Gruppenalgebren

Gruppenalgebren, also known as group algebras or group rings, are algebraic structures formed from a group and a ring. Specifically, a group algebra of a group G over a ring R is constructed by taking R as the set of coefficients and the elements of G as the basis. The multiplication of basis elements is defined by the group operation in G, and the distributive property extends this to the entire algebra.

Formally, if G is a group and R is a ring, the group algebra RG is the set of all formal finite sums of the form $\sum_{g \in G} r_g g$, where $r_g \in R$ and only a finite number of $r_g$ are non-zero. Addition is defined component-wise, and multiplication is defined by extending the group law and the distributive property: $(\sum r_g g)(\sum s_h h) = \sum_{g,h \in G} (r_g s_h) (gh)$. The ring R is typically taken to be a field, such as the field of real numbers or complex numbers, in which case RG becomes a vector space over R.

Group algebras are fundamental objects in the study of group representations, abstract algebra, and mathematical physics. They provide a way to study group properties through the lens of algebra. For instance, the representation theory of a group G can be studied by examining the module theory of its group algebra RG. Properties of the group G, like its commutativity or nilpotency, often translate into specific algebraic properties of RG. The concept has applications in various areas, including the study of finite groups, infinite groups, and topological groups.