Graded vector space

Lie Algebras

Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006

[FLM88, Sect. 1.10] Let V, W be A-graded vector spaces with graded dimensions. Then V ⊕ W is a graded vector space with graded dimension and gr dim(V ⊕ W) = gr dim(V)+gr dim(W). If W is a subspace of V and Wα ⊆ Vα for all α ∈ A, then the quotient space V/W has graded dimension and gr dim(V/W) = gr dim(V) − gr dim(W). If {(α, β) ∈ A × A|Vα, Wβ ≠ {0}, α + β = γ} is finite for all γ ∈ A, then V ⊗ W is a graded vector space with graded dimension, where (V⊗W)γ=∑α+β=γVα⊗Wβ (where we set tαtβ = tα+β).

Collectively coincidence-type results and applications

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Journal Information

Published in Applicable Analysis, 2023

Donal O'Regan

Now we describe the maps considered in this paper. Let H be the ech homology functor with compact carriers and coefficients in the field of rational numbers K from the category of Hausdorff topological spaces and continuous maps to the category of graded vector spaces and linear maps of degree zero. Thus (here X is a Hausdorff topological space) is a graded vector space, being the q-dimensional ech homology group with compact carriers of X. For a continuous map , is the induced linear map where . A space X is acyclic if X is nonempty, for every , and .

Graded vector space

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Lie Algebras

Collectively coincidence-type results and applications