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.

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.

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