In theoretical physics, the superconformal algebra is a graded Lie algebra or superalgebra that combines the conformal algebra and supersymmetry. In two dimensions, the superconformal algebra is infinite-dimensional. In higher dimensions, superconformal algebras are finite-dimensional and generate the superconformal group (in two Euclidean dimensions, the Lie superalgebra does not generate any Lie supergroup).

The conformal group of the ${\displaystyle (p+q)}$-dimensional space ${\displaystyle \mathbb {R} ^{p,q}}$ is ${\displaystyle SO(p+1,q+1)}$ and its Lie algebra is ${\displaystyle {\mathfrak {so}}(p+1,q+1)}$. The superconformal algebra is a Lie superalgebra containing the bosonic factor ${\displaystyle {\mathfrak {so}}(p+1,q+1)}$ and whose odd generators transform in spinor representations of ${\displaystyle {\mathfrak {so}}(p+1,q+1)}$. Given Kac's classification of finite-dimensional simple Lie superalgebras, this can only happen for small values of ${\displaystyle p}$ and ${\displaystyle q}$. A (possibly incomplete) list is

According to the superconformal algebra with ${\displaystyle {\mathcal {N}}}$ supersymmetries in 3+1 dimensions is given by the bosonic generators ${\displaystyle P_{\mu }}$, ${\displaystyle D}$, ${\displaystyle M_{\mu \nu }}$, ${\displaystyle K_{\mu }}$, the U(1) R-symmetry ${\displaystyle A}$, the SU(N) R-symmetry ${\displaystyle T_{j}^{i}}$ and the fermionic generators ${\displaystyle Q^{\alpha i}}$, ${\displaystyle {\overline {Q}}_{i}^{\dot {\alpha }}}$, ${\displaystyle S_{i}^{\alpha }}$ and ${\displaystyle {\overline {S}}^{{\dot {\alpha }}i}}$. Here, ${\displaystyle \mu ,\nu ,\rho ,\dots }$ denote spacetime indices; ${\displaystyle \alpha ,\beta ,\dots }$ left-handed Weyl spinor indices; ${\displaystyle {\dot {\alpha }},{\dot {\beta }},\dots }$ right-handed Weyl spinor indices; and ${\displaystyle i,j,\dots }$ the internal R-symmetry indices.

The Lie superbrackets of the bosonic conformal algebra are given by

where η is the Minkowski metric; while the ones for the fermionic generators are:

The bosonic conformal generators do not carry any R-charges, as they commute with the R-symmetry generators:

But the fermionic generators do carry R-charge:

Under bosonic conformal transformations, the fermionic generators transform as: