Nonassociative rings and algebras Definition

Lie-admissible algebra

Builds on Nonassociative algebra Lie algebra Commutator of a ring

Definition

Let $A$ be a nonassociative algebra over a field $k$ , with product $(x, y) \mapsto xy$ . Let $[x, y] = xy - yx$ be the commutator of $x$ and $y$ . The algebra $A$ is Lie-admissible if the vector space $A$ with the bracket $[\cdot, \cdot]$ is a Lie algebra. Since $[x, x] = 0$ always holds, this means that $[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0$ for all $x, y, z \in A$ . The resulting Lie algebra is written $A^-$ .

Remarks

Example: every associative algebra is Lie-admissible. Expanding $[x, [y, z]] + [y, [z, x]] + [z, [x, y]]$ gives twelve products of three factors, and by associativity they cancel in pairs.

Non-example: let $\operatorname{char} k \neq 2$ , and let $A$ have basis $x, y, z$ with $xy = y$ , $yx = -y$ , $yz = x$ , $zy = -x$ , and all other products of basis vectors zero. The product is anticommutative, so $[u, v] = 2uv$ . Then $[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 4\big(x x + y \cdot 0 + z y\big) = -4x \neq 0$ , so $A$ is not Lie-admissible.

An anticommutative algebra is Lie-admissible exactly when it is a Lie algebra, in characteristic not $2$ .

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Tag	`TA34ZM234N`
URI	`m2://TA34ZM234N`
Kind	definition
Writing	W0 stub
MSC	17D25
Defines	Lie-admissible algebra
Uses	Nonassociative algebra, Lie algebra, Commutator of a ring
Changed	3 November 2026

Lie-admissible algebra

Definition

Remarks

Reading path

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