In mathematics, a super vector space is a Z 2 {\displaystyle \mathbb {Z} _{2}} -graded vector space, that is, a vector space over a field K {\displaystyle \mathbb {K} } with a given decomposition of subspaces of grade 0 {\displaystyle 0} and grade 1 {\displaystyle 1} . The study of super vector spaces and their generalizations is sometimes called super linear algebra. These objects find their principal application in theoretical physics where they are used to describe the various algebraic aspects of supersymmetry.
Definitions A super vector space is a Z 2 {\displaystyle \mathbb {Z} _{2}} -graded vector space with decomposition
V = V 0 ⊕ V 1 , 0 , 1 ∈ Z 2 = Z / 2 Z . {\displaystyle V=V_{0}\oplus V_{1},\quad 0,1\in \mathbb {Z} _{2}=\mathbb {Z} /2\mathbb {Z} .}
Vectors that are elements of either V 0 {\displaystyle V_{0}} or V 1 {\displaystyle V_{1}} are said to be homogeneous. The parity of a nonzero homogeneous element, denoted by | x | {\displaystyle |x|} , is 0 {\displaystyle 0} or 1 {\displaystyle 1} according to whether it is in V 0 {\displaystyle V_{0}} or V 1 {\displaystyle V_{1}} ,
| x | = { 0 x ∈ V 0 1 x ∈ V 1 {\displaystyle |x|={\begin{cases}0&x\in V_{0}\\1&x\in V_{1}\end{cases}}}
Vectors of parity 0 {\displaystyle 0} are called even and those of parity 1 {\displaystyle 1} are called odd. In theoretical physics, the even elements are sometimes called Bose elements or bosonic, and the odd elements Fermi elements or fermionic. Definitions for super vector spaces are often given only in terms of homogeneous elements and then extended to nonhomogeneous elements by linearity. If V {\displaystyle V} is finite-dimensional and the dimensions of V 0 {\displaystyle V_{0}} and V 1 {\displaystyle V_{1}} are p {\displaystyle p} and q {\displaystyle q} respectively, then V {\displaystyle V} is said to have dimension p | q {\displaystyle p|q} . The standard super coordinate space, denoted K p | q {\displaystyle \mathbb {K} ^{p|q}} , is the ordinary coordinate space K p + q {\displaystyle \mathbb {K} ^{p+q}} where the even subspace is spanned by the first p {\displaystyle p} coordinate basis vectors and the odd space is spanned by the last q {\displaystyle q} . A homogeneous subspace of a super vector space is a linear subspace that is spanned by homogeneous elements. Homogeneous subspaces are super vector spaces in their own right (with the obvious grading). For any super vector space V {\displaystyle V} , one can define the parity reversed space Π V {\displaystyle \Pi V} to be the super vector space with the even and odd subspaces interchanged. That is,
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