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The geometric Whitehead product of based maps $\alpha :S^p\to Y$, $\beta :S^q\to Y$ is their wedge sum map composed with the attaching map $S^{p+q-1}\to S^p\vee S^q$ of the top cell of $S^p\times S^q$. With the usual geometric signs it satisfies $[\alpha ,\beta {]}_W=(-1{)}^{pq}[\beta ,\alpha {]}_W$. Regrading ${\pi }_{p}Y$ to degree $p-1$ and setting $\{\alpha ,\beta \}=(-1{)}^p[\alpha ,\beta {]}_W$ gives the graded Lie bracket transported from the Samelson product on the loop space. In degrees $p,q\ge 2$ it is bilinear; fundamental-group inputs instead record commutators and the action of ${\pi }_1$ on higher homotopy.

Over $\mathbb{C}$, the Witt algebra is the Lie algebra of algebraic derivations of the Laurent polynomial ring $\mathbb{C}[z,z^{-1}]$. It has basis ${\ell }_n=-z^{n+1}d/dz$ for $n\in \mathbb{Z}$. The commutator of these differential operators isIndeed, the second-derivative terms cancel in the commutator, leaving the derivative of the coefficient of each operator. The span of ${\ell }_{-1},{\ell }_0,{\ell }_1$ is a copy of the sl2 Lie algebra: take $e={\ell }_{-1}$, $h=2{\ell }_0$, $f=-{\ell }_1$.

The Witt algebra models finite Fourier series of complexified vector fields on the circle. Its nontrivial one-dimensional central extension of a Lie algebra is the Virasoro algebra, whose central term changes the representation theory relevant to conformal field theory.

For a free massless pseudoclassical spinning particle, let $p_m$ be the canonical momentum and ${\psi }^m$ odd coordinates in a Grassmann algebra. Choose the Dirac bracket conventionwith all $p_m$ commuting with the ${\psi }^n$ and with each other. The odd supercharge $\mathcal{Q}$ then obeysThe first equality follows immediately by contracting the elementary Dirac bracket with $p_m{p}_n$. It expresses the closure of two odd symmetry transformations into an even worldline translation.

In units $\hslash =1$, canonical quantization replaces the graded Dirac bracket by the corresponding graded commutator multiplied by $i$. Thus $\{{\hat{\psi }}^m,{\hat{\psi }}^n\}={\eta }^{mn}$ andConsequently the physical-state constraint $\hat{\mathcal{Q}}|\Psi ⟩=0$ implies $\hat{\mathcal{H}}|\Psi ⟩=0$, the massless mass-shell condition. Specifying these bracket and unit conventions fixes the normalization of the worldline supersymmetry algebra.