# nLab special lambda-ring

John Baez: I believe ‘special $\lambda$-ring’ is an old-fashioned term for what almost everyone now calls a lambda-ring, the ‘nonspecial’ ones having been found to be too general. This, at least, is what Hazewinkel says in his article cited on our page about lambda-rings. So I believe this page here should be folded in with lambda-ring.

At the very least, we should give both definitions of $\lambda$-ring — special and unspecial — over at lambda-ring. When I last checked, that page did not include a definition.

For any commutative ring $A$ we can consider the set $\Phi \left(A\right)$ of power series in an indeterminate $t$ with coefficients in $A$ whose constant term is $1$:

$f\left(t\right)=1+{a}_{1}t+{a}_{2}{t}^{2}+\dots$

These form an abelian group under multiplication with the constant power-series $1$ as unit. What may be less familiar is that there is a commutative associative binary operation $\circ$ on this set that distributes over multiplication making $\Phi \left(A\right)$ into a commutative ring, for which the function ${ϵ}_{A}:\Phi \left(A\right)\to A$ taking $f\left(t\right)$ to $f\prime \left(0\right)$ is a homomorphism. So what is usually called multiplication of power-series becomes addition in this ring, with $1$ as zero; very confusing. Of course, $\Phi$ is a functor from rings to rings and $ϵ$ is a natural transformation. In fact it is the counit of a comonad.

How is $\circ$ defined? We impose the condition

$\left(1+at\right)\circ f\left(t\right)=f\left(at\right)$

for $a\in A$ and $f\left(t\right)\in \Phi \left(A\right)$. It is now clear from distributivity that if $g\left(t\right)={\Pi }_{j}\left(1+{a}_{j}t\right)$ then $g\left(t\right)\circ f\left(t\right)={\Pi }_{j}f\left({a}_{j}t\right)$. But what happens if $g\left(t\right)$ is not a product of linear factors? Newton’s theorem on symmetric polynomials comes to the rescue. For we note that the coefficient of ${t}^{n}$ in ${\Pi }_{j}f\left({a}_{j}t\right)$ is a symmetric function in the ${a}_{j}$ and so can be expressed as a polynomial over the integers in the first $n$ coefficients of $g\left(t\right)$ and of $f\left(t\right)$. In this way we have indicated that a universal formula for multiplication in $\Phi \left(A\right)$ exists, though we may not have written it down explicitly.

We use the same trick in defining the comultiplication

${\mu }_{A}:\Phi \left(A\right)\to \Phi \left(\Phi \left(A\right)\right)$

but now our old-fashioned notation and use of indeterminates starts to cause trouble. A power-series in $\Phi \left(A\right)$ is really just a sequence $\left({a}_{1},{a}_{2},\dots \right)$. We demand that

${\mu }_{A}\left(a,0,0,\dots \right)=\left(\left(a,0,0,\dots \right),\left(0,0,\dots \right),\dots \right)$

Again, to define ${\mu }_{A}$ on an arbitrary power-series, factorize it formally into linear factors, apply the rule and distributivity, and apply Newton’s theorem.

A special $\lambda$-ring is a coalgebra for the comonad $\Phi$ above. If $\xi :A\to \Phi \left(A\right)$ is the costructure map for such a coalgebra, we define unary operations ${\lambda }^{n}$ on $A$ by the formula

$\xi \left(a\right)=1+{\lambda }^{1}\left(a\right)t+{\lambda }^{2}\left(a\right){t}^{2}+\dots$

The counit condition forces ${\lambda }^{1}\left(a\right)=a$. It is also traditional to denote $\xi \left(a\right)$ by ${\lambda }_{t}\left(a\right)$. Note that ${\lambda }_{t}\left({a}_{1}+{a}_{2}\right)={\lambda }_{t}\left({a}_{1}\right){\lambda }_{t}\left({a}_{2}\right)$ and that ${\lambda }_{t}\left({a}_{1}{a}_{2}\right)={\lambda }_{t}\left({a}_{1}\right)\circ {\lambda }_{t}\left({a}_{2}\right)$.

The functor $\Phi$ is representable by a commutative Hopf algebra $\Lambda$, and so has a left adjoint. The underlying ring of $\Lambda$ is $ℤ\left[{\lambda }^{1},{\lambda }^{2},\dots \right]$, the free special $\lambda$-ring on one generator (${\lambda }^{1}$).

In the terminology of bimodels $\Phi =\Lambda ⇒?$ and its left adjoint is $\Lambda \otimes ?$. So the theory of special $\lambda$-rings is a monadic extension of the theory of rings.

Revised on July 29, 2009 00:09:08 by Toby Bartels (71.104.230.172)