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Difference between revisions of "Character group"

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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Borel,  "Linear algebraic groups" , Benjamin  (1969)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.A. Morris,  "Pontryagin duality and the structure of locally compact Abelian groups" , ''London Math. Soc. Lecture Notes'' , '''29''' , Cambridge Univ. Press  (1977)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.S. Pontryagin,  "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L. Fuchs,  "Infinite abelian groups" , '''1''' , Acad. Press  (1970)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.E. Humphreys,  "Linear algebraic groups" , Springer  (1975)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Borel,  "Linear algebraic groups" , Benjamin  (1969) {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.A. Morris,  "Pontryagin duality and the structure of locally compact Abelian groups" , ''London Math. Soc. Lecture Notes'' , '''29''' , Cambridge Univ. Press  (1977) {{MR|0442141}} {{ZBL|0446.22006}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.S. Pontryagin,  "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian) {{MR|0201557}} {{ZBL|0022.17104}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L. Fuchs,  "Infinite abelian groups" , '''1''' , Acad. Press  (1970) {{MR|0255673}} {{ZBL|0209.05503}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.E. Humphreys,  "Linear algebraic groups" , Springer  (1975) {{MR|0396773}} {{ZBL|0325.20039}} </TD></TR></table>
  
  
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Hewitt,  K.A. Ross,  "Abstract harmonic analysis" , '''1''' , Springer  (1963)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Spectral theories" , Addison-Wesley  (1977)  (Translated from French)</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Hewitt,  K.A. Ross,  "Abstract harmonic analysis" , '''1''' , Springer  (1963) {{MR|0156915}} {{ZBL|0115.10603}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Spectral theories" , Addison-Wesley  (1977)  (Translated from French) {{MR|0583191}} {{ZBL|1106.46004}} </TD></TR></table>

Revision as of 10:02, 24 March 2012

of a group

The group of all characters of (cf. Character of a group) with values in an Abelian group , under the operation

induced by the operation in . When , then

where are quasi-cyclic groups, one for each prime number . This group is algebraically compact (see Pure subgroup). If is Abelian, then is a divisible group if and only if is torsion free and it is a reduced group if and only if is periodic [4].

The character group of a topological group is the group of all continuous homomorphisms , equipped with the compact-open topology. It is a Hausdorff Abelian topological group. If is locally compact, then so is ; if is compact, then is discrete, and if is discrete, then is compact.

Examples of character groups:

for any finite discrete Abelian group .

With every continuous homomorphism of topological groups there is associated a homomorphism of the character groups . Here the correspondence , , is a contravariant functor from the category of topological groups into the category of Abelian topological groups. If the category is restricted to locally compact Abelian groups , then this functor determines an equivalence of that category and its dual category (see Pontryagin duality).

The character group of an algebraic group over a field is the group of all rational characters . If is an Abelian affine algebraic group, then generates the space (that is, is a basis in this space) if and only if is a diagonalizable algebraic group, i.e. is isomorphic to a closed subgroup of a certain torus . Here is a finitely generated Abelian group (without -torsion if ), and is the group algebra of over , which makes it possible to define a duality between the categories of diagonalizable groups and that of finitely generated Abelian groups (without -torsion if ), cf. [1]. When is a finite group (regarded as a -dimensional algebraic group) and , then this duality is the same as the classical duality of finite Abelian groups.

For any connected algebraic group , the group is torsion free. In particular, a diagonalizable group is a torus if and only if .

References

[1] A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[2] S.A. Morris, "Pontryagin duality and the structure of locally compact Abelian groups" , London Math. Soc. Lecture Notes , 29 , Cambridge Univ. Press (1977) MR0442141 Zbl 0446.22006
[3] L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) MR0201557 Zbl 0022.17104
[4] L. Fuchs, "Infinite abelian groups" , 1 , Acad. Press (1970) MR0255673 Zbl 0209.05503
[5] J.E. Humphreys, "Linear algebraic groups" , Springer (1975) MR0396773 Zbl 0325.20039


Comments

In the article above denotes the circle group. A periodic group is also called a torsion group. An Abelian group is reduced if it contains no non-trivial divisible subgroups.

Above, the phrase "character" is of course strictly used in its narrowest meaning of a (continuous) homomorphism and not in the sense of the character of some representation.

The character groups of many locally Abelian groups can be found in [a1].

References

[a1] E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 1 , Springer (1963) MR0156915 Zbl 0115.10603
[a2] N. Bourbaki, "Elements of mathematics. Spectral theories" , Addison-Wesley (1977) (Translated from French) MR0583191 Zbl 1106.46004
How to Cite This Entry:
Character group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Character_group&oldid=21823
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article