Coalgebras and bialgebras

Let R be any ring. By a coalgebra over R (or R-coalgebra) we mean a module $\modmap C \from R \to R$ together with module morphisms

$$\map \delta \from C \to {C \xR C} \qquad\hbox{and}\qquad \map \varepsilon \from C \to R$$

such that

$$\begin{eqnarray*} &&\spreaddiagramcolumns {1pc}\hbox{\diagram C \rto^-{\delta\;... ...<-.5ex>_-{1_{\!X}^{} \xo \varepsilon} & C\rlap{~.} \enddiagram } \end{eqnarray*}$$

We call $\delta$ the comultiplication and $\varepsilon$ the counit. This structure provides a module with ``formal diagonals''. There is a uniquely determined

$$\map \delta \from C \to {C \xR C \xR \cdots\, \xR C}$$

where for each $c \in C$ we have $\delta (c) = \sum\limits_i c_{1i}^{}\xO\cdots\xO c_{ni}^{}$. The notation

$$\delta(c) \;=\; \sum_{(c)} c_{(1)}^{}\xO\,\cdots\,\xO c_{(n)}^{}$$

is sometimes used even though the representation of $\delta(c)$ in the tensor product is not uniquely determined--we act as though a choice of this representation has been made for each $c \in C$. Given a multilinear function $\map f \from C \times \cdots \times C \to A$ we also write

$$f\big(\delta(c)\big) \;\;=\;\; \sum_{(c)}\,f(c_{(1)}^{}\,,\,\ldots\,,\,c_{(n)}^{})\rlap{~~.}$$

In terms of this notation the axioms can be rewritten as

$$\displaylines { \sum_{(c)} \delta (c_{(1)}^{}) \xO c_{(2)}^... ...(c)} c_{(1)}^{}\xO\varepsilon(c_{(2)}^{}) \;. \qquad\qquad\cr}$$

Suppose C and D are coalgebras. A coalgebra morphism $\map f \from C \to D$ is a module morphism such that

$$\spreaddiagramrows{.5pc}\spreaddiagramcolumns{1pc}\diagram ... ...rrto^-{f\;} & & D \dlto^-{\;\varepsilon} \\ & R & \enddiagram$$

that is,

$$\begin{eqnarray*} \sum_{(c)} f(c_{(1)}^{}) \xO f(c_{(2)}^{}) & = & \sum_{(f(c... ...uad \\ \varepsilon(f(c))\qquad & = & \qquad\varepsilon(c) \;. \end{eqnarray*}$$

We write $\Cog_R(C\,,D\,)$ for the set of coalgebra morphisms from C to D.

Suppose R is commutative. A coalgebra C over R is cocommutative when

$$\diagram & C \dlto_-{\delta} \drto^-{\delta} & \\ C \xR C \rrto^-{\sigma\;} & & C \xR C \;. \enddiagram$$

Return now to a general ring R. Suppose that A is an R-algebra and C is an R-coalgebra. Then $\Hom_R(C\,,A\,)$ becomes an R-algebra under the following convolution structure.

$$\spreaddiagramcolumns{2pc}\diagram \Hom_R(C\,,A\,)\xR\Hom_R(C... ...rc\varepsilon}\\ \Hom_R(C\,,A\,) & \Hom_R(C\,,A\,) \enddiagram$$

In terms of elements, for left R-module morphisms $\map f\,,\,g \from C \to A$ their convolution product is given by

$$f*g\;=\;\mu\circ(f\xO g)\circ\delta \qquad\hbox{and}\qquad 1\;=\; \eta\circ\varepsilon \;.$$

Using the notation for comultiplication this becomes the formula

$$(f * g) (c) \;\;=\;\; \sum_{(c)} f(c_{(1)}^{})\,g(c_{(2)}^{}) \;.$$

In particular, with A = R each R-coalgebra C gives rise to a convolution R-algebra structure on the dual $C^* = \Hom_R(C\,,R\,)$. However, we prefer to regard C^* as an R-algebra via the multiplication $\map\mu\from C^*\xR C^* \to {C^*}$ defined by

$$\diagram C^* \xR C^* \xR C \rrto^-{\mu\xo1}\dto_-{1\xo1\xo\de... ...\drto_-{1\xo e\xo 1} && R \\ & C^* \xR C \urto_e & \enddiagram$$

(This works even for non-commutative R.)

