Inmathematics, asuper vector space is a${\displaystyle {\mathbb{Z}}_2}$-graded vector space, that is, avector space over afield${\displaystyle \mathbb{K}}$ with a givendecomposition of subspaces of grade${\displaystyle 0}$ and grade${\displaystyle 1}$. The study of super vector spaces and their generalizations is sometimes calledsuper linear algebra. These objects find their principal application intheoretical physics where they are used to describe the various algebraic aspects ofsupersymmetry.

A super vector space is a${\displaystyle {\mathbb{Z}}_2}$-graded vector space with decomposition^[1]

${\displaystyle V=V_0\oplus V_1,0,1\in {\mathbb{Z}}_2=\mathbb{Z}/2\mathbb{Z}.}$

Vectors that are elements of either${\displaystyle V_0}$ or${\displaystyle V_1}$ are said to behomogeneous. Theparity of a nonzero homogeneous element, denoted by${\displaystyle |x|}$, is${\displaystyle 0}$ or${\displaystyle 1}$ according to whether it is in${\displaystyle V_0}$ or${\displaystyle V_1}$,

Vectors of parity${\displaystyle 0}$ are calledeven and those of parity${\displaystyle 1}$ are calledodd. In theoretical physics, the even elements are sometimes calledBose elements orbosonic, and the odd elementsFermi 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${\displaystyle V}$ isfinite-dimensional and the dimensions of${\displaystyle V_0}$ and${\displaystyle V_1}$ are${\displaystyle p}$ and${\displaystyle q}$ respectively, then${\displaystyle V}$ is said to havedimension${\displaystyle p|q}$. The standard super coordinate space, denoted${\displaystyle {\mathbb{K}}^{p|q}}$, is the ordinarycoordinate space${\displaystyle {\mathbb{K}}^{p+q}}$ where the even subspace is spanned by the first${\displaystyle p}$ coordinate basis vectors and the odd space is spanned by the last${\displaystyle q}$.

Ahomogeneous subspace of a super vector space is alinear 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${\displaystyle V}$, one can define theparity reversed space${\displaystyle \mathrm{\Pi }V}$ to be the super vector space with the even and odd subspaces interchanged. That is,

Ahomomorphism, amorphism in thecategory of super vector spaces, from one super vector space to another is a grade-preservinglinear transformation. A linear transformation${\displaystyle f:V\to W}$ between super vector spaces is grade preserving if

${\displaystyle f(V_i)\subset W_i,i=0,1.}$

That is, it maps the even elements of${\displaystyle V}$ to even elements of${\displaystyle W}$ and odd elements of${\displaystyle V}$ to odd elements of${\displaystyle W}$. Anisomorphism of super vector spaces is abijective homomorphism. The set of all homomorphisms${\displaystyle V\to W}$ is denoted${\displaystyle \mathrm{H}\mathrm{o}\mathrm{m}(V,W)}$.^[2]

Every linear transformation, not necessarily grade-preserving, from one super vector space to another can be written uniquely as the sum of a grade-preserving transformation and a grade-reversing one—that is, a transformation${\displaystyle f:V\to W}$ such that

${\displaystyle f(V_i)\subset W_{1-i},i=0,1.}$

Declaring the grade-preserving transformations to beeven and the grade-reversing ones to beodd gives the space of all linear transformations from${\displaystyle V}$ to${\displaystyle W}$, denoted${\displaystyle \mathbf{H}\mathbf{o}\mathbf{m}(V,W)}$ and calledinternal${\displaystyle \mathrm{H}\mathrm{o}\mathrm{m}}$, the structure of a super vector space. In particular,^[3]

${\displaystyle {\left(\mathbf{H}\mathbf{o}\mathbf{m}(V,W)\right)}_0=\mathrm{H}\mathrm{o}\mathrm{m}(V,W).}$

A grade-reversing transformation from${\displaystyle V}$ to${\displaystyle W}$ can be regarded as a homomorphism from${\displaystyle V}$ to the parity reversed space${\displaystyle \mathrm{\Pi }W}$, so that

${\displaystyle \mathbf{H}\mathbf{o}\mathbf{m}(V,W)=\mathrm{H}\mathrm{o}\mathrm{m}(V,W)\oplus \mathrm{H}\mathrm{o}\mathrm{m}(V,\mathrm{\Pi }W)=\mathrm{H}\mathrm{o}\mathrm{m}(V,W)\oplus \mathrm{H}\mathrm{o}\mathrm{m}(\mathrm{\Pi }V,W).}$

The usual algebraic constructions for ordinary vector spaces have their counterpart in the super vector space setting.

Thedual space${\displaystyle V^{\ast }}$ of a super vector space${\displaystyle V}$ can be regarded as a super vector space by taking the evenfunctionals to be those that vanish on${\displaystyle V_1}$ and the odd functionals to be those that vanish on${\displaystyle V_0}$.^[4] Equivalently, one can define${\displaystyle V^{\ast }}$ to be the space of linear maps from${\displaystyle V}$ to${\displaystyle {\mathbb{K}}^{1|0}}$ (the base field${\displaystyle \mathbb{K}}$ thought of as a purely even super vector space) with the gradation given in the previous section.

