Coalgebra

In mathematics, coalgebras or cogebras are structures that are dual (in the category theory sense of reversing arrows) to unital algebra associative algebras. The of unital associative algebras can be formulated in terms of commutative diagrams. Turning all arrows around, one obtains the axioms of coalgebras. Every coalgebra, by (vector space) Dual space, gives rise to an algebra, but not in general the other way. In finite dimensions, this duality goes in both directions (see below).

Coalgebras occur naturally in a number of contexts (for example, representation theory, universal enveloping algebras and ).

There are also , with important applications in computer science.

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Informal discussion One frequently recurring example of coalgebras occurs in representation theory, and in particular, in the representation theory of the rotation group. A primary task, of practical use in physics, is to obtain combinations of systems with different states of angular momentum and spin. For this purpose, one uses the Clebsch–Gordan coefficients. Given two systems $A,B$ with angular momenta $j\_A$ and $j\_B$, a particularly important task is to find the total angular momentum $j\_A\; +\; j\_B$ given the combined state $|A\backslash rangle\backslash otimes\; |B\backslash rangle$. This is provided by the total angular momentum operator, which extracts the needed quantity from each side of the tensor product. It can be written as an "external" tensor product

$\backslash mathbf\{J\}\; \backslash equiv\; \backslash mathbf\{j\}\; \backslash otimes\; 1\; +\; 1\; \backslash otimes\; \backslash mathbf\{j\}$

The word "external" appears here, in contrast to the "internal" tensor product of a tensor algebra. A tensor algebra comes with a tensor product (the internal one); it can also be equipped with a second tensor product, the "external" one, or the coproduct, having the form above. That they are two different products is emphasized by recalling that the internal tensor product of a vector and a scalar is just simple scalar multiplication. The external product keeps them separated. In this setting, the coproduct is the map

$\backslash Delta:\; J\backslash to\; J\backslash otimes\; J$

that takes

$\backslash Delta:\; \backslash mathbf\{j\}\; \backslash mapsto\; \backslash mathbf\{j\}\; \backslash otimes\; 1\; +\; 1\; \backslash otimes\; \backslash mathbf\{j\}$

For this example, $J$ can be taken to be one of the spin representations of the rotation group, with the fundamental representation being the common-sense choice. This coproduct can be lifted to all of the tensor algebra, by a simple lemma that applies to : the tensor algebra is a free algebra, therefore, any homomorphism defined on a subset can be extended to the entire algebra. Examining the lifting in detail, one observes that the coproduct behaves as the shuffle product, essentially because the two factors above, the left and right $\backslash mathbf\{j\}$ must be kept in sequential order during products of multiple angular momenta (rotations are not commutative).

The peculiar form of having the $\backslash mathbf\{j\}$ appear only once in the coproduct, rather than (for example) defining $\backslash mathbf\{j\}\; \backslash mapsto\; \backslash mathbf\{j\}\; \backslash otimes\; \backslash mathbf\{j\}$ is in order to maintain linearity: for this example, (and for representation theory in general), the coproduct must be linear. As a general rule, the coproduct in representation theory is reducible; the factors are given by the Littlewood–Richardson rule. (The Littlewood–Richardson rule conveys the same idea as the Clebsch–Gordan coefficients, but in a more general setting).

The formal definition of the coalgebra, below, abstracts away this particular special case, and its requisite properties, into a general setting.

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Formal definition Formally, a coalgebra over a field K is a vector space C over K together with multilinear map Δ: C → C ⊗ C and ε: C → K such that

1. $(\backslash mathrm\{id\}\_C\; \backslash otimes\; \backslash Delta)\; \backslash circ\; \backslash Delta\; =\; (\backslash Delta\; \backslash otimes\; \backslash mathrm\{id\}\_C)\; \backslash circ\; \backslash Delta$
2. $(\backslash mathrm\{id\}\_C\; \backslash otimes\; \backslash varepsilon)\; \backslash circ\; \backslash Delta\; =\; \backslash mathrm\{id\}\_C\; =\; (\backslash varepsilon\; \backslash otimes\; \backslash mathrm\{id\}\_C)\; \backslash circ\; \backslash Delta$.

(Here ⊗ refers to the tensor product over K and id is the identity function.)

Equivalently, the following two diagrams commute:

In the first diagram, C ⊗ ( C ⊗ C) is identified with ( C ⊗ C) ⊗ C; the two are naturally Isomorphism. Similarly, in the second diagram the naturally isomorphic spaces C, C ⊗ K and K ⊗ C are identified.

The first diagram is the dual of the one expressing associativity of algebra multiplication (called the coassociativity of the comultiplication); the second diagram is the dual of the one expressing the existence of a multiplicative identity element. Accordingly, the map Δ is called the comultiplication (or coproduct) of C and ε is the of C.

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Examples Take an arbitrary set S and form the K-vector space C = K^{( S)} with basis S, as follows. The elements of this vector space C are those functions from S to K that map all but finitely many elements of S to zero; identify the element s of S with the function that maps s to 1 and all other elements of S to 0. Define

Δ( s) = s ⊗ s and ε( s) = 1 for all s in S.

