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Thedirect sumis anoperationbetweenstructuresinabstract algebra,a branch ofmathematics.It is defined differently, but analogously, for different kinds of structures. As an example, the direct sum of two abelian groupsandis another abelian groupconsisting of the ordered pairswhereand.To add ordered pairs, we define the sumto be;in other words addition is defined coordinate-wise. For example, the direct sum,whereisreal coordinate space,is theCartesian plane,.A similar process can be used to form the direct sum of twovector spacesor twomodules.
We can also form direct sums with any finite number of summands, for example,providedandare the same kinds of algebraic structures (e.g., all abelian groups, or all vector spaces). This relies on the fact that the direct sum isassociativeup toisomorphism.That is,for any algebraic structures,,andof the same kind. The direct sum is alsocommutativeup to isomorphism, i.e.for any algebraic structuresandof the same kind.
The direct sum of finitely many abelian groups, vector spaces, or modules iscanonically isomorphicto the correspondingdirect product.This is false, however, for some algebraic objects, like nonabelian groups.
In the case where infinitely many objects are combined, the direct sum and direct product are not isomorphic, even for abelian groups, vector spaces, or modules. As an example, consider the direct sum and direct product of (countably) infinitely many copies of the integers. An element in the direct product is an infinite sequence, such as (1,2,3,...) but in the direct sum, there is a requirement that all but finitely many coordinates be zero, so the sequence (1,2,3,...) would be an element of the direct product but not of the direct sum, while (1,2,0,0,0,...) would be an element of both. Often, if a + sign is used, all but finitely many coordinates must be zero, while if some form of multiplication is used, all but finitely many coordinates must be 1. In more technical language, if the summands are,the direct sumis defined to be the set of tupleswithsuch thatfor all but finitely manyi.The direct sumis contained in thedirect product,but is strictly smaller when theindex setis infinite, because an element of the direct product can have infinitely many nonzero coordinates.[1]
Examples
editThexy-plane, a two-dimensionalvector space,can be thought of as the direct sum of two one-dimensional vector spaces, namely thexandyaxes. In this direct sum, thexandyaxes intersect only at the origin (the zero vector). Addition is defined coordinate-wise, that is,which is the same as vector addition.
Given two structuresand,their direct sum is written as.Given anindexed familyof structures,indexed with,the direct sum may be written.EachAiis called adirect summandofA.If the index set is finite, the direct sum is the same as the direct product. In the case of groups, if the group operation is written asthe phrase "direct sum" is used, while if the group operation is writtenthe phrase "direct product" is used. When the index set is infinite, the direct sum is not the same as the direct product since the direct sum has the extra requirement that all but finitely many coordinates must be zero.
Internal and external direct sums
editA distinction is made between internal and external direct sums, though the two are isomorphic. If the summands are defined first, and then the direct sum is defined in terms of the summands, we have an external direct sum. For example, if we define the real numbersand then definethe direct sum is said to be external.
If, on the other hand, we first define some algebraic structureand then writeas a direct sum of two substructuresand,then the direct sum is said to be internal. In this case, each element ofis expressible uniquely as an algebraic combination of an element ofand an element of.For an example of an internal direct sum, consider(the integers modulo six), whose elements are.This is expressible as an internal direct sum.
Types of direct sum
editDirect sum of abelian groups
editThedirect sum ofabelian groupsis a prototypical example of a direct sum. Given two suchgroupsandtheir direct sumis the same as theirdirect product.That is, the underlying set is theCartesian productand the group operationis defined component-wise: This definition generalizes to direct sums of finitely many abelian groups.
For an arbitrary family of groupsindexed bytheirdirect sum[2] is thesubgroupof the direct product that consists of the elementsthat have finitesupport,where by definition,is said to havefinite supportifis the identity element offor all but finitely many[3] The direct sum of an infinite familyof non-trivial groups is aproper subgroupof the product group
Direct sum of modules
editThedirect sum of modulesis a construction which combines severalmodulesinto a new module.
The most familiar examples of this construction occur when consideringvector spaces,which are modules over afield.The construction may also be extended toBanach spacesandHilbert spaces.
Direct sum in categories
editAnadditive categoryis an abstraction of the properties of the category of modules.[4][5]In such a category, finite products andcoproductsagree and the direct sum is either of them, cf.biproduct.
General case:[2] Incategory theorythedirect sumis often, but not always, thecoproductin thecategoryof the mathematical objects in question. For example, in the category of abelian groups, direct sum is a coproduct. This is also true in the category of modules.
