Addition of Time: Coalgebras from Constant-Coefficient ODEs

From Polynomial Quotients to Constant-Coefficient ODEs: A Coalgebra on the Solution SpaceProof of the IsomorphismTransporting the Coalgebra StructureThe Coalgebra Axioms as Addition of TimeExample 1: $p(x)=x^2$Example 2: $p(x)=x-\lambda$Example 3: $p(x)=(x-\lambda)^2$Example 4: $D^2+1$

 

From Polynomial Quotients to Constant-Coefficient ODEs: A Coalgebra on the Solution Space

Let $k$$k$ be a field of characteristic $0$$0$, and let $p(x)\in k[x]$$p(x)\in k[x]$ be a nonzero polynomial of degree $d$$d$. Define

$$A_p=k[x]/(p(x)),$$

and write $\overline{x}$$\bar x$ for the image of $x$$x$ in $A_p$$A_p$. Then

$$p(\overline{x})=0,{\dim }_k{A}_p=d,$$

and

$$1,\overline{x},\dots ,{\overline{x}}^{d-1}$$

form a basis of $A_p$$A_p$.

Since $A_p$$A_p$ is a finite-dimensional algebra, its dual space

$$A_p^{\ast }={\mathrm{Hom}}_k(A_p,k)$$

carries a natural coalgebra structure. If

$$m:A_p\otimes A_p\to A_p$$

is multiplication, then its dual gives

$$\mathrm{\Delta }=m^{\ast }:A_p^{\ast }\to A_p^{\ast }\otimes A_p^{\ast },$$

so explicitly,

$$\mathrm{\Delta }(\phi )(a\otimes b)=\phi (ab).$$

The counit is

$$\epsilon (\phi )=\phi (1).$$

On the other hand, consider the constant-coefficient differential equation

$$p(D)F=0,D=\frac{d}{dt},$$

and let

$$V_p=\{F\in k[[t]]:p(D)F=0\}$$

be its space of formal power series solutions.

Theorem. There is a natural isomorphism

$$\mathrm{\Theta }:A_p^{\ast }\to V_p$$

defined by

$$\mathrm{\Theta }(\phi )=\sum _{n\ge 0}\phi ({\overline{x}}^n)\frac{t^n}{n!}.$$

Its inverse is given as follows: for $F\in V_p$$F\in V_p$, define

$${\phi }_F(\overline{q})=(q(D)F)(0),q\in k[x].$$

Under this isomorphism, the coalgebra structure on $V_p$$V_p$ is

$$(\mathrm{\Delta }F)(s,t)=F(s+t)$$

and

$$\epsilon (F)=F(0).$$

Proof of the Isomorphism

We first show that $\mathrm{\Theta }$$\Theta$ indeed takes values in $V_p$$V_p$.

For every $j\ge 0$$j\ge0$,

$$D^j\mathrm{\Theta }(\phi )=\sum _{n\ge 0}\phi ({\overline{x}}^{n+j})\frac{t^n}{n!}.$$

Hence

$$p(D)\mathrm{\Theta }(\phi )=\sum _{n\ge 0}\phi ({\overline{x}}^{n}p(\overline{x}))\frac{t^n}{n!}.$$

But

$$p(\overline{x})=0$$

in $A_p$$A_p$, so

$$p(D)\mathrm{\Theta }(\phi )=0.$$

Therefore

$$\mathrm{\Theta }(\phi )\in V_p.$$

Now we check that the inverse map is well defined.

Suppose

$$q-q^{\prime }=pr.$$

Then

$$(q(D)-q^{\prime }(D))F=p(D)r(D)F.$$

Since constant-coefficient differential operators commute,

$$p(D)r(D)F=r(D)p(D)F=0.$$

Therefore

$$(q(D)F)(0)=(q^{\prime }(D)F)(0),$$

so ${\phi }_F$$\varphi_F$ depends only on the class of $q$$q$ in $A_p$$A_p$.

Moreover,

$${\phi }_F({\overline{x}}^n)=F^{(n)}(0).$$

Hence

$$\mathrm{\Theta }({\phi }_F)=\sum _{n\ge 0}{F}^{(n)}(0)\frac{t^n}{n!}=F.$$

Conversely,

$${\phi }_{\mathrm{\Theta }(\phi )}({\overline{x}}^n)=\mathrm{\Theta }(\phi {)}^{(n)}(0)=\phi ({\overline{x}}^n).$$

Since

$$1,\overline{x},\dots ,{\overline{x}}^{d-1}$$

span $A_p$$A_p$, it follows that

$${\phi }_{\mathrm{\Theta }(\phi )}=\phi .$$

Thus

$$\mathrm{\Theta }:A_p^{\ast }\stackrel{\sim }{\to }{V}_p$$

is a linear isomorphism.

