Group ring - Citizendia

This page discusses the algebraic group ring of a discrete group; for the case of a topological group see group algebra, and for a general group see Group Hopf algebra. In Mathematics, the group algebra is any of various constructions to assign to a Locally compact group an Operator algebra (or more generally a Banach In mathematics the group Hopf algebra of a given group is a certain construct related to the symmetries of Group actions.

In mathematics, a group ring is a ring R[G] constructed from a ring R and a group G (written multiplicatively). Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element Sometimes the group ring is written simply as RG.

As an R-module, the ring R[G] is the free module over R on the elements G. In Mathematics, a free module is a Free object in the category of modules Given a set S, a free module on S is a (particular construction If R is a field K, the group ring is called a group algebra; it is a vector space over K, with the basis vectors given by the elements of G. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, the group algebra is any of various constructions to assign to a Locally compact group an Operator algebra (or more generally a Banach In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Basis vector redirects here For basis vector in the context of crystals see Crystal structure. The elements of the group ring are finite linear combinations of elements of G with coefficients in R. Multiplication is defined by the group operation in G extended by linearity and distributivity, and the requirement that elements of R commute with elements of G. The identity element of G is the multiplicative identity of the ring R[G].

If R is commutative, then R[G] is an associative algebra over R, and it is also called the group algebra. In Mathematics, specifically in Ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the

1 Definition
2 Two simple examples
3 Properties
4 Group algebra over a finite group
5 Group rings over an infinite group
6 Representations of a group ring
7 Category theory
8 Generalizations
9 References

Definition

Let G be a group and R a ring. We first define the set R[G] to be one of the following:

The set of all formal R-linear combinations of elements of G.
The free R-module with basis G. In Mathematics, a free module is a Free object in the category of modules Given a set S, a free module on S is a (particular construction
The set of all functions f: G → R with f(g) = 0 for all but finitely many g in G.

No matter which definition is used, we can write elements of R[G] in the form $\sum_{g \in G} a_g g$ , with all but finitely many of the a_g being 0, and an addition is defined on R[G] (by addition of formal linear combinations, addition in the module, or addition of functions, respectively). Multiplication of elements of R[G] is defined by setting

$(\sum_{g \in G} a_g g )( \sum_{h \in G} b_h h ) \ = \ \sum_{g,h \in G} (a_g b_h ) gh.$

If R has a unit element, this is the unique bilinear multiplication for which (1 g)(1 h) = (1 gh). In Mathematics, an algebra is unital (some authors say unitary) if it contains a multiplicative Identity element (or unit) i In this case, G is commonly identified with the elements 1 g of R[G]. The identity element of G then serves as the 1 in R[G].

R is commonly a commutative ring with unit, or even a field.

Two simple examples

Let G = Z₃, the cyclic group of three elements with generator a. In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an An element r of C[G] may be written as

$r = z_1 + z_2 a + z_3 a^2\,$

where z₁, z₂ and z₃ are in C, the complex numbers. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Writing a different element s as

$s=w_1 +w_2 a +w_3 a^2\,$

their sum is

$r + s = z_1+w_1 + (z_2+w_2) a + (z_3+w_3) a^2\,$

and their product is

$rs = z_1w_1 + z_2w_3 + z_3w_2 +(z_1w_2 + z_2w_1 + z_3w_3)a +(z_1w_3 + z_3w_1 + z_2w_2)a^2\,$

When G is a non-commutative group, one must be careful to preserve the order of the group elements (and not accidentally commute them) when multiplying the terms.

A different example is that of the Laurent polynomials: these are nothing more or less than the group ring of the infinite cyclic group Z. In Mathematics, a Laurent polynomial in one variable over a ring R is a Linear combination of positive and negative powers of the variable with In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an

Properties

If R and G are both commutative (i. e. , R is commutative and G is an abelian group), R[G] is commutative. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the

If H is a subgroup of G, then R[H] is a subring of R[G]. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, a subring is a Subset of a ring, which contains the Multiplicative identity and is itself a ring under the same Binary operations Similarly, if S is a subring of R, S[G] is a subring of R[G].

Group algebra over a finite group

Group algebras occur naturally in the theory of group representations of finite groups. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of In Mathematics, a finite group is a group which has finitely many elements The group algebra K[G] over a field K is essentially the group ring, with the field K taking the place of the ring. As a set and vector space, it is the free vector space over the field, with the elements being formal sums:

$x=\sum_{g\in G} a_g g$

The algebra structure on the vector space defined by the multiplication in the group:

$g \cdot h = gh,$

where on the left, g and h indicate elements of the group algebra, while the multiplication on the right is the group operation (written as multiplication). In Mathematics, a free module is a Free object in the category of modules Given a set S, a free module on S is a (particular construction In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with

Because the above multiplication can be confusing, one can also write the basis vectors of K[G] as e_g (instead of g), in which case the multiplication is written as:

