nLab graded vector space

Definition

Given a set $G$, an $G$-graded vector space is a map $V$ assigning to each element $g \in G$ a vector space $V_g$. Given $G$-graded vector spaces $V$ and $W$, a morphism $f\colon V \to W$ assigns to each element $g \in G$ a linear operator $f_g\colon V_g \to W_g$. That is, the category of $G$-graded vector spaces is the functor category $Vect^G$.

We can just as easily talk about a $G$-graded module or a $G$-graded object in any category. However, a graded algebra has additional requirements (using a monoid structure on $G$, as below).

In other words, a $G$-graded vector space is a functor $V\colon G \to Vect$, where the set $G$ is treated as a discrete category, and Vect is the category of vector spaces. Similarly, a morphism of $G$-graded vector spaces is a natural transformation between such functors. In short, the category of $G$-graded vector spaces is the functor category $Vect^G$.

People are usually interested in $G$-graded vector spaces when the set $G$ is equipped with extra structure. If the set $G$ is a monoid, $Vect^G$ is a monoidal category, with the tensor product given by

$$(V \otimes W)_g = \bigoplus_{h \cdot k = g} V_h \otimes W_k.$$

The unit is given by having the underlying field $k$ as $V_e$ and the trivial vector space for other gradings. If $G$ is a commutative monoid, then $Vect^G$ is a symmetric monoidal category.

If $G$ is a group, every finite-dimensional $G$-graded vector space has a left dual and a right dual. And if $G$ is an abelian group, these duals coincide.

By far the most widely-used examples are $G = \mathbb{Z}$ and $G = \mathbb{N}$. Indeed, the term graded vector space is often used to mean a $G$-graded vector space with one of these choices of $G$. The case $G = \mathbb{Z}/2$ is also important: a $\mathbb{Z}/2$-graded vector space is also called a supervector space. However, in this case one often uses a different braiding on $Vect^G$, one which uses the ring structure of $\mathbb{N}$; see Wikipedia.

Explicit definition

Let $G$ be a set with decidable equality. Then, given a commutative ring $R$ (usually a field $F$), a $G$-graded $R$-module is an $R$-module $V$ (usually a vector space $V$) with a binary function $\langle - \rangle_{(-)}: V \times G \to V$ called the grade projection operator such that

• for all $v:V$, $v = \sum_{g:G} \langle v \rangle_g$

• for all $a:R$, $b:R$, $v:V$, $w:V$, and $g:G$, $\langle a v + b w \rangle_g = a \langle v \rangle_g + b \langle w \rangle_g$

• for all $v:V$ and $g:G$, $\langle \langle v \rangle_g \rangle_g = \langle v \rangle_g$

• for all $v:V$, $g:G$, and $h:G$, $(g \neq h) \Rightarrow (\langle \langle v \rangle_g \rangle_h = 0)$

For an element $g:G$, the image of $\langle - \rangle_g$ under $V$ is denoted as $\langle V \rangle_g$.

$$\langle V \rangle_g \coloneqq \mathrm{im}(\langle - \rangle_g)$$

Remarks

For the case that $G$ is a group, this means that the category of $G$-graded vector spaces is a categorification of the group algebra of $G$, where numbers are replaced by vector spaces. Recalling from the remark in category algebra that the group algebra of a group can be identified with the monoid of spans of the form

$$\array{ && G \\ & \swarrow && \searrow \\ pt &&\Rightarrow&& pt \\ & \searrow && \swarrow \\ && Vect } \,,$$

where $pt \to Vect$ goes to the ground field $k$, the monoidal category of $G$-graded vector spaces can be identified with the monoid of spans of the form

$$\array{ && G \\ & \swarrow && \searrow \\ pt &&\Rightarrow&& pt \\ & \searrow && \swarrow \\ && 2Vect } \,.$$

Here $2Vect$ denotes some version of the category of 2-vector spaces with the property that the category $Vect$ is one of its objects and such that $End_{2Vect}(Vect) \simeq Vect$ (in analogy to how $End_{Vect}(k) \simeq k$) and $pt \to 2Vect$ maps to $Vect$. Possible choices for $2Vect$ is the 2-category of Kapranov-Voevodsky 2-vector spaces or the bigger bicategory Bimod of algebras and bimodules.

More details on this perspective on graded vector spaces are in Nonabelian cocycles and their quantum symmetries.

In general, adding algebraic structure onto $G$ will add categorical structure onto $\text{Vect}^G$. A table keeping track of these structures is below:

Structure on $G$	Structure on $\text{Vect}^G$
Set	Category
Monoid	Monoidal category
Finite group	Fusion category
Finite abelian group	Braided fusion category

Special case of $\mathbb{Z}$-graded vector spaces

The case $G = \mathbb{Z}$ serves as a base for many other applications of the same basic idea. It has some of its own ‘traditional’ terminology and structure that links it to differential objects, so that a ‘differential graded vector space’ is a chain complex of vector spaces. We will use ‘gvs’ as an abbreviation for this sort of graded vector space and ‘dgvs’ for the differential form. (Of course, the theory easily adapts to handle graded modules over a ring, and with some restriction, to graded groups.) Basing algebras on dgvs gives differential graded algebras (dg-algebra) and so on.

Concepts

The entry here will be a sort of lexicon of some terms which are taken from a source on rational homotopy theory. This will be more or less ‘as-is’ from the source (except translating it from the original French that is!), i.e. without too much editing. This means that there may be conflicts with other entries, which will need resolving later. Some links to other entries have been given but more could be made. There WILL initially be some duplication but that will be eliminated later on.

The lexicon will be spread over a number of entries with links given in the table of contents ion the right hand side at the top.