Transporting the Coalgebra Structure

Now let

$$F=\mathrm{\Theta }(\phi ).$$

The coalgebra structure on $A_p^{\ast }$$A_p^*$ is determined by

$$\mathrm{\Delta }(\phi )(a\otimes b)=\phi (ab).$$

In particular,

$$\mathrm{\Delta }(\phi )({\overline{x}}^m\otimes {\overline{x}}^n)=\phi ({\overline{x}}^{m+n}).$$

Therefore

$$(\mathrm{\Theta }\otimes \mathrm{\Theta })\mathrm{\Delta }(\phi )=\sum _{m,n\ge 0}\phi ({\overline{x}}^{m+n})\frac{s^m}{m!}\frac{t^n}{n!}.$$

Since

$$\phi ({\overline{x}}^{m+n})=F^{(m+n)}(0),$$

this becomes

$$(\mathrm{\Theta }\otimes \mathrm{\Theta })\mathrm{\Delta }(\phi )=\sum _{m,n\ge 0}{F}^{(m+n)}(0)\frac{s^m}{m!}\frac{t^n}{n!}.$$

On the other hand, Taylor expansion gives

$$F(s+t)=\sum _{r\ge 0}{F}^{(r)}(0)\frac{(s+t{)}^r}{r!}.$$

Using

$$\frac{(s+t{)}^r}{r!}=\sum _{m+n=r}\frac{s^m}{m!}\frac{t^n}{n!},$$

we obtain

$$F(s+t)=\sum _{m,n\ge 0}{F}^{(m+n)}(0)\frac{s^m}{m!}\frac{t^n}{n!}.$$

Hence

$$(\mathrm{\Theta }\otimes \mathrm{\Theta })\mathrm{\Delta }(\phi )=F(s+t).$$

Thus, after transporting the coalgebra structure from $A_p^{\ast }$$A_p^*$ to $V_p$$V_p$,

$$(\mathrm{\Delta }F)(s,t)=F(s+t).$$

The counit behaves similarly:

$$\epsilon (F)=F(0).$$

Indeed,

$$F(0)=\mathrm{\Theta }(\phi )(0)=\phi (1).$$

The Coalgebra Axioms as Addition of Time

In this form, the coalgebra axioms become almost tautological.

Coassociativity says

$$(\mathrm{\Delta }\otimes \mathrm{id})\mathrm{\Delta }=(\mathrm{id}\otimes \mathrm{\Delta })\mathrm{\Delta }.$$

On a solution $F$$F$, the two sides are

$$((\mathrm{\Delta }\otimes \mathrm{id})\mathrm{\Delta }F)(r,s,t)=F((r+s)+t)$$

and

$$((\mathrm{id}\otimes \mathrm{\Delta })\mathrm{\Delta }F)(r,s,t)=F(r+(s+t)).$$

Since addition is associative,

$$(r+s)+t=r+(s+t),$$

the two expressions agree.

Similarly, the counit axioms are simply

$$F(t+0)=F(t)=F(0+t).$$

Thus the coalgebra structure on the solution space records the addition law of the time variable.

Example 1: $p(x)=x^2$$p(x)=x^2$

In this case,

$$V_p=\mathrm{span}\{1,t\}.$$

Since

$$1(s+t)=1,$$

we have

$$\mathrm{\Delta }(1)=1\otimes 1.$$

And since

$$s+t=s+t,$$

we obtain

$$\mathrm{\Delta }(t)=t\otimes 1+1\otimes t.$$

Thus $t$$t$ is a primitive element.

Example 2: $p(x)=x-\lambda$$p(x)=x-\lambda$

Here

$$V_p=\mathrm{span}\{e^{\lambda t}\}.$$

Since

$$e^{\lambda (s+t)}=e^{\lambda s}{e}^{\lambda t},$$

we get

$$\mathrm{\Delta }(e^{\lambda t})=e^{\lambda t}\otimes e^{\lambda t}.$$

Thus $e^{\lambda t}$$e^{\lambda t}$ is group-like.

Example 3: $p(x)=(x-\lambda {)}^2$$p(x)=(x-\lambda)^2$

Now

$$V_p=\mathrm{span}\{e^{\lambda t},t{e}^{\lambda t}\}.$$

Write

$$g(t)=e^{\lambda t},v(t)=t{e}^{\lambda t}.$$

Then

$$\mathrm{\Delta }(g)=g\otimes g.$$

Moreover,

$$\begin{array}{rl}v(s+t)& =(s+t)e^{\lambda (s+t)}\\ & =s{e}^{\lambda s}{e}^{\lambda t}+e^{\lambda s}t{e}^{\lambda t}\\ & =v(s)g(t)+g(s)v(t).\end{array}$$

Hence

$$\mathrm{\Delta }(v)=v\otimes g+g\otimes v.$$

Thus $v$$v$ is $g$$g$-primitive.

Example 4: $D^2+1$$D^2+1$

Take

$$p(x)=x^2+1.$$

The corresponding differential equation is

$$(D^2+1)F=0,$$

that is,

$$F^{″}+F=0.$$

Its solution space is

$$V_p=\mathrm{span}\{\cos t,\sin t\}.$$

Using

$$(\mathrm{\Delta }F)(s,t)=F(s+t),$$

we obtain

$$\cos (s+t)=\cos s\cos t-\sin s\sin t,$$

and hence

$$\mathrm{\Delta }(\cos t)=\cos t\otimes \cos t-\sin t\otimes \sin t.$$

Similarly,

$$\sin (s+t)=\sin s\cos t+\cos s\sin t,$$

so

$$\mathrm{\Delta }(\sin t)=\sin t\otimes \cos t+\cos t\otimes \sin t.$$

The counit is

$$\epsilon (\cos t)=1,\epsilon (\sin t)=0.$$

If $k$$k$ contains $i$$i$, we may instead use the basis

$$e^{it},e^{-it}.$$

Then

$$\mathrm{\Delta }(e^{it})=e^{it}\otimes e^{it},$$

and

$$\mathrm{\Delta }(e^{-it})=e^{-it}\otimes e^{-it}.$$

Thus both $e^{it}$$e^{it}$ and $e^{-it}$$e^{-it}$ are group-like.

In this sense, the classical addition formulas for sine and cosine are precisely the comultiplication formulas of the coalgebra $V_{x^2+1}$$V_{x^2+1}$.