Abstract
A multiplicative Lie algebra is a group together with a bracket function that satisfies the basic properties of the commutator function. This paper investigates the construction of such functions.
Highlights
In his paper [1], Graham Ellis defined the concept of a multiplicative Lie algebra
Definition 1.1 A multiplicative Lie algebra consists of a group G together with a bracket function {, } : G×G → G satisfying the following identities for all x, y, z ∈ G : 1. {x, x} = 1, 2. {x, yz} = {x, y} y{x, z}, 3. {xy, z} =x {y, z}{x, z}, 4. {{y, x}, xz}{{x, z}, zy}{{z, y}, yx} = 1, 5. z{x, y} = {zx,z y}
Yx is short for yxy−1, [x, y] is the commutator xyx−1y−1, and (iv) is a Jacobi–Witt–Hall type identity
Summary
In his paper [1], Graham Ellis defined the concept of a multiplicative Lie algebra. According to his definition we have the following. Yx is short for yxy−1 , [x, y] is the commutator xyx−1y−1 , and (iv) is a Jacobi–Witt–Hall type identity The study of such properties began in the papers by MacDonald and Neumann ([2], [3]), who were interested in the interrelationships between various commutator laws. Graham Ellis was interested in showing that any universal commutator identity was a consequence of the identities in the above definition.
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