Note With $\mathbb{Z}$-graded vector spaces (and sometimes with other examples as well), some authors work with a direct sum of the various vector spaces instead of using an indexed family.

(Pre-)graded vector spaces

A pre-$\mathbb{Z}$-graded vector space (pre-gvs) is a direct sum $V = \bigoplus_{p\in \mathbb{Z}} V_p$. The elements of $V_p$ are said to be homogeneous of degree $p$. If $x \in V_p$, write $|x| = p$.

Sometimes it may be convenient to write $\bar{x} = (-1)^{|x|}x$ and $V_+ = \bigoplus_{p\gt 0}V_p$. Another very useful piece of notation is $V^p = V_{-p}$. In this case we will refer to an ‘upper grading’ with, in contrast, the other notation being a ‘lower grading’. These are merely for convenience and have little or no mathematical significance.

For the purposes of this lexicon:

A graded vector space (gvs) is a positively or negatively graded pre-gvs that is either $V = \sum_{p\geq 0}V_p$ or $V = \sum_{p\leq 0}V^p$. (This effectively restricts from a $\mathbb{Z}$-grading to one over $\mathbb{N}$

We consider the field $k$ to be a pre-gvs with $(k)_0 = k$, and $(k)_p =0$ if $p\neq 0$. We say $V$ is of finite type if $dim(V_p) \lt \infty$ for all $p$.

A linear map $f :V\to W$ between pre-gvs is of degree $p$ if $f(V_q) \subseteq W_{p+q}$ for all $q$. (Note this may also occur as $f(V^q) \subseteq W^{q-p}$.)

A morphism $f : V\to W$ is a linear map of degree zero. Pregraded vector spaces and the morphisms between them define the category ${pre GVS}$.

The set of all linear maps of degree $p$ from $V$ to $W$ will be denoted $Hom_p(V,W)$ and we set

$$Hom(V,W) = \bigoplus_p Hom_p(V,W).$$

Of course, we now have two notations for the same object, ${pre GVS}(V,W) = Hom_0(V,W)$.

Suspension

If $r\in \mathbb{Z}$, the $r$-suspension of $V$ is given by $(s^r V)_n = V_{n-r}.$

We will need $s$, the 1-suspension, and $s^{-1}$ in particular. Of course, $(s^{-1}V)_n = V_{n+1}$. It is also useful to note $(s^r V)^p = V^{p+r}$. Again, of course, $s^r: V\to s^r V$ is an isomorphism of degree $-r$ having $s^{-r}$ as its inverse.

(This is the basic example of the suspension functor discussed in triangulated category.)

Duals

The dual of a (pre-)gvs $V$ is $\#V$ defined by

$$\begin{aligned} (\#V)_p &: = Hom_p(V,k)\\ &\cong{Vect}(V_{-p},k)\\ &\cong\#(V_{-p})\\ & =\#(V^p). \end{aligned}$$

If $f : V\to W$ is of degree $|f|$, then its transpose

$$^t f : \#W \to \#V$$

is given by

$$(^t f)(\psi)(x) = (-1)^{|f||\psi|}\psi f(x),$$

for $\psi \in \#W$ and $x\in V$. Thus if $V\stackrel{f}{\to}W\stackrel{g}{\to}X$, then

$^t (g\circ f) = (-1)^{|f||g|}(^t f\circ {}^t g).$} In particular, for $f$ an isomorphism

$$( ^t f)^{-1} = (-1)^{|f|} ^t (f^{-1}).$$

Duality

Let $V$ be a gvs, by convention in the duality

$$\langle \quad ; \quad \rangle: V \leftrightarrow \#V,$$

we will usually assume $V$ is non-negatively graded (so $V = \bigoplus_{p\geq 0}V_p$), whilst the right hand side is non-positively graded.

If $V$ is of finite type then $\#\#V \cong V$, of course. The suspension of the dual $s(\#V)$ can be identified with $\#(s^{-1}V)$ and similarly $s^{-1}(\#V) = \#s(V)$. These identifications are via the rules:

$$\begin{aligned} \langle s^{-1}z;su\rangle &= (-1)^{|z|} \langle z;u \rangle,\\ \langle s z;s^{-1}u \rangle &= (-1)^{|z|+1}\langle z;u \rangle.\end{aligned}$$

This sign convention is needed to ensure that $s s^{-1} = id$.

Tensor products

The tensor product of two pre-gvs, $V$ and $W$, is $V\otimes W$, where

$$(V\otimes W)_n = \bigoplus_{p + q = n}V_p\otimes W_q.$$

On morphisms

$$(f\otimes g) (v\otimes w) = (-1)^{|g||f|}(f(v) \otimes g(w))$$

and is of degree $|f| + |g|$.

In particular there is a natural injection $(\#V)\otimes (\#W) \to \#(V\otimes W)$, and this is an isomorphism if either $V$ or $W$ is of finite type.

This tensor product makes the category of graded vector spaces into a monoidal category. This becomes a symmetric monoidal category with braiding the evident braiding inherited from the underlying vector spaces.

Notice that given suitable cocycles on $G$, then $G$-graded vector spaces may be equiped with a non-trivial braiding. For instance on the monoidal category of $\mathbb{Z}/2$-graded vector spaces there is one non-trivial symmetric braiding, up to equivalence. Imposing this yields super vector spaces.

Related concepts

• graded set, graded object

• super vector space

• chain complex

• differential graded algebra

References

• G. Aragón, J.L. Aragón, M.A. Rodríguez (1997), Clifford Algebras and Geometric Algebra, Advances in Applied Clifford Algebras Vol. 7 No. 2, pg 91–102, doi:10.1007/BF03041220, S2CID:120860757