By linearity, both Δ and ε can then uniquely be extended to all of C. The vector space C becomes a coalgebra with comultiplication Δ and counit ε.

As a second example, consider the polynomial ring K 'X in one indeterminate X. This becomes a coalgebra (the divided power coalgebra 'See also Raianu, Serban. Coalgebras from Formulas , p. 2.) if for all n'' ≥ 0 one defines:

$\backslash Delta(X^n)\; =\; \backslash sum\_\{k=0\}^n\; \backslash dbinom\{n\}\{k\}\; X^k\backslash otimes\; X^\{n-k\},$

$\backslash varepsilon(X^n)=\backslash begin\{cases\}$

1& \mbox{if } n=0\\ 0& \mbox{if } n>0 \end{cases} Again, because of linearity, this suffices to define Δ and ε uniquely on all of K X. Now KX is both a unital associative algebra and a coalgebra, and the two structures are compatible. Objects like this are called , and in fact most of the important coalgebras considered in practice are bialgebras.

Examples of coalgebras include the tensor algebra, the exterior algebra, and . Unlike the polynomial case above, none of these are commutative. Therefore, the coproduct becomes the shuffle product, rather than the divided power structure given above. The shuffle product is appropriate, because it preserves the order of the terms appearing in the product, as is needed by non-commutative algebras.

The singular homology of a topological space forms a graded coalgebra whenever the Künneth isomorphism holds, e.g. if the coefficients are taken to be a field.

If C is the K-vector space with basis { s, c}, consider Δ: C → C ⊗ C is given by

Δ( s) = s ⊗ c + c ⊗ s

Δ( c) = c ⊗ c − s ⊗ s

and ε: C → K is given by

ε( s) = 0

ε( c) = 1

In this situation, ( C, Δ, ε) is a coalgebra known as trigonometric coalgebra.See also , and , Ex. 1.1.5.Raianu, Serban. Coalgebras from Formulas , p. 1.

For a locally finite poset P with set of intervals J, define the incidence coalgebra C with J as basis. The comultiplication and counit are defined as

$\backslash Deltax,z\; =\; \backslash sum\_\{y\; \backslash in\; x,z\}\; x,y\; \backslash otimes\; y,z\; \backslash text\{\; for\; \}\; x\; \backslash leq\; z\; \backslash \; .$

$\backslash varepsilonx,y\; =\; \backslash begin\{cases\}$

1 & \text{if } x=y, \\ 0 & \text{if } x \ne y. \end{cases}

The intervals of length zero correspond to points of P and are group-like elements.Montgomery (1993) p.61

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Finite dimensions In finite dimensions, the duality between algebras and coalgebras is closer: the dual of a finite-dimensional (unital associative) algebra is a coalgebra, while the dual of a finite-dimensional coalgebra is a (unital associative) algebra. In general, the dual of an algebra may not be a coalgebra.

The key point is that in finite dimensions, and are isomorphic.

To distinguish these: in general, algebra and coalgebra are dual notions (meaning that their axioms are dual: reverse the arrows), while for finite dimensions, they are also dual objects (meaning that a coalgebra is the dual object of an algebra and conversely).

If A is a finite- unital associative K-algebra, then its K-dual A^∗ consisting of all K-linear maps from A to K is a coalgebra. The multiplication of A can be viewed as a linear map , which when dualized yields a linear map . In the finite-dimensional case, is naturally isomorphic to , so this defines a comultiplication on A^∗. The counit of A^∗ is given by evaluating at 1.

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Sweedler notation When working with coalgebras, a certain notation for the comultiplication simplifies the formulas considerably and has become quite popular. Given an element c of the coalgebra ( C, Δ, ε), there exist elements c and c in C such that

$\backslash Delta(c)=\backslash sum\_i\; c\_\{(1)\}^\{(i)\}\backslash otimes\; c\_\{(2)\}^\{(i)\}$

Note that neither the number of terms in this sum, nor the exact values of each $c\_\{(1)\}^\{(i)\}$ or $c\_\{(2)\}^\{(i)\}$, are uniquely determined by $c$; there is only a promise that there are finitely many terms, and that the full sum of all these terms $c\_\{(1)\}^\{(i)\}\backslash otimes\; c\_\{(2)\}^\{(i)\}$ have the right value $\backslash Delta(c)$.

In Sweedler's notation,Underwood (2011) p.35 (so named after Moss Sweedler), this is abbreviated to

$\backslash Delta(c)=\backslash sum\_\{(c)\}\; c\_\{(1)\}\backslash otimes\; c\_\{(2)\}.$

The fact that ε is a counit can then be expressed with the following formula

$c=\backslash sum\_\{(c)\}\; \backslash varepsilon(c\_\{(1)\})c\_\{(2)\}\; =\; \backslash sum\_\{(c)\}\; c\_\{(1)\}\backslash varepsilon(c\_\{(2)\}).\backslash ;$

Here it is understood that the sums have the same number of terms, and the same lists of values for $c\_\{(1)\}$ and $c\_\{(2)\}$, as in the previous sum for $\backslash Delta(c)$.

