Group Algebra

see also:

• Group Representation Theory
• Decomposition

The group algebra is a unifying structure in mathematics that blends group theory with linear algebra, providing a powerful framework for studying groups and their representations. It plays a crucial role in representation theory, harmonic analysis, and other areas where groups and vector spaces intersect.

Definition

Given a group $G$ and a field $F$, the group algebra $F[G]$ is a vector space over $F$ with a basis consisting of the elements of $G$. This vector space is also equipped with a multiplication operation that extends the group operation linearly.

Formally, elements of $F[G]$ are formal sums of the form

$$\sum _{g\in G}{a}_{g}g,$$

where $a_g\in F$ for all $g\in G$, and only finitely many $a_g$ are non-zero. The addition in $F[G]$ is defined component-wise, and the multiplication is defined by distributively extending the group operation, making $F[G]$ an algebra over $F$.

Properties

• Algebra Structure: $F[G]$ is an associative algebra with the identity element being the identity of the group $G$ with a coefficient of 1. It’s associative because the group operation is associative, and it respects the scalar multiplication of the field $F$.

• Dimensionality: If $G$ is a finite group of order $n$, then $F[G]$ is an $n$-dimensional vector space over $F$. Each group element corresponds to a basis vector in this vector space.

• Center of $F[G]$: The Center Of The Group Algebra, denoted $Z(F[G])$, consists of all elements in $F[G]$ that commute with every element of $F[G]$. These often correspond to sums over conjugacy classes in $G$ and have special significance in the study of representations.

Applications

• Representation Theory: The group algebra forms the foundational setting for studying group representations. Representations of $G$ can be equivalently viewed as algebra homomorphisms from $F[G]$ to the algebra of matrices over $F$. This perspective is particularly useful for classifying and decomposing representations.

• Harmonic Analysis: In harmonic analysis, especially when $G$ is a finite group or a compact Lie group, the group algebra serves as a domain for the Fourier transform on $G$, allowing for the analysis and synthesis of functions on $G$.

• Non-commutative Geometry and Quantum Groups: In more advanced settings, group algebras, especially those associated with infinite or quantum groups, play a role in non-commutative geometry, where they help generalize geometric concepts to algebraic settings lacking a conventional space.

Example

For a simple example, consider the group $\mathbb{Z}/2\mathbb{Z}=\{0,1\}$ with addition modulo 2, and let $F=\mathbb{R}$. The group algebra $\mathbb{R}[\mathbb{Z}/2\mathbb{Z}]$ consists of all real linear combinations of the elements $0$ and $1$, which can be thought of as the set of all vectors $\{a\cdot 0+b\cdot 1|a,b\in \mathbb{R}\}$. Multiplication in this algebra reflects the group structure, so, for example, $1\cdot 1=0$ because $1+1=0\mathrm{mod}2$.

The group algebra encapsulates both the algebraic structure of the group and the linear structure of the field, serving as a bridge between abstract group theory and the more concrete world of linear algebra and vector spaces.