Let $M$ be the Möbius band and $\partial M$ its boundary. Consider an embedding $g:M\to S^3$. I have to determine the homomorphism $i_*:H_1(S^3-g(M))\to H_1(S^3-g(\partial M))$ induced by the inclusion.
Firs of all, I've computed both homology groups. For $H_1(S^3-g(M))$ I divided $M$ into rect...10
A quaternion algebra is a generalization of the classical Hamiltonian quaternions. It is a $4$-dimensional algebra over a field $F$ with basis $1,i,j,ij$ subject to the relations
$$
i^2 = a, \quad j^2 = b, \quad ji = -ij
$$
for some $a,b \in F^\times$. (This is the definition for $\operatorname...
15
I agree with course of action (1): Expand the scope of quaternions to include quaternion algebras.
Partially because they are closely related, partially because this matches existing usage patterns, and partially because there are not enough questions of this type to merit a new tag.
Complete...
In mathematics, a quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes the matrix algebra by extending scalars (equivalently, tensoring with a field extension), i.e. for a suitable field extension K of F,
A
⊗
F
K
{\displaystyle A\otimes _{F}K}
is isomorphic to the 2×2 matrix algebra over K.
The notion of a quaternion algebra can be seen as a generalization of Hamilton's quaternions to an arbitrary base field...