The coassociativity of Δ can be expressed as

$\backslash sum\_\{(c)\}c\_\{(1)\}\backslash otimes\backslash left(\backslash sum\_\{(c\_\{(2)\})\}(c\_\{(2)\})\_\{(1)\}\backslash otimes\; (c\_\{(2)\})\_\{(2)\}\backslash right)\; =\; \backslash sum\_\{(c)\}\backslash left(\; \backslash sum\_\{(c\_\{(1)\})\}(c\_\{(1)\})\_\{(1)\}\backslash otimes\; (c\_\{(1)\})\_\{(2)\}\backslash right)\; \backslash otimes\; c\_\{(2)\}.$

In Sweedler's notation, both of these expressions are written as

$\backslash sum\_\{(c)\}\; c\_\{(1)\}\backslash otimes\; c\_\{(2)\}\backslash otimes\; c\_\{(3)\}.$

Some authors omit the summation symbols as well; in this sumless Sweedler notation, one writes

$\backslash Delta(c)=c\_\{(1)\}\backslash otimes\; c\_\{(2)\}$

and

$c=\backslash varepsilon(c\_\{(1)\})c\_\{(2)\}\; =\; c\_\{(1)\}\backslash varepsilon(c\_\{(2)\}).\backslash ;$

Whenever a variable with lowered and parenthesized index is encountered in an expression of this kind, a summation symbol for that variable is implied.

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Further concepts and facts A coalgebra is called co-commutative if $\backslash sigma\backslash circ\backslash Delta\; =\; \backslash Delta$, where is the K-linear map defined by for all c, d in C. In Sweedler's sumless notation, C is co-commutative if and only if

$c\_\{(1)\}\backslash otimes\; c\_\{(2)\}=c\_\{(2)\}\backslash otimes\; c\_\{(1)\}$

for all c in C. (It's important to understand that the implied summation is significant here: it is not required that all the summands are pairwise equal, only that the sums are equal, a much weaker requirement.)

A group-like element (or set-like element) is an element x such that and . Contrary to what this naming convention suggests the group-like elements do not always form a group and in general they only form a set. The group-like elements of a Hopf algebra do form a group. A primitive element is an element x that satisfies . The primitive elements of a Hopf algebra form a Lie algebra.

Eiichi Abe (2025). 9780521604895, Cambridge University Press. ISBN 9780521604895

If and are two coalgebras over the same field K, then a coalgebra morphism from C₁ to C₂ is a K-linear map such that $(f\backslash otimes\; f)\backslash circ\backslash Delta\_1\; =\; \backslash Delta\_2\backslash circ\; f$ and $\backslash varepsilon\_2\backslash circ\; f\; =\; \backslash varepsilon\_1$. In Sweedler's sumless notation, the first of these properties may be written as:

$f(c\_\{(1)\})\backslash otimes\; f(c\_\{(2)\})=f(c)\_\{(1)\}\backslash otimes\; f(c)\_\{(2)\}.$

The composition of two coalgebra morphisms is again a coalgebra morphism, and the coalgebras over K together with this notion of morphism form a category theory.

A linear subspace I in C is called a coideal if and . In that case, the quotient space C/ I becomes a coalgebra in a natural fashion.

A subspace D of C is called a subcoalgebra if ; in that case, D is itself a coalgebra, with the restriction of ε to D as counit.

The kernel of every coalgebra morphism is a coideal in C₁, and the image is a subcoalgebra of C₂. The common isomorphism theorems are valid for coalgebras, so for instance C₁/ker( f) is isomorphic to im( f).

If A is a finite-dimensional unital associative K-algebra, then A^∗ is a finite-dimensional coalgebra, and indeed every finite-dimensional coalgebra arises in this fashion from some finite-dimensional algebra (namely from the coalgebra's K-dual). Under this correspondence, the commutative finite-dimensional algebras correspond to the cocommutative finite-dimensional coalgebras. So in the finite-dimensional case, the theories of algebras and of coalgebras are dual; studying one is equivalent to studying the other. However, relations diverge in the infinite-dimensional case: while the K-dual of every coalgebra is an algebra, the K-dual of an infinite-dimensional algebra need not be a coalgebra.

Every coalgebra is the sum of its finite-dimensional subcoalgebras, something that is not true for algebras. Abstractly, coalgebras are generalizations, or duals, of finite-dimensional unital associative algebras.

Corresponding to the concept of representation for algebras is a corepresentation or comodule.

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See also

• Cofree coalgebra
• Measuring coalgebra
• Dialgebra

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Further reading

• .
• Chapter III, section 11 in
  Nicolas Bourbaki (1989). 9780387193731, Springer-Verlag. ISBN 9780387193731

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External links

• William Chin: A brief introduction to coalgebra representation theory

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