Superconformal algebra

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 Kač'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

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