Whitehead product

References

• Serre, Jean-Pierre (2006). Lie Algebras and Lie Groups. 1964 lectures given at Harvard University. Lecture Notes in Mathematics. 1500 (Corrected 5th printing of the 2nd (1992) ed.). Berlin: Springer-Verlag. doi:10.1007/978-3-540-70634-2. ISBN 3-540-55008-9.