  We study the algebraic structure of electron density operators in gapless
Weyl fermion systems in $d=3,5,7,\cdots$ spatial dimensions and in topological
insulators (without any protecting symmetry) in $d=4,6,8,\cdots$ spatial
dimensions. These systems are closely related by the celebrated bulk-boundary
correspondence. Specifically, we study the higher bracket -- a generalization
of commutator for more than two operators -- of electron density operators in
these systems. For topological insulators, we show that the higher-bracket
algebraic structure of density operators structurally parallels with the
Girvin-MacDonald-Platzman algebra (the $W_{1+\infty}$ algebra), the algebra of
electron density operators projected onto the lowest Landau level in the
quantum Hall effect. By the bulk-boundary correspondence, the bulk
higher-bracket structure mirrors its counterparts at the boundary.
Specifically, we show that the density operators of Weyl fermion systems, once
normal-ordered with respect to the ground state, their higher bracket acquires
a c-number part. This part is an analog of the Schwinger term in the commutator
of the fermion current operators. We further identify this part with a cyclic
cocycle, which is a topological invariant and an element of Connes'
noncommutative geometry.
