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In
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, a λ-ring or lambda ring is a
commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
together with some operations λ''n'' on it that behave like the
exterior power In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector ...
s of
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
s. Many rings considered in
K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ...
carry a natural λ-ring structure. λ-rings also provide a powerful formalism for studying an action of the
symmetric function In mathematics, a function of n variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f\left(x_1,x_2\right) of two arguments is a symmetric function if and only if f\left(x_1,x_2\right) = f\ ...
s on the
ring of polynomials In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often ...
, recovering and extending many classical results (). λ-rings were introduced by . For more about λ-rings see , , and .


Motivation

If ''V'' and ''W'' are finite-
dimensional In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
vector spaces over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
''k'', then we can form the
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...
''V'' ⊕ ''W'', the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
''V'' ⊗ ''W'', and the ''n''-th
exterior power In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector ...
of ''V'', Λ''n''(''V''). All of these are again finite-dimensional vector spaces over ''k''. The same three operations of direct sum, tensor product and exterior power are also available when working with ''k''-linear representations of a
finite group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...
, when working with
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to eve ...
s over some
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
, and in more general situations. λ-rings are designed to abstract the common algebraic properties of these three operations, where we also allow for formal inverses with respect to the direct sum operation. (These formal inverses also appear in
Grothendieck group In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a group homomorp ...
s, which is why the underlying additive groups of most λ-rings are Grothendieck groups.) The addition in the ring corresponds to the direct sum, the multiplication in the ring corresponds to the tensor product, and the λ-operations to the exterior powers. For example, the
isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
:\Lambda^2(V\oplus W)\cong \Lambda^2(V)\oplus\left(\Lambda^1(V)\otimes\Lambda^1(W)\right)\oplus\Lambda^2(W) corresponds to the formula :\lambda^2(x+y)=\lambda^2(x)+\lambda^1(x)\lambda^1(y)+\lambda^2(y) valid in all λ-rings, and the isomorphism : \Lambda^1(V\otimes W)\cong \Lambda^1(V)\otimes\Lambda^1(W) corresponds to the formula :\lambda^1(xy)=\lambda^1(x)\lambda^1(y) valid in all λ-rings. Analogous but (much) more complicated formulas govern the higher order λ-operators.


Motivation with Vector Bundles

If we have a
short exact sequence In mathematics, an exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Im ...
of vector bundles over a
smooth scheme In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a sm ...
X
0 \to \mathcal'' \to \mathcal \to \mathcal' \to 0,
then locally, for a small enough
open neighborhood In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a po ...
U we have the isomorphism :\bigwedge^n \mathcal, _U \cong \bigoplus_ \bigwedge^i \mathcal', _U \otimes\bigwedge^j\mathcal'', _U Now, in the
Grothendieck group In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a group homomorp ...
K(X) of X (which is actually a ring), we get this local equation globally for free, from the defining
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equ ...
s. So :\begin \left bigwedge^n \mathcal \right&= \left bigoplus_ \bigwedge^i \mathcal' \otimes\bigwedge^j\mathcal''\right\\ &= \sum_ \left \bigwedge^i \mathcal' \rightcdot \left \bigwedge^j \mathcal'' \right\end demonstrating the basic relation in a λ-ring, that :\lambda^n(x+y) = \sum_\lambda^i(x)\lambda^j(y).


Definition

A λ-ring is a commutative ring ''R'' together with operations λ''n'' : ''R'' → ''R'' for every non-negative
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
''n''. These operations are required to have the following properties valid for all ''x'', ''y'' in ''R'' and all ''n, m'' ≥ 0: *λ0(''x'') = 1 *λ1(''x'') = x *λ''n''(1) = 0 if ''n'' â‰¥ 2 *λ''n''(''x'' + ''y'') = Σ''i''+''j''=''n'' Î»''i''(''x'') λ''j''(''y'') *λ''n''(''xy'') = ''P''''n''(λ1(''x''), ..., λ''n''(''x''), λ1(''y''), ..., λ''n''(''y'')) *λ''n''(λ''m''(''x'')) = ''P''''n'',''m''(λ1(''x''), ..., λ''mn''(''x'')) where ''P''''n'' and ''Pn,m'' are certain universal polynomials with integer coefficients that describe the behavior of exterior powers on tensor products and under composition. These polynomials can be defined as follows. Let ''e''1, ..., ''e''''mn'' be the
elementary symmetric polynomial In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary sy ...
s in the variables ''X''1, ..., ''X''''mn''. Then ''P''''n'',''m'' is the unique polynomial in ''nm'' variables with integer coefficients such that ''Pn,m''(''e''1, ..., ''e''''mn'') is the coefficient of ''t''''n'' in the expression :\prod_ (1+tX_X_\cdots X_)   (Such a polynomial exists, because the expression is symmetric in the ''Xi'' and the elementary symmetric polynomials generate all symmetric polynomials.) Now let ''e''1, ..., ''e''''n'' be the elementary symmetric polynomials in the variables ''X''1, ..., ''X''''n'' and ''f''1, ..., ''f''''n'' be the elementary symmetric polynomials in the variables ''Y''1, ..., ''Y''''n''. Then ''P''''n'' is the unique polynomial in 2''n'' variables with integer coefficients such that is the coefficient of ''t''''n'' in the expression :\prod_^n (1+tX_iY_j)


Variations

The λ-rings defined above are called "special λ-rings" by some authors, who use the term "λ-ring" for a more general concept where the conditions on λ''n''(1), λ''n''(''xy'') and λ''n''(λ''m''(''x'')) are dropped.


