special lambda-ring

John Baez: I believe ‘special λ\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 λ\lambda-ring — special and unspecial — over at lambda-ring. When I last checked, that page did not include a definition.

For any commutative ring AA we can consider the set Φ(A)\Phi(A) of power series in an indeterminate tt with coefficients in AA whose constant term is 11:

f(t)=1+a 1t+a 2t 2+f(t) = 1 + a_1t + a_2t^2 + \ldots

These form an abelian group under multiplication with the constant power-series 11 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 Φ(A)\Phi(A) into a commutative ring, for which the function ϵ A:Φ(A)A\epsilon_A: \Phi(A) \rightarrow A taking f(t)f(t) to f(0)f'(0) is a homomorphism. So what is usually called multiplication of power-series becomes addition in this ring, with 11 as zero; very confusing. Of course, Φ\Phi is a functor from rings to rings and ϵ\epsilon is a natural transformation. In fact it is the counit of a comonad.

How is \circ defined? We impose the condition

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

for aAa\in A and f(t)Φ(A)f(t)\in \Phi(A). It is now clear from distributivity that if g(t)=Π j(1+a jt)g(t) = \Pi_j(1+a_j t) then g(t)f(t)=Π jf(a jt)g(t)\circ f(t) = \Pi_j f(a_j t). But what happens if g(t)g(t) 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 nt^n in Π jf(a jt)\Pi_j f(a_j t) is a symmetric function in the a ja_j and so can be expressed as a polynomial over the integers in the first nn coefficients of g(t)g(t) and of f(t)f(t). In this way we have indicated that a universal formula for multiplication in Φ(A)\Phi(A) exists, though we may not have written it down explicitly.

We use the same trick in defining the comultiplication

μ A:Φ(A)Φ(Φ(A))\mu_A : \Phi(A) \rightarrow \Phi(\Phi(A))

but now our old-fashioned notation and use of indeterminates starts to cause trouble. A power-series in Φ(A)\Phi(A) is really just a sequence (a 1,a 2,)(a_1, a_2, \ldots). We demand that

μ A(a,0,0,)=((a,0,0,),(0,0,),)\mu_A (a,0,0, \ldots) = ((a,0,0, \ldots),(0,0, \ldots), \ldots )

Again, to define μ A\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 λ\lambda-ring is a coalgebra for the comonad Φ\Phi above. If ξ:AΦ(A)\xi : A \rightarrow \Phi(A) is the costructure map for such a coalgebra, we define unary operations λ n\lambda^n on AA by the formula

ξ(a)=1+λ 1(a)t+λ 2(a)t 2+\xi(a) = 1 + \lambda^1(a)t + \lambda^2(a)t^2 + \ldots

The counit condition forces λ 1(a)=a\lambda^1(a) = a. It is also traditional to denote ξ(a)\xi(a) by λ t(a)\lambda_t (a). Note that λ t(a 1+a 2)=λ t(a 1)λ t(a 2)\lambda_t(a_1 + a_2) = \lambda_t(a_1)\lambda_t(a_2) and that λ t(a 1a 2)=λ t(a 1)λ t(a 2)\lambda_t(a_1 a_2) = \lambda_t(a_1)\circ\lambda_t(a_2).

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

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

Last revised on July 29, 2009 at 00:09:08. See the history of this page for a list of all contributions to it.