Lambda-ring

A pre-$\lambda$-ring is a commutative ring $R$ with identity element $1$ and a set of mappings $\lambda^n : R \rightarrow R$, $n = 0,1,2,\ldots$ such that

i) $\lambda^0(x) = 1$ for all $x \in R$;

ii) $\lambda^1(x) = x$ for all $x \in R$;

iii) $\lambda^n(x+y) = \sum_{i+j=n} \lambda^i(x) \lambda^j(y)$.

Examples are, for instance, the topological $K$-groups $K(M)$ and $K_G(M)$, $G$ a compact Lie group (cf. $K$-theory), and the complex representation ring $R(G)$ of a finite group $G$ (cf. Representation of a compact group). In all these cases the $\lambda^n$ are induced by taking exterior powers. For instance, for $M = \text{pt}$, $K(M) = \mathbf{Z}$ and the $\lambda$-structure is given by $\lambda^n(m) = \binom{m}{n}$ (binomial coefficients; the formula $\binom{m_1+m_2}{n} = \sum_{i+j=n} \binom{m_1}{i} \binom{m_2}{j}$ follows by the binomial expansion theorem from $(X+Y)^{m_1+m_2} = (X+Y)^{m_1} (X+Y)^{m_2}$.

Let $R$ be any commutative ring with unit element 1. Consider the set $\Lambda(R) = 1 + t R[[t]]$ of power series in $t$ over $R$ with constant term 1. Multiplication of power series turns $\Lambda(R)$ into an Abelian group. A pre-$\lambda$-ring structure on $\Lambda(R)$ defines a homomorphism of Abelian groups $\lambda_t : R \rightarrow \Lambda(R)$, $\lambda_t(x) = \lambda^0(x) + \lambda^1(x) t + \lambda^2(x) t^2 + \cdots$, and vice versa.

Let $\alpha(t) = 1 + a_1 t + a_2 t^2 + \cdots$, $\beta(t) = 1 + b_1 t + b_2 t^2 + \cdots$ be two elements of $\Lambda(R)$. Formally, write $$ \alpha(t) = \prod_{i=1}^\infty \left({ 1 - \xi_i t }\right) $$ $$ \beta(t) = \prod_{i=1}^\infty \left({ 1 - \eta_i t }\right) $$ and consider the expressions $$ \prod_{i,j=1}^\infty \left({ 1 - \xi_i \eta_j t }\right) = 1 + P_1 t + P_2 t^2 + \cdots \ , $$ $$ \prod_{i_1 < i_2 < \cdots < i_n}^\infty \left({ 1 - \xi_{i_1} \xi_{i_2} \cdots \xi_{i_n} t }\right) = 1 + L_{1,n} t + L_{2,n} t^2 + \cdots \ . $$

The $P_i$ and $L_{i,n}$ are symmetric polynomial expressions in the $\xi$'s and $\eta$'s and hence can be written as universal polynomial expressions in the $a$'s and $b$'s. Now define a multiplication on $\Lambda(R)$ by $$ \alpha(t) * \beta(t) = 1 + P_1(a,b) t + P_2(a,b) t^2 + \cdots $$ ($a = (a_1,a_2,\ldots)$, $b = (b_1,b_2,\ldots)$), and define operations (mappings) $\lambda^n : \Lambda(R) \rightarrow \Lambda(R)$ by $$ \lambda^n \alpha(t) = 1 + L_{1,n}(a,b) t + L_{2,n}(a,b) t^2 + \cdots \ . $$

The ring $\Lambda(R)$ with these operations is a pre-$\lambda$-ring. Given two pre-$\lambda$-rings $R_1$, $R_2$, a $\lambda$-ring homomorphism $\phi : R_1 \rightarrow R_2$ is a homomorphism of rings such that $\phi(\lambda^n(x)) = \lambda^n(\phi(x))$ for all $x \in R_1$, $n = 0,1,2,\ldots$.

A pre-$\lambda$-ring $R$ is a $\lambda$-ring if $\lambda_{-t} : R \rightarrow \Lambda(R)$, $\lambda_{-t}(x) = 1 - \lambda^1(x) t + \lambda^2(x) t^2 - \cdots$, is a homomorphism of pre-$\lambda$-rings. The ring $\Lambda(R)$ is always a $\lambda$-ring and so are the standard examples $K(M)$, $K_G(M)$, $R(G)$ of pre-$\lambda$-rings mentioned above.

