In algebra, the fixed-point subring

R^f

of an

automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...

''f'' of a ring ''R'' is the

subring In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those wh ...

of the fixed points of ''f'', that is, :

R^f = \.

More generally, if ''G'' is a group

acting Acting is an activity in which a story is told by means of its enactment by an actor or actress who adopts a character—in theatre, television, film, radio, or any other medium that makes use of the mimetic mode. Acting involves a broad r ...

on ''R'', then the subring of ''R'' :

R^G = \

is called the fixed subring or, more traditionally, the ring of invariants under . If ''S'' is a set of automorphisms of ''R'', the elements of ''R'' that are fixed by the elements of ''S'' form the ring of invariants under the group generated by ''S''. In particular, the fixed-point subring of an automorphism ''f'' is the ring of invariants of the cyclic group generated by ''f''. In Galois theory, when ''R'' is a field and ''G'' is a group of field automorphisms, the fixed ring is a subfield called the fixed field of the automorphism group; see Fundamental theorem of Galois theory. Along with a module of covariants, the ring of invariants is a central object of study in invariant theory. Geometrically, the rings of invariants are the coordinate rings of (affine or projective) GIT quotients and they play fundamental roles in the constructions in geometric invariant theory. Example: Let

R = k_1, \dots, x_n /math> be a polynomial ring in ''n'' variables. The symmetric group S

_''n'' acts on ''R'' by permuting the variables. Then the ring of invariants

R^G = k_1, \dots, x_n

is the

ring of symmetric polynomials In algebra and in particular in algebraic combinatorics, the ring of symmetric functions is a specific limit of the rings of symmetric polynomials in ''n'' indeterminates, as ''n'' goes to infinity. This ring serves as universal structure in whi ...

. If a reductive algebraic group ''G'' acts on ''R'', then the fundamental theorem of invariant theory describes the generators of ''R''^''G''. Hilbert's fourteenth problem asks whether the ring of invariants is finitely generated or not (the answer is affirmative if ''G'' is a reductive algebraic group by Nagata's theorem.) The finite generation is easily seen for a

finite group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...

''G'' acting on a finitely generated algebra ''R'': since ''R'' is integral over ''R''^''G'',Given ''r'' in ''R'', the polynomial

\prod_ (t - g \cdot r)

is a monic polynomial over ''R''^''G'' and has ''r'' as one of its roots. the

Artin–Tate lemma In algebra, the Artin–Tate lemma, named after Emil Artin and John Tate, states: :Let ''A'' be a commutative Noetherian ring and B \sub C commutative algebras over ''A''. If ''C'' is of finite type over ''A'' and if ''C'' is finite over ''B'', ...

implies ''R''^''G'' is a finitely generated algebra. The answer is negative for some

unipotent group In mathematics, a unipotent element ''r'' of a ring ''R'' is one such that ''r'' − 1 is a nilpotent element; in other words, (''r'' − 1)''n'' is zero for some ''n''. In particular, a square matrix ''M'' is a unipote ...

s. Let ''G'' be a finite group. Let ''S'' be the symmetric algebra of a finite-dimensional ''G''-module. Then ''G'' is a reflection group if and only if

S

is a

free module In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in t ...

(of finite rank) over ''S''^''G'' (Chevalley's theorem). In

differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

, if ''G'' is a

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...

and

\mathfrak = \operatorname(G)

its

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...

, then each principal ''G''-bundle on a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

''M'' determines a graded algebra homomorphism (called the Chern–Weil homomorphism) :

G \to \operatorname^(M; \mathbb)

where

\mathbb mathfrak /math> is the ring of polynomial functions on \mathfrak and ''G'' acts on \mathbb mathfrak /math> by adjoint representation .

Notes

References

* * {{Citation , last=Springer , first=Tonny A. , title=Invariant theory , series=Lecture Notes in Mathematics , volume=585 , publisher=Springer , year=1977 Ring theory

See also

Notes

References