Lie ring

In mathematics a Lie ring is a structure related to Lie algebras that can arise as a generalisation of Lie algebras, or through the study of the lower central series of groups.

Formal definition

A Lie ring is defined as a nonassociative ring with multiplication that is anticommutative and satisfies the Jacobi identity. More specifically we can define a Lie ring $L$ to be an abelian group with an operation $[cdot,cdot]$ that has the following properties:

* Bilinearity:

::$[x\; +\; y,\; z]\; =\; [x,\; z]\; +\; [y,\; z]\; ,\; quad\; [z,\; x\; +\; y]\; =\; [z,\; x]\; +\; [z,\; y]$

:for all "x", "y", "z" ∈ "L".

* The "Jacobi identity":

:: $[x,\; [y,z]\; +\; [y,\; [z,x]\; +\; [z,\; [x,y]\; =\; 0\; quad$

:for all "x", "y", "z" in "L".

* For all "x" in "L".

::$[x,x]\; =0\; quad$

Examples

* Any Lie algebra over a general ring instead of a field is an example of a Lie ring.

* Any associative ring can be made into a Lie ring by defining a bracket operator $[x,y]\; =\; xy\; -\; yx$.

* For an example of a Lie ring arising from the study of groups, let $G$ be a group, and let $G\; =\; G\_0\; supseteq\; G\_1\; supseteq\; G\_2\; supseteq\; cdots\; supseteq\; G\_n\; supseteq\; cdots$ be a central series in $G$ - that is $[G\_i,G\_j]\; subseteq\; G\_\{i+j\}$ for any $i,j$. Then

::

:is a Lie ring with addition supplied by the group operation (which will be commutative in each homogeneous part), and the bracket operation given by

:: $[xG\_i,\; yG\_j]\; =\; [x,y]\; G\_\{i+j\}$

:extended linearly.