Jacobi identity

Definition

The Jacobi identity is one of the defining identities for the definition of Lie ring (and also for the definition of Lie algebra over any commutative unital ring), and plays a role in Lie rings similar to the role that associativity plays in groups. For a Lie ring $L$, the identity states that for any $x,y,z\in L$, the following are true:

• Left-normed version: $[[x,y],z]+[[y,z],x]+[[z,x],y]=0$
• Right-normed version: $[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0$

Note that the left-normed and right-normed versions of the Jacobi identity are equivalent for an alternating ring and more generally a skew-symmetric ring.

The Jacobi identity can also be viewed as one of the two generators in a two-element generating set for the Lie operad (applicable in characteristic zero). The other generator is the skew symmetry identity.

Verification

Linearity of the identity

Suppose $L$ is a (non-associative) algebra over a commutative unital ring $R$. Then, the expressions for the left-normed and right-normed Jacobi identity are $R$-trilinear in all three inputs. Explicitly, the expressions:

• $J_l(x,y,z)=[[x,y],z]+[[y,z],x]+[[z,x],y]$
• $J_r(x,y,z)=[x,[y,z]]+[y,[z,x]]+[z,[x,y]]$

are $R$-linear in each coordinate as well as cyclically symmetric in the three inputs.

Further, if the bracket operation is skew-symmetric, then $J_l(x,y,z)=-J_r(x,y,z)$.

Verification using a module generating set

Suppose $L$ is a (non-associative) algebra over a commutative unital ring $R$ and $S$ is a generating set for the additive group of $L$ as a $R$-module. Then the following are true:

• The left-normed Jacobi identity is true for all $x,y,z\in L$ if and only if it is true for all $x,y,z\in S$.
• The right-normed Jacobi identity is true for all $x,y,z\in L$ if and only if it is true for all $x,y,z\in S$.

Under the skew symmetry assumption, the left-normed Jacobi identity is equivalent to the right-normed Jacobi identity.

Description in terms of structure constants

Suppose $L$ is a (non-associative) algebra over a commutative unital ring $R$ and $e_i,i\in I$ form a generating set for $L$. We can define the $R$-bilinear map $[,]:L\times L\to L$ by its structure constants $\underset{ij}{\overset{k}{\lambda }}$ where:

$[e_i,e_j]={\sum }_{k\in I}\underset{ij}{\overset{k}{\lambda }}{e}_k$

In the case that $L$ is a free $R$-modules, the structure constants are uniquely determined, and the Jacobi identity reduces to verifying the following:

The left-normed Jacobi identity says that for all $i,j,k,l\in I$, the following quadratic condition holds:

${\sum }_{m\in I}(\underset{ij}{\overset{m}{\lambda }}\underset{mk}{\overset{l}{\lambda }}+\underset{jk}{\overset{m}{\lambda }}\underset{mi}{\overset{l}{\lambda }}+\underset{ki}{\overset{m}{\lambda }}\underset{mj}{\overset{l}{\lambda }})=0$

The right-normed Jacobi identity says that for all $i,j,k,l\in I$, the following quadratic condition holds:

${\sum }_{m\in I}(\underset{im}{\overset{l}{\lambda }}\underset{jk}{\overset{m}{\lambda }}+\underset{jm}{\overset{l}{\lambda }}\underset{ki}{\overset{m}{\lambda }}+\underset{km}{\overset{l}{\lambda }}\underset{ij}{\overset{m}{\lambda }})=0$

Under the skew symmetry assumption, $\underset{ij}{\overset{k}{\lambda }}=-\underset{ji}{\overset{k}{\lambda }}$, and thus, the above two quadratic conditions are equivalent.

Verification for a graded ring

Suppose $R$ is a non-associative ring graded over an abelian group $G$:

$R={⨁}_{g\in G}{R}_g$

Then, to verify that the left-normed Jacobi identity holds in $R$ as a whole, it suffices to verify that for all $g,h,k\in G$ (possibly equal, possibly distinct), and all $x\in R_g,y\in R_h,z\in R_k$, the left-normed Jacobi identity holds for $x,y,z$. Similarly for the right-normed Jacobi identity.