$e_g \cdot e_h = e_{gh}.$

Thinking of the free vector space as K-valued functions on G, the algebra multiplication is convolution of functions. Basis vector redirects here For basis vector in the context of crystals see Crystal structure. In Mathematics, a free module is a Free object in the category of modules Given a set S, a free module on S is a (particular construction In Mathematics and in particular Functional analysis, convolution is a mathematical operation on two functions f and

Regular representation

The group algebra is an algebra over itself; under the correspondence of representations over R and R[G] modules, it is the regular representation of the group. In Mathematics, and in particular the theory of Group representations the regular representation of a group G is the Linear representation afforded

Written as a representation, it is the representation g $\mapsto$ ρ_g with the action given by $\rho(g)\cdot e_h = e_{gh}$ , or

$\rho(g)\cdot r = \sum_{h\in G} k_h \rho(g)\cdot e_h = \sum_{h\in G} k_h e_{gh}$

Properties

The dimension of the vector space K[G] is just equal to the number of elements in the group. The field K is commonly taken to be the complex numbers C or the reals R, so that one discusses the group algebras C[G] or R[G].

The group algebra C[G] of a finite group over the complex numbers is a semisimple ring. In Mathematics, especially in the area of Abstract algebra known as Module theory, a semisimple module or completely reducible module is a type This result, Maschke's theorem, allows us to understand C[G] as a finite product of matrix rings with entries in C. Maschke's theorem is a theorem in Group representation theory which concerns the decomposition of representations of a Finite group into irreducible pieces In Mathematics, it is possible to combine several rings into one large product ring. In Abstract algebra the matrix ring M( n, R) is the set of all n × n matrices over an arbitrary ring

Representations of a group algebra

Taking K[G] to be an abstract algebra, one may ask for concrete representations of the algebra over a vector space V. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of Such a representation

$\tilde{\rho}:K[G]\rightarrow \mbox{End} (V)$ .

is an algebra homomorphism from the group algebra to the set of endomorphisms on V. In Mathematics, an endomorphism is a Morphism (or Homomorphism) from a mathematical object to itself Taking V to be an abelian group, with group addition given by vector addition, such a representation in fact a left K[G]-module over the abelian group V. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars That this is so is exhibited below, where each axiom of a module is demonstrated.

Pick r ∈ K[G] so that

$\tilde{\rho}(r) \in \mbox{End}(V)$ .

Then $\tilde{\rho}(r)$ is a homomorphism of abelian groups, in that

$\tilde{\rho}(r) \cdot (v_1 +v_2) = \tilde{\rho}(r) \cdot v_1 + \tilde{\rho}(r) \cdot v_2$

for any v₁, v₂ ∈ V. Next, one notes that the set of endomorphisms of an abelian group is an endomorphism ring. In Abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object The representation $\tilde{\rho}$ is a ring homomorphism, in that one has

$\tilde{\rho}(r+s)\cdot v = \tilde{\rho}(r)\cdot v + \tilde{\rho}(s)\cdot v$

for any two r, s ∈ K[G] and v ∈ V. Similarly, under multiplication,

$\tilde{\rho}(rs)\cdot v = \tilde{\rho}(r)\cdot \tilde{\rho}(s)\cdot v$

Finally, one has that the unit is mapped to the identity:

$\tilde{\rho}(1)\cdot v = v$

where 1 is the multiplicative unit of K[G]; that is,

$1 = e_e\,$

is the vector corresponding to the identity element e in G.

The last three equations show that $\tilde{\rho}$ is a ring homomorphism from K[G] taken as a group ring, to the endomorphism ring. The first identity showed that individual elements are group homomorphisms. Thus, a representation $\tilde{\rho}$ is a left K[G]-module over the abelian group V.

Note that given a general K[G]-module, a vector-space structure is induced on V, in that one has an additional axiom

$\tilde{\rho}(ar) \cdot v_1 + \tilde{\rho}(br) \cdot v_2 = a \tilde{\rho}(r) \cdot v_1 + b \tilde{\rho}(r) \cdot v_2 = \tilde{\rho}(r) \cdot (av_1 +bv_2)$

for scalar a, b ∈ K.

Any group representation

$\rho:G\rightarrow \mbox{Aut}(V)$ ,

with V a vector space over the field K, can be extended linearly to an algebra representation

$\tilde{\rho}:K[G]\rightarrow \mbox{End}(V)$ ,

simply by mapping $\rho(g) \mapsto \tilde{\rho}(e_g)$ . Thus, representations of the group correspond exactly to representations of the algebra, and so, in a certain sense, talking about the one is the same as talking about the other.