Direct sums versus coproducts in category of groups
editHowever, the direct sum(defined identically to the direct sum of abelian groups) isnota coproduct of the groupsandin thecategory of groups.So for this category, a categorical direct sum is often simply called a coproduct to avoid any possible confusion.
Direct sum of group representations
editThedirect sum of group representationsgeneralizes thedirect sumof the underlyingmodules,adding agroup actionto it. Specifically, given agroupand tworepresentationsandof(or, more generally, two-modules), the direct sum of the representations iswith the action ofgiven component-wise, that is, Another equivalent way of defining the direct sum is as follows:
Given two representationsandthe vector space of the direct sum isand the homomorphismis given bywhereis the natural map obtained by coordinate-wise action as above.
Furthermore, ifare finite dimensional, then, given a basis of,andare matrix-valued. In this case,is given as
Moreover, if we treatandas modules over thegroup ring,whereis the field, then the direct sum of the representationsandis equal to their direct sum asmodules.
Direct sum of rings
editSome authors will speak of the direct sumof two rings when they mean thedirect product,but this should be avoided[6]sincedoes not receive natural ring homomorphisms fromand:in particular, the mapsendingtois not a ring homomorphism since it fails to send 1 to(assuming thatin). Thusis not a coproduct in thecategory of rings,and should not be written as a direct sum. (The coproduct in thecategory of commutative ringsis thetensor product of rings.[7]In the category of rings, the coproduct is given by a construction similar to thefree productof groups.)
Use of direct sum terminology and notation is especially problematic when dealing with infinite families of rings: Ifis an infinite collection of nontrivial rings, then the direct sum of the underlying additive groups can be equipped with termwise multiplication, but this produces arng,that is, a ring without a multiplicative identity.
Direct sum of matrices
editFor any arbitrary matricesand,the direct sumis defined as theblock diagonal matrixofandif both are square matrices (and to an analogousblock matrix,if not).
Direct sum of topological vector spaces
editAtopological vector space(TVS)such as aBanach space,is said to be atopological direct sumof two vector subspacesandif the addition map is anisomorphism of topological vector spaces(meaning that thislinear mapis abijectivehomeomorphism), in which caseandare said to betopological complementsin This is true if and only if when considered asadditivetopological groups(so scalar multiplication is ignored),is thetopological direct sum of the topological subgroupsand If this is the case and ifisHausdorffthenandare necessarilyclosedsubspaces of
Ifis a vector subspace of a real or complex vector spacethen there always exists another vector subspaceofcalled analgebraic complement ofinsuch thatis thealgebraic direct sumofand(which happens if and only if the addition mapis avector space isomorphism). In contrast to algebraic direct sums, the existence of such a complement is no longer guaranteed for topological direct sums.
A vector subspaceofis said to be a (topologically)complemented subspaceofif there exists some vector subspaceofsuch thatis the topological direct sum ofand A vector subspace is calleduncomplementedif it is not a complemented subspace. For example, every vector subspace of a Hausdorff TVS that is not a closed subset is necessarily uncomplemented. Every closed vector subspace of aHilbert spaceis complemented. But everyBanach spacethat is not a Hilbert space necessarily possess some uncomplemented closed vector subspace.
Homomorphisms
editThe direct sumcomes equipped with aprojectionhomomorphismfor eachjinIand acoprojectionfor eachjinI.[8]Given another algebraic structure(with the same additional structure) and homomorphismsfor everyjinI,there is a unique homomorphism,called the sum of thegj,such thatfor allj.Thus the direct sum is thecoproductin the appropriatecategory.
See also
editNotes
edit- ^Thomas W. Hungerford,Algebra,p.60, Springer, 1974,ISBN0387905189
- ^abDirect Sumat thenLab
- ^Joseph J. Rotman,The Theory of Groups: an Introduction,p. 177, Allyn and Bacon, 1965
- ^""p.45""(PDF).Archived fromthe original(PDF)on 2013-05-22.Retrieved2014-01-14.
- ^"Appendix"(PDF).Archived fromthe original(PDF)on 2006-09-17.Retrieved2014-01-14.
- ^Math StackExchangeon direct sum of rings vs. direct product of rings.
- ^Lang 2002,section I.11
- ^Heunen, Chris (2009).Categorical Quantum Models and Logics.Pallas Proefschriften. Amsterdam University Press. p. 26.ISBN978-9085550242.
References
edit- Lang, Serge(2002),Algebra,Graduate Texts in Mathematics,vol. 211 (Revised third ed.), New York: Springer-Verlag,ISBN978-0-387-95385-4,MR1878556,Zbl0984.00001