On the other hand, consider a finite group $G$. A finite $G$-set is a finite set together with a group action of $G$. Using disjoint union and Cartesian products with diagonal action, the isomorphism classes of finite $G$-sets form a semi-ring, $A^+(G)$. The associated Grothendieck ring $A(G)$ is called the Burnside ring. On $A^+(G)$, define operations $\lambda^n : A^+(G) \rightarrow A^+(G)$ by taking $\lambda^n(S)$ to be the set of$n$-element subsets of $S$ with the natural induced $G$-action. This generalizes the $\lambda$-operations $\lambda^n$ on $\mathbf{N} \subset \mathbf{Z}$, $\lambda^n(m) = \binom{m}{n}$. Using iii), the $\lambda^n$ extend to $A(G)$, making the Burnside ring into a pre-$\lambda$-ring. As a rule this pre-$\lambda$-ring is not a $\lambda$-ring, [a9].

Instead of pre-$\lambda$-ring and $\lambda$-ring one also finds, respectively, the phrases $\lambda$-ring and special $\lambda$-ring in the literature.

Let $R$ be a pre-$\lambda$-ring. One defines new operations $\psi^i : R \rightarrow R$ by the formula $$ -t \frac{d}{dt} \log \lambda_{-t} (x) = \sum_{i=1}^\infty \psi^i(x) t^i \ . $$

These operations are called the Adams operations on the pre-$\lambda$-ring $R$. They were introduced in the case $R = K(M)$ by J.F. Adams ([a10]).

The Adams operations satisfy

iv) $\Psi^1(x) = x$;

v) $\Psi^n(x+y) = \Psi^n(x) + \Psi^n(y)$.

Let $R$ be a torsion-free pre-$\lambda$-ring; then $R$ is a $\lambda$-ring if and only if the Adams operations satisfy in addition

vi) $\Psi^i(1) = 1$;

vii) $\Psi^n(xy) = \Psi^n(x) \Psi^n(y)$;

viii) $\Psi^{ij}(x) = \Psi^i(\Psi^j(x))$.

A ring $R$ with operations $\Psi^i$ satisfying iv)–viii) is sometimes called a $\Psi$-ring.

The ring $\Lambda(R)$ is isomorphic to the ring $W(R)$ of (big) Witt vectors (cf. (the editorial comments to) Witt vector): $$ \bar E : W(R) \rightarrow \Lambda(R) $$ $$ (a_1,a_2,\ldots) \mapsto \prod_{i=1}^\infty \left({ 1 - a_i t^i }\right) $$

Under this isomorphism the Adams operations $\Psi^n$ on $\Lambda(R)$ correspond to the Frobenius operations $\mathbf{f}_n : W(R) \rightarrow W(R)$.

The $\lambda$-structures on the rings $\Lambda(R)$ define a functorial morphism of ring-valued functors $\lambda_{-t}(\cdot) : \Lambda(\cdot) \rightarrow \Lambda(\Lambda(\cdot)) $. Together with $\Lambda(R) \rightarrow R$, $1 + a_1 t + a_2 t^2 + \cdots \mapsto a_1$, this defines a co-triple structure on the functor $\Lambda$, and the $\lambda$-rings are precisely the co-algebras of this co-triple.

Via the isomorphism $\bar E$ one finds "exponential homomorphisms" $$ E : W(R) \rightarrow W(W(R)) $$ $$ E' : W(R) \rightarrow \Lambda(W(R)) $$ which should be seen as (generalizing) the so-called Artin–Hasse exponential ([a11], [a12]).

Let $w_n(R) : W(R) \rightarrow R$ be the ring homomorphism $$ w_n (a_1,a_2,\ldots) = \sum_{d|n} d a_d^{n/d} \ . $$ Then the Artin–Hasse exponential $E(R) : W(R) \rightarrow W(W(R))$ is functorially characterized by $$ w_n(W(R)) \circ E(R) = \mathbf{f}_n(R) $$ where $\mathbf{f}_n$ is the Frobenius homomorphism.

