This page is about finite semisimple \(\mathbb{R}\)-algebra.
(Artin-Wedderburn) Suppose \(A\) is a semisimple algebra over a field \(k\). Then for some finite-dimensional \(k\)-division algebras \(D_1, D_2, \dots, D_r\) and natural numbers \(n_1, \dots , n_r\), we get the isomorphism \[\begin{align} A \simeq M_{n_1}(D_1) \oplus \dots \oplus M_{n_r}(D_r). \label{eq:semisimple} \end{align}\]
The right side of the Equation is always semisimple for any choice of finitely many finite-dimensional \(k\)-division algebras. Thus, any reader who is not familiar with these objects could take the definition of semisimple \(k\)-algebras as the object on the right side.
(Frobenius) The only finite-dimensional \(\mathbb{R}\)-divison algebras (up to isomorphism) are \(\mathbb{R}\), \(\mathbb{C}\) and \(\mathbb{H}\).
The three \(\mathbb{R}\)-division algebras all have a special conjugation involution that is compatible with the canonical inclusion \(\mathbb{R} \hookrightarrow \mathbb{C} \hookrightarrow \mathbb{H}\). The map \(\overline{(\ )}:\mathbb{H} \rightarrow \mathbb{H}\) given as \(a + i b + j c + k d \mapsto a - i b - j c - k d\) (\(a,b,c,d \in \mathbb{R}\) and \(i,j,k\) canonically span \(\mathbb{H}\)) satisfies that for any \(x,y \in \mathbb{H}\) we have \(\overline{x.y} = \overline{y}. \overline{x}\). When restricted to \(\mathbb{C}\), this is the usual complex conjugation and when restricted to \(\mathbb{R}\), this is the idenitity map. Another important property is that for any \(a + i b + j c + k d =x \in \mathbb{H}\), \(\overline{x} x = a^{2} + b^{2} + c^{2} + d^{2} \in \mathbb{R}_{\ge 0}\).
The two theorems stated above give rise to the following proposition.
Any semisimple \(\mathbb{R}\)-algebra is isomorphic to one of products of matrix algebras over \(\mathbb{R}\), \(\mathbb{C}\) and \(\mathbb{H}\).
Matrix algebras over \(\mathbb{R}\), \(\mathbb{C}\) and \(\mathbb{H}\) are well understood. One important property is that the conjugation map defined above can be extended to a conjugate transpose involution on such matrices by simply defining the mapping \([x_{ij}]^{*} = [\overline{x_{ji}}]\). With this, we can also define a positive definite quadratic form on these matrix algebras by sending \(a \mapsto \mathop{\mathrm{tr}}( a^{*} a)\).
On a given finite-dimensional algebra over \(\mathbb{R}\), it is possible to define the trace map \(\mathop{\mathrm{tr}}_{A}: A \rightarrow \mathbb{R}\) and the norm map \(\mathop{\mathrm{N}}_{A}: A \rightarrow \mathbb{R}\) as the trace and the determinant of the matrix of the left-multiplication operation induced by any element. Similarly, it is also possible to generalize the above involution simply by taking direct sums of the respective involutions for matrix rings over \(\mathbb{R}, \mathbb{C}\) or \(\mathbb{H}\). We will omit the subscripts in \(\mathop{\mathrm{tr}}_{A}\) and \(\mathop{\mathrm{N}}_{A}\) when \(A\) is clear from the context.
Any semisimple \(\mathbb{R}\)-algebra \(A\) admits an involution \((\ )^{*} : A \rightarrow A\) such that the following conditions are satisfied.
Simply take the direct sum of the conjugate transpose operation defined above on each matrix component of the semisimple algebra \(A\). It is then to be seen that the trace function on \(A\) is a sum of traces on matrices rings over \(\mathbb{R},\mathbb{C},\mathbb{H}\) when they are realized as real matrix algebras. For instance, we must see \(M_1(\mathbb{C})\) as a \(2\)-dimensional matrix algebra under the mapping \(a+ib \mapsto \left[ \begin{smallmatrix} a & -b \\ b & a\end{smallmatrix}\right]\).
Any involution \(A \rightarrow A\) satisfying the two properties of Corollary is said to be a positive involution on \(A\).
Suppose \((\ )^{*}: A \rightarrow A\) is a positive involution. Then
The proofs are very enjoyable, so we leave all of them for the reader. The third one will require the use of semisimplicity of \(A\), which implies that the left-multiplication trace and right-multiplication trace are the same.
The notions of symmetric and positive definiteness can also be defined for \((A,(\ ) ^{*})\).
Given a finite-dimensional semisimple \(\mathbb{R}\)-algebra and an involution \((\ )^{*}\) as mentioned in Corollary , we shall call an element \(a \in A\)
The first is a trivial verification.
For the second, note that if \(a\) is a zero divisor then there exists some non-zero \(x \in A\) such that \(ax=0 \Rightarrow \mathop{\mathrm{tr}}(x^{*} a x) = 0\) which contradicts the positive definiteness of \(a\). Finally \(\mathop{\mathrm{tr}}(a) = \mathop{\mathrm{tr}}(1^{*}_{A} a 1_{A}) > 0\).
See Norm-Trace.
A type of Gram-Schmidt process can be done for the real-semisimple algebra. This is in fact, related to the Cholesky decomposition too.
For any finite-dimensional division algebra \(D\) over \(\mathbb{Q}\), \(D \otimes_{\mathbb{Q}} \mathbb{R} = D_{\mathbb{R}}\) is a real semisimple algebra.
Why is \(D_{\mathbb{R}}\) semisimple? The trace form \((a,b)\mapsto \mathop{\mathrm{tr}}(ab)\) is clearly non-degenerate on \(D\). It is classically known that the trace form on a finite-dimensional \(k\)-algebra is non-degenerate if and only if it is absolutely semisimple, i.e. \(A \otimes_{k} L\) is semisimple for any field extension \(L\) of \(k\).
See division algebra for more info.
This page was updated on October 12, 2021.
Main Page