Lorentz algebra ($\mathfrak{so}(1,3)$)

The Lorentz algebra has rotation generators $J_i$ and boost generators $K_i$ satisfying

Lorentz generators produce infinitesimal Lorentz transformations. In four dimensions they split into three rotation and three boost generators. Their action on supercharges uses the appropriate spinor representation, while their action on translations is the mixed Poincare algebra commutator. Signs in component formulas must agree with whether the generators act on states, operator columns or dual rows.

For the Lorentz algebra convention $[M_{\mu \nu },M_{\rho \sigma }]=i(M_{\mu \sigma }{\eta }_{\nu \rho }+M_{\nu \rho }{\eta }_{\mu \sigma }-M_{\mu \rho }{\eta }_{\nu \sigma }-M_{\nu \sigma }{\eta }_{\mu \rho })$, choose $\eta =\mathrm{diag}(s,-s,-s,-s)$, $s=\pm 1$, ${ϵ}_{123}=1$, $J_i={ϵ}_{ijk}{M}_{jk}/2$, and $K_i=M_{0i}$. The resulting rotation bracket is $[J_i,J_j]=si{ϵ}_{ijk}{J}_k$ as well as the two displayed brackets. Changing the Minkowski metric signature while holding this generator convention fixed reverses all coefficients. A simultaneous redefinition of the generators can instead preserve the familiar form. Conventions must be specified before comparing algebra signs.

Over the complex numbers, $A_i=(J_i+i{K}_i)/2$ and $B_i=(J_i-i{K}_i)/2$ generate commuting copies of ${\mathfrak{sl}}_2(\mathbb{C})$, givingFinite-dimensional irreducible representations are labelled $(j_L,j_R)$.

For Hermitian rotations $J_i$ and boosts $K_i$, the complex combinations $A_i=(J_i+i{K}_i)/2$ and $B_i=(J_i-i{K}_i)/2$ are adjoints of one another. Although their complex brackets split into two commuting ${\mathfrak{sl}}_2$ factors, they are not two independent Hermitian compact rotation algebras. The real Lorentz spin group is $SL(2,\mathbb{C})$; $SU(2)\times SU(2)$ is a different compact real form.

The complexified Lorentz algebra splits into two commuting ${\mathfrak{sl}}_2$ factors. Extending two homogeneous polynomial representations of SU2 therefore gives every finite-dimensional irreducible complex group representation of the Lorentz spinor double cover. The kernel sign permits descent precisely when $j_L+j_R$ is an integer. These field group representations are generally nonunitary for a positive-definite inner product; infinite-dimensional unitary Lorentz group representations belong to a different construction.

Parity fixes $J_i$, negates $K_i$, and therefore exchanges the two chiral factors. It sends $(j_L,j_R)$ to $(j_R,j_L)$.

The quadratic contraction $M_{\mu \nu }{M}^{\mu \nu }$ is parity even, while $M_{\mu \nu }{\tilde{M}}^{\mu \nu }$ is parity odd. Their linear combinations give the quadratic Casimirs of the two chiral ${\mathfrak{sl}}_2$ factors.