Multiplicative Lie ring

Definition

Left action convention

A multiplicative Lie ring is a (possibly abelian, possibly non-abelian) group $L$ equipped with a function $\{\ ,\ \}:\times L\to L$ such that the following hold. For convenience, we denote by $c_{g}(h)$ the element $ghg^{-1}$, obtained by conjugation of $h$ by $g$. The following conditions must hold for all $x,x',y,y'\in L$.

• $\{x,x\}=1$ (here, $1$ denotes the identity elemnet of $L$)
• $\{x,yy'\}=\{x,y\}c_{y}(\{x,y'\})$
• $\{xx',y\}=c_{x}(\{x',y\})\{x,y\}$
• $\{\{x,y\},c_{y}(z)\}\{\{y,z\},c_{z}(x)\}\{\{z,x\},c_{x}(y)\}=1$
• $c_{z}(\{x,y\})=\{c_{z}(x),c_{z}(y)\}$

Viewpoints

• The multiplicative Lie rings form a variety of algebras.
• We can therefore also talk of the category of multiplicative Lie rings.

Particular cases

• Every group can naturally be viewed as a multiplicative Lie ring, namely, the multiplicative Lie ring where the bracket is the commutator in the group. The second and third identity follow from the formula for commutator of element and product of two elements. The fourth identity is a variation of Witt's identity, while the fifth identity follows from the fact that group acts as automorphisms by conjugation.
• Every group equipped with the trivial bracket is a multiplicative Lie ring. This particular multiplicative Lie ring structure is termed the abelian multiplicative Lie ring.
• Every Lie ring is a multiplicative Lie ring. In fact, Lie rings are precisely the multiplicative Lie rings whose underlying groups are abelian groups.

References

Journal references

• On five well-known commutator identities by Graham Ellis, Journal of the Australian Mathematical Society, Volume 54, Page 1 - 19(Year 1993): ^{More info}
• Homology of multiplicative Lie rings by A. Bak, G. Donadze and N. Inassaridze, Journal of Pure and Applied Algebra, Volume 208, Page 761 - 777(Year 2007): ^{Ungated copy (PDF)More info}