Example 8.1  

Suppose $\cal X$ is any category which admits finite products. Suppose $\map F \from \scrX \to {\Mod_R^R}$ is a functor into the category of modules from R to R. Suppose there are natural module morphisms

$$\edef\eldots{,\,\ldots\,,\,} \map \phisub{X_1^{}\eldots X_n^{... ...X_n^{})\; \to {\;FX_1^{}\xR FX_2^{}\xR\,\cdots\,\xR FX_n^{}}$$

compatible with the canonical associativity isomorphisms for product and tensor product. Then for each object X of $\scrX$ we obtain a coalgebra FX, with comultiplication and counit

$$\spreaddiagramrows{-1.5pc}\spreaddiagramcolumns{1pc}\diagram ... ...varepsilon\;} & F1 \rto^-{\phisub0\;} & R\;\;.\quad \enddiagram$$

If furthermore R is commutative and F is compatible with the twists, then this coalgebra is cocommutative.

Subexamples (a), (b) suggest two definitions that we can make for any coalgebra C.

• Say that $c \in C$ is set-like when $\delta (c) = c \xO c$ and $\varepsilon (c) = 1$. (In the case of $\,C = \Free_R(X)$ the set-like elements are precisely the elements of X.) Write $\scrD(C)$ for the set of set-like elements of C.

• Say that $c \in C$ is primitive when
  
  $$\delta (c) = c \xO 1 + 1 \xO c \;\;.$$
  
  
  (In the case of $C = \Univ(L)$ each element of L is primitive.)
  Write $\scrP(C)$ for the submodule of primitive elements of C.

Proposition 8.2 (with R a field.)    

The set-like elements of any coalgebra C form a linearly independent subset $\scrD(C)$.

We shall come back to set-like and primitive elements in the context of bialgebras.

Example 8.3  

(with R commutative.) 

Let $C = \Free_R \N$ be the free R-module on the countable set $\N$. Define

$$\delta (n) \;\;=\;\; \sum_{p + q = n} p \xO q \qquad\hbox{\e... ... {1 & {\em for\/} $n = 0$\ \cr 0 & {\em for\/} $n > 0 \;$.\cr}$$

This defines a cocommutative coalgebra structure on C.

Take an R-algebra A and look at an example of convolution with this coalgebra C. The convolution structure transports across the R-module isomorphism

$$\qquad \Hom_R(C\,,A\,)\;\;\isom\;\;A^{\N} \qquad\rlap{\rm (\,sequences in $A$\,)}$$

to give the multiplication

$$a\,b \;\;=\;\; \big(\sum_{p + q = n} a_p^{}\,b_q^{}\big)$$

for sequences $a = (a_n^{}) = (a_0^{}\,,a_1^{}\,,\ldots \, )$ and $b = (b_n^{}) = (b_0^{}\,,b_1^{}\,,\ldots \, )$ in A. The unit sequence is $(1\,,0\,,0\,,0\,,\ldots\,)$. A precise definition of indeterminate can be taken to mean the sequence

$$x \;\;=\;\; (0\,,1\,,0\,,0\,,0\,,0\,, \ldots \,) \;\in\; A^{\N}$$

in A. Each $u \in A$ is identified with $u\,1 = (u\,,0\,,0\,,0\,,\ldots \,) \in A^{\N}$. Then each $a \in A^{\N}$ can be written as a formal (no convergence requirements!) power series

$$a \;\;=\;\; \sum^{\infty}_{n = 0} a_n^{}\,x^n \;.$$

Write A[![x]!] for $A^{\N}$ with this algebra structure. It is the R- algebra of formal power series in A. If A is commutative so is A[![x]!]. In particular when A = R we obtain the commutative R-algebra C^* = R[![x]!].

