二维 $\mathbb R$-代数分类

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$\mathbb R^2$上的通常加法为$(a,b)+(c,d)=(a+c,b+d)$, 我们知道$(\mathbb R^2,+)$是一个群.
问题: 有几种$\times$的定义, 使得$(\mathbb R^2,+,\times)$是一个环?

• 直积 $(a,b)×(c,d)=(ac,bd)$
• $\Bbb C$的$(a,b)×(c,d)=(ac-bd,ad+bc)$
• Dual number  $(a,b)(c,d)=(ac,ad+bc)$

这是3种$\times$的定义.

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对$\Bbb R$上的任意不可约二次多项式$x^2+bx+c$
则$\Bbb R[x]/⟨x^2+bx+c⟩$到$\Bbb R^2$的线性映射$1\mapsto(1,0),x\mapsto(0,1)$给出$\Bbb R^2$上的一个乘法, $\Bbb R^2$关于此乘法成为一个环.

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Czhang271828 发表于 2023-1-28 07:10
最后的结果确实是上述三种...

又看到一个split-complex number与$\mathbb R\oplus\mathbb R$环同构
Relativistic dot products and complex numbers | NJ Wildberger
$$(a,b)(c,d)=(ac+bd,ad+bc)$$

Isomorphism
On the basis {e, e*} it becomes clear that the split-complex numbers are ring-isomorphic to the direct sum R⊕R with addition and multiplication defined pairwise.

$\mathbb R[x]/(x^2-1)$.

From the definition it is apparent that the ring of split-complex numbers is isomorphic to the group ring $ \mathbb {R} [C_2] $ of the cyclic group $C_2$ over the real numbers $ \mathbb {R} . $

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对应于 对角矩阵$\pmatrix{a\\&b}$、对称矩阵$\pmatrix{a&b\\b&a}$、反对称矩阵$\pmatrix{a&-b\\b&a}$均为$GL(2,\mathbb R)$的二维子环?

Czhang271828

hbghlyj 发表于 2023-3-15 22:15
对应于 对角矩阵$\pmatrix{a\\&b}$、对称矩阵$\pmatrix{a&b\\b&a}$、反对称矩阵$\pmatrix{a&-b\\b&a}$均为$ ...

Let $\{a,b\}$ be a basis of $A$, a $2$-dimensional $\mathbb R$-algebra. Here we can also regard $a$ and $b$ as $2\times 2$ real matrices acting on some fixed basis. Denote $\chi _M$ as characteristic polynomial of matrix $M$.  

If $\dim \chi_a=1$, then we set $a=1$ without the loss of generality. Since $\{a,b\}$ are linearly independent, $\dim \chi _b=2$. Such $A$ is isomorphic to $\mathbb R[X]/(X^2+cX+d)$, which is isomorphic to

• $\mathbb R[X]/(X^2+1)\simeq \mathbb C$ for $\Delta<0$,
• $\mathbb R[X]/(X^2)\simeq \text{dual number}$ for $\Delta=0$,
• $\mathbb R[X]/(X^2-1)\simeq \text{split }\mathbb C$ for $\Delta >0$.
  

Consider the case $\dim \chi _a=\dim \chi _b=2$. Whenever both are non-diagonalisable, or both are diagonalisable, then there exists $\mu,\lambda \in \mathbb R$ such that $\lambda a+\mu b=1$. This is what we have discussed previously.  

Now we focus on $\dim \chi _a=\dim \chi _b=2$, and $a$ is diagonalisable yet $b$ is not. Since $a\in \mathrm{span}_{\mathbb R}(b,b^2)$, $a$ commutes with $b$. If we write $a=\lambda b+\mu b^2$, then $\lambda =0$. Thus $A\simeq \mathbb C$.

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lectures18.pdf

$\left\{\left(\begin{array}{ll}a & b \\ b & a\end{array}\right): a, b \in \mathbb{R}\right\} \cong\Bbb R^2$

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Associative Composition Algebra

For a sample of AC algebra, the following is offered: Let A = (R², xy) be the real plane with quadratic form xy. Further, let A be equipped with component-wise addition and multiplication, making it a real algebra. Denote N(x,y) = xy in this case. Then

$\displaystyle N(x_{1},y_{1})N(x_{2},y_{2})=(x_{1}y_{1})(x_{2}y_{2})=x_{1}x_{2}y_{1}y_{2}=N((x_{1},y_{1})(x_{2},y_{2})).$

Thus N is said to compose over the multiplication in A, and A might be called a composition algebra. However, in this text, AC algebras have an involution called a conjugation, written x*, used to define N by N(x) = x x*. Nevertheless, the algebra A constructed above is very closely related to split-binarions described in the next chapter. The split-binarions are a normalized form of A, where the multiplicative identity is a unit distance from the origin, and it has some formal correspondences with the complex field C. In A, the quadratic form can be interpreted as a weight, so that a transformation leaving it invariant is an isobaric transformation, a name used in 1999 by Peter Olver (Classical Invariant Theory, page 217) to describe a squeeze mapping.

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Extension Ring
$\mathbb{C}$ is the only extension of $\mathbb{R}$ of degree 2