Lambda-ring
A pre--ring is a commutative ring
with identity element and a set of mappings
,
such that
i) for all
;
ii) for all
;
iii) .
Examples are, for instance, the topological -groups
and
,
a compact Lie group (cf.
-theory), and the complex representation ring
of a finite group
(cf. Representation of a compact group). In all these cases the
are induced by taking exterior powers. For instance, for
,
and the
-structure is given by
(binomial coefficients; the formula
follows by the binomial expansion theorem from
.
Let be any commutative ring with unit element 1. Consider the set
of power series in
over
with constant term 1. Multiplication of power series turns
into an Abelian group. A pre-
-ring structure on
defines a homomorphism of Abelian groups
,
, and vice versa.
Let ,
be two elements of
. Formally, write
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and consider the expressions
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The and
are symmetric polynomial expressions in the
's and
's and hence can be written as universal polynomial expressions in the
's and
's. Now define a multiplication on
by
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(,
), and define operations (mappings)
by
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The ring with these operations is a pre-
-ring. Given two pre-
-rings
,
, a
-ring homomorphism
is a homomorphism of rings such that
for all
,
.
A pre--ring
is a
-ring if
,
, is a homomorphism of pre-
-rings. The ring
is always a
-ring and so are the standard examples
,
,
of pre-
-rings mentioned above.
On the other hand, consider a finite group . A finite
-set is a finite set together with an action of
on it. Using disjoint union and Cartesian products with diagonal action, the isomorphism classes of finite
-sets form a semi-ring,
. The associated Grothendieck ring
(cf. Grothendieck group) is called the Burnside ring. On
, define operations
by
(set of
-element subsets of
) (with the natural induced
-action). This generalizes the
-operations
on
,
. Using iii), the
extend to
, making the Burnside ring into a pre-
-ring. As a rule this pre-
-ring is not a
-ring, [a9].
Instead of pre--ring and
-ring one also finds, respectively, the phrases
-ring and special
-ring in the literature.
Let be a pre-
-ring. One defines new operations
by the formula
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These operations are called the Adams operations on the pre--ring
. They were introduced in the case
by J.F. Adams ([a10]).
The Adams operations satisfy
iv) ;
v) .
Let be a torsion-free pre-
-ring; then
is a
-ring if and only if the Adams operations satisfy in addition
vi) ;
vii) ;
viii) .
A ring with operations
satisfying iv)–viii) is sometimes called a
-ring.
The ring is isomorphic to the ring
of (big) Witt vectors (cf. (the editorial comments to) Witt vector):
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Under this isomorphism the Adams operations on
correspond to the Frobenius operations
.
The -structures on the rings
define a functorial morphism of ring-valued functors
. Together with
,
, this defines a co-triple structure on the functor
, and the
-rings are precisely the co-algebras of this co-triple (cf. Triple).
Via the isomorphism one finds "exponential homomorphisms"
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which should be seen as (generalizing) the so-called Artin–Hasse exponential ([a11], [a12]).
Let be the ring homomorphism
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Then the Artin–Hasse exponential is functorially characterized by
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where is the Frobenius homomorphism.
Let be the forgetful functor. Then the functor
is right adjoint (cf. Adjoint functor) to
:
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(cf. [a5], p. 20).
There are (besides the identity) three natural automorphisms of the Abelian group , given by the substitution
, the "inversion"
, and the combination of the two. Correspondingly there are four natural ways to introduce a ring structure on
; the corresponding unit elements are
,
,
,
. All four occur in the literature. The most frequently occurring have
or
as their unit element — here, in the above,
is the unit element —, and
seems to be the most rare case.
-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 is one-dimensional, i.e.
for
, the terminology derives from the case
or
; one has
, whence the name power operations for the
. On the
the operations
are directly defined by
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References
[a1] | M.F. Atiyah, D.O. Tall, "Group representations, ![]() ![]() |
[a2] | A. Grothendieck, "La théorie des classes de Chern" Bull. Soc. Math. France , 86 (1958) pp. 137–154 |
[a3] | A. Grothendieck, "Classes de faisceaux et théorème de Riemann–Roch" , Sem. Géom. Algébrique , 6 , Springer (1972) pp. 20–77 |
[a4] | M. Hazewinkel, "Formal groups and applications" , Acad. Press (1978) pp. 144ff |
[a5] | D. Knutson, "![]() |
[a6] | P. Berthelot, "Généralités sur les ![]() |
[a7] | W. Fulton, S. Lang, "Riemann–Roch algebra" , Springer (1985) |
[a8] | T. tom Dieck, "Transformation groups and representation theory" , Springer (1979) |
[a9] | C. Siebeneicher, "![]() |
[a10] | J.F. Adams, "Vectorfields on spheres" Ann. of Math. , 75 (1962) pp. 603–632 |
[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 |
[a12] | E. Artin, H. Hasse, "Die beide Ergänzungssätze zum Reciprozitätsgesetz der ![]() ![]() |
[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 |
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