# Decompositions of $\ast$-Algebras

Every finite-dimensional $K$-algebra $A$ has a decomposition $A=J\oplus W$, where $J$ is the Jacobson radical of $A$ and $W$ is a semisimple subring of $A$. We refer to such decompositions as *Wedderburn decompositions*. The procedure that computes Wedderburn decompositions is adapted from an analogous Magma function written by W. de Graaf for algebras defined by structure constants.

If $A$ is a $*$-algebra and the characteristic of $K$ is not 2, then it follows from a result of Taft [[Taft, 1957](../../references.md#cite-taft)] that $A$ has a Wedderburn decomposition of the form $A=J\oplus T$ in which $T$ is invariant under the involution of $A$. We refer to such decompositions as *Taft decompositions*. The procedure that computes Taft decompositions is based on Taft’s original proof, and is described in [[Brooksbank and Wilson, 2012](../../references.md#cite-brooksbankwilson1), Proposition 4.3].

## `WedderburnDecomposition(A): AlgMat -> AlgMat, AlgMat`

## `WedderburnDecomposition(A): AlgGrp -> AlgGrpSub, AlgGrpSub`

A Wedderburn decomposition is constructed for the $*$-algebra $A$. Specifically, the Jacobson radical, $J$, of $A$, and a semisimple complement, $W$, to $J$ in $A$ are computed. Here $A$ may be either a matrix algebra or a group algebra over any field.

## `TaftDecomposition(A): AlgMat -> AlgMat, AlgMat`

## `TaftDecomposition(A): AlgGrp -> AlgGrpSub, AlgGrpSub`

A Taft decomposition is constructed for the $*$-algebra $A$. Specifically, the Jacobson radical, $J$, of $A$ and a $*$-invariant Wedderburn complement to $J$ in $A$ are computed. This function requires that the base ring of $A$ has characteristic different from 2. Here $A$ may be either a matrix $*$-algebra or a group algebra.

## `Example: Taft Decomposition (ex-d6a904)`

We compute a Wedderburn decomposition of the group algebra ${\rm GF (5)}[A_5]$ equipped with its natural involution.

```magma
> K := GF(5);
> G := AlternatingGroup(5);
> A := GroupAlgebraAsStarAlgebra(K, G);
> J, W := WedderburnDecomposition(A);

```

We check dimensions and the $*$-invariance of $T$.

```magma
> Dimension(J); Dimension(W);
25
35
> forall { i : i in [1..Ngens (W)] | W.i@Star(A) in W };
false

```

Now find a $*$-invariant decomposition.

```magma
> J, T := TaftDecomposition(A);
> Dimension(J); Dimension(W);
25
35
> forall { i : i in [1..Ngens(T)] | T.i@Star(A) in T };
true

```