# Graded algebras

The exterior algebra is an example of a graded algebra, which means that it has a decomposition, or gradation (AKA grading), into a direct sum of vector subspaces $${\bigoplus V_{g}}$$ where each $${V_{g}}$$ corresponds to a weight (AKA degree), an element $${g}$$ of a monoid $${G}$$ (e.g. $${\mathbb{N}}$$ under $${+}$$) such that $${V_{g}V_{h}=V_{g+h}}$$. The tensor algebra is a $${\mathbb{N}}$$-graded algebra, since $${T^{j}V\otimes T^{k}V=T^{j+k}V}$$, as is the exterior algebra of $${\mathbb{R}^{n}}$$ (although $${V_{j}}$$ vanishes for $${j>n}$$). The property $${A\wedge B=\left(-1\right)^{jk}B\wedge A}$$ is then called graded commutativity (AKA graded anti-commutativity), whose definition can be generalized to other monoids. In this book we will assume that gradation weights take integer values.

A graded Lie algebra also obeys graded versions of the Jacobi identity and anti-commutativity. If we indicate the weight of $${v}$$ by $${\left|v\right|}$$, the graded Lie bracket becomes $${\left[u,v\right]=\left(-1\right)^{\left|u\right|\left|v\right|+1}\left[v,u\right]}$$, and the graded Jacobi identity is $${\left(-1\right)^{\left|u\right|\left|w\right|}\left[\left[u,v\right],w\right]+\left(-1\right)^{\left|v\right|\left|u\right|}\left[\left[v,w\right],u\right]+\left(-1\right)^{\left|w\right|\left|v\right|}\left[\left[w,u\right],v\right]=0}$$. A Lie superalgebra (AKA super Lie algebra) is a $${\mathbb{Z_{\textrm{2}}}}$$-graded Lie algebra $${V_{0}\oplus V_{1}}$$ that is used to describe supersymmetry in physics.