Graded Vector Space - Linear Maps

Linear Maps

For general index sets I, a linear map between two I-graded vector spaces f:V→W is called a graded linear map if it preserves the grading of homogeneous elements:

for all i in I.

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

for all j in I,

where "+" denotes the monoid operation. If moreover I satisfies the cancellation property so that it can be embedded into a commutative group A which 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). In particular for i in I a linear map will be homogeneous of degree −i if

for all j in I, while

if j−i 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.