Example 8.4  

Let $\nn = \{ 1,\,2, \,\ldots\,, n \}$ and put $C = \Free_R(\nn \times \nn)$. Then C becomes an R-coalgebra on defining

$$\delta(i\,,j) \;=\; \sum^{n}_{k=1}(i\,,k)\xO(k\,,j) \qquad\h... ...s{ 1 & {\em for\/} $i = j$\ \cr 0 & {\em otherwise\,.} \cr}$$

Given any R-algebra A, the convolution structure simply transports across the R-module isomorphism

$$\Hom_R(C\,,A\,) \;\;\isom\;\; A^{\nn \times \nn}_{} \qquad \rlap{\em ( $n \times n$\ matrices in $A$\,)}$$

to give the usual matrix multiplication

$$(a_{ij}^{})\,(b_{ij}^{})\;\;=\;\;\big(\sum^n_{k=1} a_{ik}^{}\,b_{kj}^{}\big) \; .$$

In this way we obtain the R-algebra $\Mat\!(n\,,A\,)\isom \End_R(A^n_{})$ of n x n matrices with entries in A.

Suppose that R is a commutative ring. An R- bialgebra is an R-module B together with algebra and coalgebra structures

$$\begin{eqnarray*} \hbox{\map \mu \from B \xR B \to B} &\hbox{and}& \hbox{\map \e... ... {B \xR B}} &\hbox{and}& \hbox{\map \varepsilon \from B \to R } \end{eqnarray*}$$

satisfying the conditions

$$\begin{eqnarray*} &{\spreaddiagramrows{1pc}\spreaddiagramcolumns{1.5pc}\diagram ... ... \\ R \urto^-{\eta} \rrto^-{1_{\!R}^{}\;} & & R \enddiagram } & \end{eqnarray*}$$

Notice the complete duality between $(\mu\,,\eta\,)$ and $(\delta\,,\varepsilon\,)$. When expressed in terms of elements the duality is not so apparent:

$$\displaylines {\delta (x\,,y) \;=\; \sum_{(x)}\sum_{(y)}\, x... ...;=\; 1\xO1 \quad\hbox{and}\quad \varepsilon(1)\;=\; 1 \;. \cr}$$

For R commutative, the tensor product $A \xR A'$ of R-algebras A and A' becomes an R-algebra via the multiplication

$$\spreaddiagramcolumns{1.5pc}\diagram (A \xR A') \xR (A \xR A'... ... A) \xR (A' \xR A') \rto^-{\mu\xo\mu\; } & A \xR A' \enddiagram$$

and unit

$$\spreaddiagramcolumns{2pc}\diagram R \;\isom\; R \xR R \rto^-{\eta\xo\eta\;}& A \xR A'\rlap{~~.} \enddiagram$$

Also the tensor product $C \xR C'$, of R-coalgebras C and C', becomes an R-coalgebra via the comultiplication

$$\spreaddiagramcolumns{1.5pc}\diagram C \xR C' \rto^-{\delta ... ...;}_-{1 \xo \sigma \xo 1} & (C \xR C')\xR(C \xR C') \enddiagram$$

and counit

$$\spreaddiagramcolumns{1.5pc}\diagram C \xR C' \rto^-{\vareps... ...xo \varepsilon \;} & R \xR R \;\isom\; R\rlap{~~.} \enddiagram$$

With this, we can make the observation:

Proposition 8.5  

Suppose $(\mu\,,\eta\,)$ and $(\delta\,,\varepsilon\,)$ are respectively, algebra and coalgebra structures on the R-module B. Then the following conditions are equivalent:

(i)

B is a bialgebra;

(ii)

$\relax \map \mu \from B \xR B \to B$ and $\relax \map \eta \from R \to B$ are coalgebra morphisms;

(iii)

$\relax \map \delta \from B \to {B \xR B}$ and $\relax \map \varepsilon \from B \to R$ are algebra morphisms.

For bialgebras B and B' a bialgebra morphism $\relax \map f \from B \to {B'}$ is a function which is both an algebra and coalgebra morphism. Write $\BIG_R(B\,,B'\,)$ for the set of such functions f.

Before giving examples of bialgebras we prove some extra results on the set-like and primitive elements for the bialgebra case.

Proposition 8.6  

If B is a bialgebra then the set-like elements are closed under multiplication: so $\scrD(B)$ becomes a monoid.

Proposition 8.7  

If B is a bialgebra then the set of primitive elements is closed under commutator, so $\scrP(B)$ becomes a Lie algebra. Also $\varepsilon(x) = 0$ for all $x \in \scrP(B)$.

