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 $\Phi (A)$ of power series? in an indeterminate $t$ with coefficients in $A$ whose constant term is $1$:

$f(t)=1+a_{1}t+a_2{t}^2+\dots$

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 $\Phi (A)$ into a commutative ring, for which the function ${ϵ}_A:\Phi (A)\to A$ taking $f(t)$ to $f\prime (0)$ 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

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

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

We use the same trick in defining the comultiplication

${\mu }_A:\Phi (A)\to \Phi (\Phi (A))$

but now our old-fashioned notation and use of indeterminates starts to cause trouble. A power-series in $\Phi (A)$ is really just a sequence $(a_1,a_2,\dots )$. We demand that

${\mu }_A(a\mathrm{,0,0},\dots )=((a\mathrm{,0,0},\dots ),(\mathrm{0,0},\dots ),\dots )$

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 $\xi :A\to \Phi (A)$ is the costructure map for such a coalgebra, we define unary operations ${\lambda }^n$ on $A$ by the formula

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

The counit condition forces ${\lambda }^1(a)=a$. It is also traditional to denote $\xi (a)$ by ${\lambda }_t(a)$. Note that ${\lambda }_t(a_1+a_2)={\lambda }_t(a_1){\lambda }_t(a_2)$ and that ${\lambda }_t(a_1{a}_2)={\lambda }_t(a_1)\circ {\lambda }_t(a_2)$.

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

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