Direct sum

The direct sum is an operation from abstract algebra, a branch of mathematics. For example, the direct sum formula_1, where formula_2 is real coordinate space, is the Cartesian plane, formula_3. To see how the direct sum is used in abstract algebra, consider a more elementary structure in abstract algebra, the abelian group. The direct sum of two abelian groups formula_4 and formula_5 is another abelian group formula_6 consisting of the ordered pairs formula_7 where formula_8 and formula_9. (Confusingly this ordered pair is also called the cartesian product of the two groups.) To add ordered pairs, we define the sum formula_10 to be formula_11; in other words addition is defined coordinate-wise. A similar process can be used to form the direct sum of two vector spaces or two modules.
We can also form direct sums with any finite number of summands, for example formula_12, provided formula_13 and formula_14 are the same kinds of algebraic structures (e.g., all abelian groups, or all vector spaces). This relies on the fact that the direct sum is associative up to isomorphism. That is, formula_15 for any algebraic structures formula_4, formula_5, and formula_14 of the same kind. The direct sum is also commutative up to isomorphism, i.e. formula_19 for any algebraic structures formula_4 and formula_5 of the same kind.
In the case of two summands, or any finite number of summands, the direct sum is the same as the direct product. If the arithmetic operation is written as +, as it usually is in abelian groups, then we use the direct sum. If the arithmetic operation is written as × or ⋅ or using juxtaposition (as in the expression formula_22) we use direct product.
In the case where infinitely many objects are combined, most authors make a distinction between direct sum and direct product. As an example, consider the direct sum and direct product of infinitely many real lines. An element in the direct product is an infinite sequence, such as (1,2,3...) but in the direct sum, there would be 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. More generally, 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 formula_23, the direct sum formula_24 is defined to be the set of tuples formula_25 with formula_26 such that formula_27 for all but finitely many "i". The direct sum formula_24 is contained in the direct product formula_29, but is usually strictly smaller when the index set formula_30 is infinite, because direct products do not have the restriction that all but finitely many coordinates must be zero.
Examples.
The "xy"-plane, a two-dimensional vector space, can be thought of as the direct sum of two one-dimensional vector spaces, namely the "x" and "y" axes. In this direct sum, the "x" and "y" axes intersect only at the origin (the zero vector). Addition is defined coordinate-wise, that is formula_31, which is the same as vector addition.
Given two structures formula_4 and formula_5, their direct sum is written as formula_6. Given an indexed family of structures formula_35, indexed with formula_36, the direct sum may be written formula_37. Each "Ai" is called a direct summand of "A". 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 as formula_38 the phrase "direct sum" is used, while if the group operation is written formula_39 the 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.
A distinction is made between internal and external direct sums, though the two are isomorphic. If the factors are defined first, and then the direct sum is defined in terms of the factors, we have an external direct sum. For example, if we define the real numbers formula_40 and then define formula_41 the direct sum is said to be external.
If, on the other hand, we first define some algebraic structure formula_42 and then write formula_42 as a direct sum of two substructures formula_44 and formula_45, then the direct sum is said to be internal. In this case, each element of formula_42 is expressible uniquely as an algebraic combination of an element of formula_44 and an element of formula_45. For an example of an internal direct sum, consider formula_49 (the integers modulo six), whose elements are formula_50. This is expressible as an internal direct sum formula_51.
Types of direct sum.
Direct sum of abelian groups.
The direct sum of abelian groups is a prototypical example of a direct sum. Given two abelian groups formula_52 and formula_53, their direct sum formula_54 is the same as their direct product. That is, the underlying set is the Cartesian product formula_55 and the group operation formula_56 is defined component-wise:
This definition generalizes to direct sums of finitely many abelian groups.
For an infinite family of abelian groups "Ai" for "i" ∈ "I", the direct sum
is a proper subgroup of the direct product. It consists of the elements formula_59 such that "ai" is the identity element of "Ai" for all but finitely many "i".
Direct sum of modules.
The "direct sum of modules" is a construction which combines several modules into a new module.
The most familiar examples of this construction occur when considering vector spaces, which are modules over a field. The construction may also be extended to Banach spaces and Hilbert spaces.
Direct sum of group representations.
The direct sum of group representations generalizes the direct sum of the underlying modules, adding a group action to it. Specifically, given a group "G" and two representations "V" and "W" of "G" (or, more generally, two "G"-modules), the direct sum of the representations is "V" ⊕ "W" with the action of "g" ∈ "G" given component-wise, i.e.
Another equivalent way of defining the direct sum is as follows:
Given two representations formula_60 and formula_61 the vector space of the direct sum is formula_62 and the homomorphism formula_63 is given by formula_64, where formula_65 is the natural map obtained by coordinate-wise action as above.
Furthermore, if formula_66 are finite dimensional, then, given a basis of formula_66, formula_68 and formula_69 are matrix-valued. In this case, formula_63 is given as
formula_71
Moreover, if we treat formula_44 and formula_45 as modules over the group ring formula_74, where formula_75 is the field, then the direct sum of the representations formula_44 and formula_45 is equal to their direct sum as formula_74 modules.
Direct sum of rings.
Some authors will speak of the direct sum formula_79 of two rings when they mean the direct product formula_80, but this should be avoided since formula_80 does not receive natural ring homomorphisms from "R" and "S": in particular, the map formula_82 sending "r" to ("r",0) is not a ring homomorphism since it fails to send 1 to (1,1) (assuming that 0≠1 in "S"). Thus formula_80 is not a coproduct in the category of rings, and should not be written as a direct sum. (The coproduct in the category of commutative rings is the tensor product of rings. In the category of rings, the coproduct is given by a construction similar to the free product of groups.)
Use of direct sum terminology and notation is especially problematic when dealing with infinite families of rings: If formula_84 is an infinite collection of nontrivial rings, then the direct sum of the underlying additive groups can be equipped with termwise multiplication, but this produces a rng, i.e., a ring without a multiplicative identity.
Direct sum in categories.
An additive category is an abstraction of the properties of the category of modules. In such a category finite products and coproducts agree and the direct sum is either of them, cf. biproduct.
General case:
In category theory the direct sum is often, but not always, the coproduct in the category of 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.
Homomorphisms.
The direct sum formula_24 comes equipped with a "projection" homomorphism formula_86 for each "j" in "I" and a "coprojection" formula_87 for each "j" in "I". Given another algebraic structure formula_5 (with the same additional structure) and homomorphisms formula_89 for every "j" in "I", there is a unique homomorphism formula_90, called the sum of the "g""j", such that formula_91 for all "j". Thus the direct sum is the coproduct in the appropriate category.