Direct sums of super vector spaces are constructed as in the ungraded case with the grading given by

${\displaystyle (V\oplus W{)}_0=V_0\oplus W_0,}$

${\displaystyle (V\oplus W{)}_1=V_1\oplus W_1.}$

One can also constructtensor products of super vector spaces. Here the additive structure of${\displaystyle {\mathbb{Z}}_2}$ comes into play. The underlying space is as in the ungraded case with the grading given by

${\displaystyle (V\otimes W{)}_i=\underset{j+k=i}{⨁}{V}_j\otimes W_k,}$

where the indices are in${\displaystyle {\mathbb{Z}}_2}$. Specifically, one has

${\displaystyle (V\otimes W{)}_0=(V_0\otimes W_0)\oplus (V_1\otimes W_1),}$

${\displaystyle (V\otimes W{)}_1=(V_0\otimes W_1)\oplus (V_1\otimes W_0).}$

Just as one may generalize vector spaces over a field tomodules over acommutative ring, one may generalize super vector spaces over a field tosupermodules over asupercommutative algebra (or ring).

A common construction when working with super vector spaces is to enlarge the field of scalars to a supercommutativeGrassmann algebra. Given a field${\displaystyle \mathbb{K}}$ let

${\displaystyle R=\mathbb{K}[{\theta }_1,\cdots ,{\theta }_N]}$

denote the Grassmann algebragenerated by${\displaystyle N}$ anticommuting odd elements${\displaystyle {\theta }_i}$. Any super vector${\displaystyle V}$ space over${\displaystyle \mathbb{K}}$ can be embedded in a module over${\displaystyle R}$ by considering the (graded) tensor product

${\displaystyle \mathbb{K}[{\theta }_1,\cdots ,{\theta }_N]\otimes V.}$

Thecategory of super vector spaces, denoted by${\displaystyle \mathbb{K}-\mathrm{S}\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}$, is thecategory whoseobjects are super vector spaces (over a fixed field${\displaystyle \mathbb{K}}$) and whosemorphisms areeven linear transformations (i.e. the grade preserving ones).

The categorical approach to super linear algebra is to first formulate definitions and theorems regarding ordinary (ungraded) algebraic objects in the language ofcategory theory and then transfer these directly to the category of super vector spaces. This leads to a treatment of "superobjects" such assuperalgebras,Lie superalgebras,supergroups, etc. that is completely analogous to their ungraded counterparts.

The category${\displaystyle \mathbb{K}-\mathrm{S}\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}$ is amonoidal category with the super tensor product as the monoidal product and the purely even super vector space${\displaystyle {\mathbb{K}}^{1|0}}$ as the unit object. The involutive braiding operator

${\displaystyle {\tau }_{V,W}:V\otimes W\to W\otimes V,}$

given by

${\displaystyle {\tau }_{V,W}(x\otimes y)=(-1{)}^{|x||y|}y\otimes x}$

on homogeneous elements, turns${\displaystyle \mathbb{K}-\mathrm{S}\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}$ into asymmetric monoidal category. This commutativity isomorphism encodes the "rule of signs" that is essential to super linear algebra. It effectively says that a minus sign is picked up whenever two odd elements are interchanged. One need not worry about signs in the categorical setting as long as the above operator is used wherever appropriate.

${\displaystyle \mathbb{K}-\mathrm{S}\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}$ is also aclosed monoidal category with theinternal Hom object,${\displaystyle \mathbf{H}\mathbf{o}\mathbf{m}(V,W)}$, given by the super vector space ofall linear maps from${\displaystyle V}$ to${\displaystyle W}$. The ordinary${\displaystyle \mathrm{H}\mathrm{o}\mathrm{m}}$ set${\displaystyle \mathrm{H}\mathrm{o}\mathrm{m}(V,W)}$ is the even subspace therein:

${\displaystyle \mathrm{H}\mathrm{o}\mathrm{m}(V,W)=\mathbf{H}\mathbf{o}\mathbf{m}(V,W{)}_0.}$

The fact that${\displaystyle \mathbb{K}-\mathrm{S}\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}$ is closed means that the functor${\displaystyle -\otimes V}$ isleft adjoint to the functor${\displaystyle \mathrm{H}\mathrm{o}\mathrm{m}(V,-)}$, given a natural bijection

${\displaystyle \mathrm{H}\mathrm{o}\mathrm{m}(U\otimes V,W)\cong \mathrm{H}\mathrm{o}\mathrm{m}(U,\mathbf{H}\mathbf{o}\mathbf{m}(V,W)).}$

Asuperalgebra over${\displaystyle \mathbb{K}}$ can be described as a super vector space${\displaystyle \mathcal{A}}$ with a multiplication map

${\displaystyle \mu :\mathcal{A}\otimes \mathcal{A}\to \mathcal{A},}$

that is a super vector space homomorphism. This is equivalent to demanding^[5]

${\displaystyle |ab|=|a|+|b|,a,b\in \mathcal{A}}$

Associativity and the existence of an identity can be expressed with the usual commutative diagrams, so that aunital associative superalgebra over${\displaystyle \mathbb{K}}$ is amonoid in the category${\displaystyle \mathbb{K}-\mathrm{S}\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}$.

• Deligne, P.;Morgan, J. W. (1999)."Notes on Supersymmetry (following Joseph Bernstein)".Quantum Fields and Strings: A Course for Mathematicians. Vol. 1.American Mathematical Society. pp. 41–97.ISBN 0-8218-2012-5 – viaIAS.

• Varadarajan, V. S. (2004).Supersymmetry for Mathematicians: An Introduction. Courant Lecture Notes in Mathematics. Vol. 11. American Mathematical Society.ISBN 978-0-8218-3574-6.

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