Graded vector space

Let \mathbb{N} be the set of non-negative integers. An \mathbb{N}-graded vector space, often called simply a graded vector space without the prefix \mathbb{N}, is a vector space together with a decomposition into a direct sum of the form : V = \bigoplus_{n \in \mathbb{N}} V_n where each V_n is a vector space. For a given n the elements of V_n are then called homogeneous elements of degree n. Graded vector spaces are common. For example the set of all polynomials in one or several variables forms a graded vector space, where the homogeneous elements of degree n are exactly the linear combinations of monomials of degree n. ==General gradation==

The subspaces of a graded vector space need not be indexed by the set of natural numbers, and may be indexed by the elements of any set I. An I-graded vector space V is a vector space together with a decomposition into a direct sum of subspaces indexed by elements i of the set I: : V = \bigoplus_{i \in I} V_i. Therefore, an \mathbb{N}-graded vector space, as defined above, is just an I-graded vector space where the set I is \mathbb{N} (the set of natural numbers). The case where I is the ring \mathbb{Z}/2\mathbb{Z} (the elements 0 and 1) is particularly important in physics. A (\mathbb{Z}/2\mathbb{Z})-graded vector space is also known as a supervector space. ==Homomorphisms==

For general index sets I, a linear map between two I-graded vector spaces is called a graded linear map if it preserves the grading of homogeneous elements. A graded linear map is also called a homomorphism (or morphism) of graded vector spaces, or homogeneous linear map: :f(V_i)\subseteq W_i for all i in I. For a fixed field and a fixed index set, the graded vector spaces form a category whose morphisms are the graded linear maps. When I is a commutative monoid (such as the natural numbers), then one may more generally define linear maps that are homogeneous of any degree i in I by the property :f(V_j)\subseteq W_{i+j} for all j in I, where "+" denotes the monoid operation. If moreover I satisfies the cancellation property so that it can be embedded into an abelian group A that it generates (for instance the integers if I is the natural numbers), then one may also define linear maps that are homogeneous of degree i in A by the same property (but now "+" denotes the group operation in A). Specifically, for i in I a linear map will be homogeneous of degree −i if :f(V_{i+j})\subseteq W_j for all j in I, while :f(V_j)=0\, if is not in I. Just as the set of linear maps from a vector space to itself forms an associative algebra (the algebra of endomorphisms of the vector space), the sets of homogeneous linear maps from a space to itself – either restricting degrees to I or allowing any degrees in the group A – form associative graded algebras over those index sets. ==Operations on graded vector spaces==

Some operations on vector spaces can be defined for graded vector spaces as well. Given two I-graded vector spaces V and W, their direct sum has underlying vector space V ⊕ W with gradation :(V ⊕ W)i = Vi ⊕ Wi . If I is a semigroup, then the tensor product of two I-graded vector spaces V and W is another I-graded vector space, V \otimes W, with gradation : (V \otimes W)_i = \bigoplus_{\left\{\left(j,k\right) \,:\; j+k=i\right\}} V_j \otimes W_k. ==Hilbert–Poincaré series==

Given a \N-graded vector space that is finite-dimensional for every n\in \N, its Hilbert–Poincaré series is the formal power series :\sum_{n\in\N}\dim_K(V_n)\, t^n. From the formulas above, the Hilbert–Poincaré series of a direct sum and of a tensor product of graded vector spaces (finite dimensional in each degree) are respectively the sum and the product of the corresponding Hilbert–Poincaré series. ==See also==