Center of a group algebra

The center of the group algebra is the set of elements that commute with all elements of the group algebra:

$Z(K[G]) := \left\{ z \in K[G] \mid zr = rz \mbox{ for all } r \in K[G]\right\}$ . In Abstract algebra, the center of a group G is the set Z ( G) of all elements in G which commute with all the

The center is equal to the set of class functions, that is the set of elements that are constant on each conjugacy class: $Z(K[G]) = \left\{ \sum_{g \in G} a_g g: \text{for all } g,h \in G, a_g = a_{h^{-1}gh}\right\}$

If $K=\mathbb{C}$ , the set of irreducible characters of $G$ forms an orthonormal basis of $Z (K [G])$ with respect to the inner product $\langle \sum_{g \in G} a_g g, \sum_{g \in G} b_g g \rangle = \frac{1}{|G|} \sum_{g \in G} \bar{a_g} b_g$ . In Mathematics, especially in the fields of Group theory and representation theory of groups, a class function is a function f on This article refers to the use of the term character theory in mathematics for the media studies definition see Character theory (Media.

Group rings over an infinite group

Much less is known in the case where G is countably infinite, or uncountable, and this is an area of active research. The case where R is the field of complex numbers is probably the one best studied. In this case, Irving Kaplansky proved that if a and b are elements of C[G] with ab = 1, then ba = 1. Irving Kaplansky ( March 22, 1917 &ndash June 25, 2006) was a Canadian Mathematician. Whether this is true if R is a field of positive characteristic remains unknown.

If G is torsion-free, it is conjectured that C[G] has no nontrivial idempotents or zero divisors; this has been proven for special cases, such as the ones where G is abelian, elementary amenable, or free. In Abstract algebra, the term torsion refers to a number of concepts related to elements of finite order in groups and to the failure of modules to be Idempotence ˌaɪdɨmˈpoʊtəns describes the property of operations in Mathematics and Computer science which means that multiple applications of the operation In Abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0 An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, a group is called elementary amenable if it can be built up from finite groups and abelian groups by a sequence of simple operations that result in In Mathematics, a group G is called free if there is a Subset S of G such that any element of G can be

The case of G being a topological group is discussed in greater detail in the article on group algebras. In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the In Mathematics, the group algebra is any of various constructions to assign to a Locally compact group an Operator algebra (or more generally a Banach

Representations of a group ring

A module M over R[G] is then the same as a linear representation of G over the field R. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of There is no particular reason to limit R to be a field here. However, the classical results were obtained first when R is the complex number field and G is a finite group, so this case deserves close attention. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted It was shown that R[G] is a semisimple ring, under those conditions, with profound implications for the representations of finite groups. In Mathematics, especially in the area of Abstract algebra known as Module theory, a semisimple module or completely reducible module is a type More generally, whenever the characteristic of the field R does not divide the order of the finite group G, then R[G] is semisimple (Maschke's theorem). In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's Maschke's theorem is a theorem in Group representation theory which concerns the decomposition of representations of a Finite group into irreducible pieces

When G is a finite abelian group, the group ring is commutative, and its structure is easy to express in terms of roots of unity. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, the n th roots of unity, or de Moivre numbers are all the Complex numbers that yield 1 when raised to a given power When R is a field of characteristic p, and the prime number p divides the order of the finite group G, then the group ring is not semisimple: it has a non-zero Jacobson radical, and this gives the corresponding subject of modular representation theory its own, deeper character. In Ring theory, a branch of Abstract algebra, the Jacobson radical of a ring R is an ideal of R which contains those elements Modular representation theory is a branch of Mathematics, and is that part of Representation theory which studies Linear representations of Finite group

Category theory

Categorically, the group ring construction is left adjoint to "group of units"; the following functors are an adjoint pair:

$\operatorname{GrpRng}\colon \mathbf{\operatorname{Grp}} \to R\mathbf{\operatorname{-Alg}}$

$\operatorname{GrpUnits}\colon R\mathbf{\operatorname{-Alg}} \to \mathbf{\operatorname{Grp}}$

where "GrpRng" takes a group to its group ring over R, and "GrpUnits" takes an R-algebra to its group of units. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i

When R = Z, this gives an adjunction between the category of groups and the category of rings. In Mathematics, the category Grp has the class of all groups for objects and Group homomorphisms for Morphisms As such In Mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity and whose Morphisms

Generalizations

The group algebra generalizes to the categorical algebra, of which another example is the incidence algebra. In Category theory, a field of Mathematics, a categorical algebra is an Associative algebra, defined for any locally finite category and In Order theory, a field of Mathematics, an incidence algebra is an Associative algebra, defined for any locally finite Partially ordered set

References

A. A. Bovdi (2001), “Group algebra”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
C. The Encyclopaedia of Mathematics is a large reference work in Mathematics. W. Curtis, I. Reiner, Representation theory of finite groups and associative algebras, Interscience (1962)
D. S. Passman, The algebraic structure of group rings, Wiley (1977)

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Contents

Definition

Two simple examples

Properties

Group algebra over a finite group

Regular representation

Properties

Representations of a group algebra

Center of a group algebra

Group rings over an infinite group

Representations of a group ring

Category theory

Generalizations

References