Exceptional symmetric spaces with quaternionic structure

Following this and this question I found following chain of exceptional symmetric spaces being quaternionic manifolds. I listed dimensions as superscripts for reader convenience. F28I⊂E40II⊂E64VI⊂E112IX It corresponds to following chain of algebras: ˜O⊂C⊗˜O⊂H⊗˜O⊂O⊗˜O ˜O is algebra of split octonions. One can see that codimensions of above three inclusions are 12, 24, 48. The above symmetric … Read more

Quaternion orders such that every proper ideal is invertible

Let $B$ be a quaternion algebra over $\mathbb{Q}$ and let $\mathcal{O} \subset B$ be an order. A lattice in $B$ is (left) proper over $\mathcal{O}$ if its left order is equal to $\mathcal{O}$. We say $\mathcal{O}$ is good if every lattice in $B$ proper over $\mathcal{O}$ is invertible (equivalently, locally principal; equivalently, projective as a … Read more

Why is this mapping surjective?

It is mentioned in wikipedia that every single orthogonal $3 \times 3$ rational matrix is of the form $$\dfrac{1}{m^2+n^2+p^2+q^2}\begin{pmatrix} m^2+n^2-p^2-q^2 & 2np-2mq & 2mp+2nq \\ 2mq+2np & m^2-n^2+p^2-q^2 & 2pq-2mn \\ 2nq-2mp & 2mn+2pq & m^2-n^2-p^2+q^2\end{pmatrix}$$ For $m,n,p,q \in \mathbb Q$. This statement refers to J. Cremona, Letter to the Editor, Amer. Math. Monthly 94. … Read more

Left- and right-sided principal ideals of quaternions have same index?

One fact about the Lipschitz integers (quaternions of the form $a + bi + cj + dk$ where $a, b, c, d$ are integers) is that the left-sided ideal generated by any element $Q$ has the same index in the additive group as does the right-sided ideal generated by $Q$. I know this is a … Read more

What does the Jacquet-Langlands correspondence say about quaternion algebras of class number one?

If F is a totally real number field of degree n, and A is a definite quaternion algebra over F, I understand (not really) the Jacquet Langlands correspondence to construct a modular form in n variables out of a linear combination of conjugacy classes of maximal orders in A. When A has class number one, … Read more

Inner forms of GL(2)GL(2)

I know that inner forms of GL(2) are quaternion algebras. However, I cannot find the proof myself. First, since quaternion algebras are forms of GL(2) by the usual embedding in matrices, they are automatically inner by Skolem-Noether theorem. But how can I prove the converse, that is to say if I have a group “inner … Read more

Polar decomposition for quaternionic matrices?

A non-zero complex number can be uniquely written in polar form as reiθ. There is an analogous result for complex matrices: any invertible complex matrix can be uniquely written as UP, where U is a unitary matrix and P is a positive definite Hermitian matrix (see e.g. the description on Wikipedia) Suppose we consider n×n … Read more

Is an associative division algebra required for this phenomenon?

For which integers d≥1 can we find real matrices R1,…,Rd of size d×d such that for any unit vector v∈Rd, R1v,…,Rdv is an orthonormal basis? Note that the chosen set of Ris has to work simultaneously for all v. (Does this phenomenon have a name?) It may not be obvious at first, but this question … Read more

Generalizing contour integration to quaternions and bicomplex numbers

I am interested in the possibility of generalizing the notion of contour integration to the quaternions or bicomplex numbers. I am aware that the Frobenius theorem prevents the construction of a true division algebra in R3. Likewise, Liouville’s theorem requires all three-dimensional conformal maps to be a composition of Mobius transforms. If I sacrificed commutativity, … Read more

Hochschild cohomology of SU(2)

I have a question about the computation of an Hochschild Cohomology. Or at least about a space which really looks like a cohomology space. All the functions i consider are assumed to be smooth. Let’s consider the right action of SU(2) on its tangent algebra (the 2×2 anti hermitian matrices, denoted by iH2) by : … Read more