Everything about Direct sum
Algebraic structure formed from a collection of algebraic structures
In mathematics, more specifically in algebra, the direct sum of a collection of abelian groups is an abelian group constructed by combining the given groups as described below. If the input abelian groups have additional structure, then the direct sum usually maintains that structure. Wikipedia →
🧠Start here: a 6-step reading path
- Read the overview: Direct sumA 5-minute orientation
- Skim a recent paper: Sum Rate Maximization for RIS-Aided NOMA With Direct LinksInfluential work since 2022
- Pick up a classic book: Module theoryThe most-reprinted book on the subject
- Read the landmark paper: Direct-sum representations of injective modulesCited 138 times
- Go sideways: K-theoryA closely related topic
- Go sideways: Complemented subspaceA closely related topic
đź“– Definitions
Noun coproduct in some categories, like abelian groups, topological spaces or modules
Noun A linear sum in which the intersection of the summands has dimension zero. Equivalently, a linear sum of two subspaces, any vector of which can be expressed uniquely as a sum of two vectors: one vector belonging to the first subspace, and the other vector belonging to the second subspace.
Noun A block diagonal matrix interpreted as having its non-zero blocks (which are square matrices) as summands.
Noun ↑ Garrett Birkhoff with Saunders Mac Lane (©1953), A Survey Of Modern Algebra, Revised edition, U.S.A.: The Macmillan Company, published 1960 (9th printing), §X.8, page 326
🗂️ The subject at a glance
🏆 Landmark papers (most cited)
Direct-sum representations of injective modules
Vector quantizers with direct sum codebooks
Direct instantons in QCD nucleon sum rules
Simplifying additivity problems using direct sum constructions
Direct-Sum Decompositions over Local Rings
Direct sum decompositions and indecomposable TQFTs
Apolarity and direct sum decomposability of polynomials
Counterexamples to Strassen’s direct sum conjecture
🆕 Influential since 2022
Sum Rate Maximization for RIS-Aided NOMA With Direct Links
Weighted sum of harmonic signals for direct imaging in magnetic particle imaging
On the direct and inverse zero-sum problems over Câ‹ŠC2
Local Derivations and Automorphisms of Direct Sum Null-Filiform Leibniz Algebras
Direct observation of oil-surface adhesion via sum frequency generation spectroscopy
đź“š Books worth reading
Module theoryAlberto Facchini · 2012
Module theoryAlberto Facchini · 1998
Direct Sum Decompositions of Torsion-Free Finite Rank Groups (Pure and Applied Mathematics)Theodore G. Faticoni · 2007- 📕Model Theory of Modules, Algebras and CategoriesAlberto Facchini, Lorna Gregory · 2019
Ranked by number of editions on Open Library, a rough measure of how lasting a book is.
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