Example 8.8  

Return to the situation of coalgebras in the first example. There are two conditions on the functor $\map F \from \scrX \to {\Mod_R}$ which gives rise to bialgebras FX.

(a) When the morphisms $\phi$ are all invertible.
Then F takes each monoid G in $\cal X$ to a bialgebra FG. The multiplication and unit for G give an algebra structure

$$\spreaddiagramcolumns{.5pc}\diagram FG \, \xR FG \;\isom\; F(... ...quad \diagram R \;\isom\; F1 \rto^-{F \eta \;} & FG \enddiagram$$

on FG. These are coalgebra morphisms since all arrows in $\scrX$ ``commute with diagonals''. By the 2nd proposition, each FG becomes a bialgebra. This is the situation for the functor $\map \Free_R \from \Set \to {\Mod_R}$, so for each monoid G the monoid algebra R(G) is a cocommutative bialgebra. Notice here that $G\isom D\big(R(G)\big)$ as monoids (see the 3rd proposition).

(b) When F lifts to $\map F \from \scrX \to {\Alg_R}$.

In this case each FX is clearly a bialgebra since the comultiplication and counit are algebra morphisms (2nd proposition). For the functor $\map\Univ\from \Lie_R \to {\Alg_R}$ this is indeed the situation. Thus we have that each universal enveloping algebra $\Univ(L)$ is a cocommutative bialgebra.

Example 8.9  

Return to the 2nd example of a coalgebra. This time, to use the symbol $\N\,$ to denote our countable set would be confusing. Instead we denote it by $E = \{ e_0^{}, e_1^{}, e_2^{}, e_3^{}, \ldots \,\}$. Then the coalgebra structure on $\Free_R(E)$ is

$$\delta (e_n^{})\;=\;\sum_{p+q=n}\,e_p^{}\xO e_q^{} \qquad\h... ...ses{0 & {\em for } $n > 0\;$,\cr 1 & {\em for } $n=0 \;$. \cr}$$

We now make $\Free_R(E)$ into an algebra via

$$e_p^{}\,e_q^{} \;=\; {(p+q)! \over p!\,q!}\; e_{p+q}^{} \qquad\hbox{\em with}\qquad e_0^{} = 1 \;.$$

(The binomial coefficient is an integer and so ``lives" in any ring R.) Then $\Free_R(E)$ is a bialgebra. If R is a field of characteristic 0 (i.e. $1+ \cdots +1 \neq 0$ in R for any non-zero number of terms) put x = e₁ so that one sees that $e_n = {1 \over n!}\,x^n$. Hence, as an algebra, $\Free_R(E)$ is isomorphic to the polynomial algebra R[x] in one variable. For general R we can think of $\Free_R(E)$ as the algebra of Hurwitz polynomials in one indeterminate:

$$\sum^k_{n=0} {a_n\, x^n \over n!} \qquad\hbox{\em with each }\quad a_n \in R \;.$$

Example 8.10  

Return to the 3rd example and form the symmetric algebra $\Symm(C)$ of the coalgebra $C = \Free_R(\nn \times \nn)$. Since using $\nn\times\nn$ can be confusing we replace it by any set $X = \{ x_{ij}^{}\;\vert\; i,j \in \nn \}$ of cardinality n². Then we identify $S(\Free_R(X))$ with the polynomial R- algebra R[(x_ij)] in n² commuting indeterminates x_ij for $i\,,j \in \nn$. In the 1st example we saw that this becomes a bialgebra by virtue of the fact that it is the universal enveloping algebra of a commutative Lie algebra $\Free_R (X)$, but this is not the structure of interest here. The coalgebra C induces the bialgebra structure

$$\delta(x_{ij}^{})\;=\;\sum_k x_{ik}^{} \xO x_{kj}^{} \qquad... ...s{1 & {\em for } $i=j$\ \cr 0 & {\em for } $i \neq j $\ \cr}$$

which we call the matrix bialgebra M!(n) over R. This must not be confused with the matrix algebra

$$\Mat\!(n\,,R)\;\;\isom\;\;\Alg_R (M\!(n)\,,R\,)$$

(which is the algebra of ``points" of M!(n) ).