Definition Definition

A graded Lie algebra is a graded vector space V = { V p } 𝑉 subscript 𝑉 𝑝 V=\{V_{p}\} italic_V = { italic_V start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT } together with a graded skew commutative bracket [ , ] : V ⊗ V → V fragments fragments [ , ] : V tensor-product V → V [\ ,\ ]:V\otimes V\rightarrow V [ , ] : italic_V ⊗ italic_V → italic_V such that V p ⊗ V q → V p + q → tensor-product subscript 𝑉 𝑝 subscript 𝑉 𝑞 subscript 𝑉 𝑝 𝑞 V_{p}\otimes V_{q}\to V_{p+q} italic_V start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ⊗ italic_V start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT → italic_V start_POSTSUBSCRIPT italic_p + italic_q end_POSTSUBSCRIPT satisfying the graded Jacobi identity:

[ u , [ v , w ] ] = [ [ u , v ] , w ] + ( - 1 ) p ⁢ q ⁢ [ v , [ u , w ] ] . 𝑢 𝑣 𝑤 𝑢 𝑣 𝑤 superscript 1 𝑝 𝑞 𝑣 𝑢 𝑤 [u,[v,w]]=[[u,v],w]+(-1)^{pq}[v,[u,w]]. [ italic_u , [ italic_v , italic_w ] ] = [ [ italic_u , italic_v ] , italic_w ] + ( - 1 ) start_POSTSUPERSCRIPT italic_p italic_q end_POSTSUPERSCRIPT [ italic_v , [ italic_u , italic_w ] ] .