Examples

*The ring Z of
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s, with the
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s \lambda^n(x)= as operations (which are also defined for negative ''x'') is a λ-ring. In fact, this is the only λ-structure on Z. This example is closely related to the case of finite-dimensional vector spaces mentioned in the Motivation section above, identifying each vector space with its dimension and remembering that \dim(\Lambda^n(k^x ))=. *More generally, any
binomial ring In mathematics, a binomial ring is a commutative ring whose additive group is torsion-free and contains all binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial ...
becomes a λ-ring if we define the λ-operations to be the binomial coefficients, λ''n''(''x'') = (). In these λ-rings, all
Adams operation In mathematics, an Adams operation, denoted ψ''k'' for natural numbers ''k'', is a cohomology operation in topological K-theory, or any allied operation in algebraic K-theory or other types of algebraic construction, defined on a pattern introd ...
s are the identity. *The
K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ...
K(''X'') of a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
''X'' is a λ-ring, with the lambda operations induced by taking exterior powers of a vector bundle. *Given a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...
''G'' and a base field ''k'', the
representation ring In mathematics, especially in the area of algebra known as representation theory, the representation ring (or Green ring after J. A. Green) of a group is a ring formed from all the (isomorphism classes of the) finite-dimensional linear represent ...
''R''(''G'') is a λ-ring; the λ-operations are induced by the exterior powers of ''k''-linear representations of the group ''G''. *The ring ΛZ of symmetric functions is a λ-ring. On the integer coefficients the λ-operations are defined by binomial coefficients as above, and if ''e''1, ''e''2, ... denote the elementary symmetric functions, we set λ''n''(''e''1) = ''e''''n''. Using the axioms for the λ-operations, and the fact that the functions ''e''''k'' are
algebraically independent In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non- trivial polynomial equation with coefficients in K. In particular, a one element set \ is algebraically i ...
and generate the ring ΛZ, this definition can be extended in a unique fashion so as to turn ΛZ into a λ-ring. In fact, this is the free λ-ring on one generator, the generator being ''e''1. (). *The
Grothendieck group In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a group homomorp ...
of any
symmetric Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
monoidal
abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category o ...
is a λ-ring (the symmetric tensor product giving rise to the commutative multiplication), generalizing most of the examples above.


Further properties and definitions

Every λ-ring has characteristic 0 and contains the λ-ring Z as a λ-subring. Many notions of
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...
can be extended to λ-rings. For example, a λ-homomorphism between λ-rings ''R'' and ''S'' is a
ring homomorphism In mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function that preserves addition, multiplication and multiplicative identity ...
''f : R → S'' such that ''f''(λ''n''(''x'')) = λ''n''(''f''(''x'')) for all ''x'' in ''R'' and all ''n'' ≥ 0. A λ-ideal in the λ-ring ''R'' is an
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...
''I'' in ''R'' such that λ''n''(''x'') ϵ ''I'' for all ''x'' in ''R'' and all ''n'' ≥ 1. If ''x'' is an element of a λ-ring and ''m'' a non-negative integer such that λ''m''(''x'') ≠ 0 and λ''n''(''x'') = 0 for all ''n'' > ''m'', we write dim(''x'') = ''m'' and call the element ''x'' finite-dimensional. Not all elements need to be finite-dimensional. We have dim(''x''+''y'') ≤ dim(''x'') + dim(''y'') and the product of elements is . Symmetric functions act on λ-rings, as follows: given a symmetric polynomial ''p'' in arbitrary many variables with integer coefficients, and an element ''x'' of a λ-ring ''R,'' there exists a unique λ-homomorphism ''f'' : ΛZ ''→ R'' with ''f''(''e1)=x,'' and we set ''p.x := f(p).''


See also

*
Chern class In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches ...
*
Symmetric Function In mathematics, a function of n variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f\left(x_1,x_2\right) of two arguments is a symmetric function if and only if f\left(x_1,x_2\right) = f\ ...
*
K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ...
*
Adams operation In mathematics, an Adams operation, denoted ψ''k'' for natural numbers ''k'', is a cohomology operation in topological K-theory, or any allied operation in algebraic K-theory or other types of algebraic construction, defined on a pattern introd ...
*
Plethystic exponential In mathematics, the plethystic exponential is a certain operator defined on (formal) power series which, like the usual exponential function, translates addition into multiplication. This exponential operator appears naturally in the theory of sym ...


References

* *Expo 0 and V of * * * * * * * {{DEFAULTSORT:Lambda-ring Ring theory K-theory