Let $V : \lambda\textsf{-Ring} \rightarrow \textsf{Ring}$ be the forgetful functor. Then the functor $\Lambda : \textsf{Ring} \rightarrow \lambda\textsf{-Ring}$ is right adjoint (cf. Adjoint functor) to $V$: $$ \textsf{Ring}(V(S),R) \equiv \lambda\textsf{-Ring}(S,\Lambda(R)) $$ (cf. [a5], p. 20).

There are (besides the identity) three natural automorphisms of the Abelian group $\Lambda(R) = 1 + t R[[t]]$, given by the substitution $t \mapsto -t$, the "inversion" $\alpha(t) \mapsto \alpha(t)^{-1}$, and the combination of the two. Correspondingly there are four natural ways to introduce a ring structure on $\Lambda(R)$; the corresponding unit elements are $1-t$, $1+t$, $(1-t)^{-1}$, $(1+t)^{-1}$. All four occur in the literature. The most frequently occurring have $1-t$ or $1+t$ as their unit element — here, in the above, $1-t$ is the unit element —, and $(1+t)^{-1}$ seems to be the most rare case.

$\lambda$-rings were introduced by A. Grothendieck in an algebraic-geometric setting [a2] and were first used in group representation theory by M.F. Atiyah and D.O. Tall ([a1]).

In case $x \in R$ is one-dimensional, i.e. $\lambda^n(x) = 0$ for $n \ge 2$, the terminology derives from the case $R = K(M)$ or $R = R(G)$; one has $\Psi^n(x) = x^n$, whence the name power operations for the $\Psi^n$. On the $\Lambda(R)$ the operations $\Psi^n$ are directly defined by $$ \Psi^n \left({ \prod_{i=1}^\infty (1 - \xi_i t) }\right) = \prod_{i=1}^\infty (1 - \xi_i^n t) \ . $$

References

[a1]	M.F. Atiyah, D.O. Tall, "Group representations, $\lambda$-rings and the $J$-homomorphism" Topology , 8 (1969) pp. 253–297 MR244387
[a2]	A. Grothendieck, "La théorie des classes de Chern" Bull. Soc. Math. France , 86 (1958) pp. 137–154 MR0116023 Zbl 0091.33201
[a3]	A. Grothendieck, "Classes de faisceaux et théorème de Riemann–Roch" , Sem. Géom. Algébrique , 6 , Springer (1972) pp. 20–77 Zbl 0229.14008
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[a5]	D. Knutson, "$\lambda$-rings and the representation theory of the symmetric group" , Springer (1974) MR0364425 Zbl 0272.20008
[a6]	P. Berthelot, "Généralités sur les $\lambda$-anneaux" , Sem. Géom. Algébrique , 6 , Springer (1972) pp. 297–365
[a7]	W. Fulton, S. Lang, "Riemann–Roch algebra" , Springer (1985) MR0801033 Zbl 0579.14011
[a8]	T. tom Dieck, "Transformation groups and representation theory" , Springer (1979) Zbl 0445.57023
[a9]	C. Siebeneicher, "$\lambda$-Ringstrukturen auf dem Burnsidering der Permutationsdarstellungen einer endlichen Gruppe" Math. Z. , 146 (1976) pp. 223–238 MR0390035 Zbl 0306.20011
[a10]	J.F. Adams, "Vectorfields on spheres" Ann. of Math. , 75 (1962) pp. 603–632 DOI 10.2307/1970213 Zbl 0112.38102
[a11]	M. Hazewinkel, "Twisted Lubin–Tate formal group laws, ramified Witt vectors and (ramified) Artin–Hasse exponentials" Trans. Amer. Math. Soc. , 259 (1980) pp. 47–63 MR0561822 Zbl 0437.13014
[a12]	E. Artin, H. Hasse, "Die beide Ergänzungssätze zum Reciprozitätsgesetz der $\ell^n$-ten Potenzreste im Körper der $\ell^n$-ten Einheitswurzeln" Abh. Math. Sem. Univ. Hamburg , 6 (1928) pp. 146–162
[a13]	G. Whaples, "Generalized local class field theory III: Second form of the existence theorem, structure of analytic groups" Duke Math. J. , 21 (1954) pp. 575–581 MR73645