# Algebra

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Related KMR-pages:

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Other relevant sources of information:

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Anchors into the text below

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Emmy Noether: The Newton of Algebra and the grandmother of Category Theory:

Emmy Noether (at Wikipedia)
Emmy Noether, BBC: In Our Time. Melvyn Bragg and guests discuss the ideas
of one of the great 20th-century mathematicians.
Emmy Noether (at famousmathematicians.net)
• The origins of mathematical abstraction, by Saunders Mac Lane, 1999.
• Category theory

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Saunders Mac Lane and Samuel Eilenberg: The fathers of category theory

Categories for the Working Mathematician, by Saunders Mac Lane, Springer Verlag, 1971.

/////// ‘algebra’ versus ‘an algebra’
‘algebra’_versus_’an algebra’

What is ‘algebra‘ and what is meant by ‘an algebra‘ ?

By the collective (= non-plurizable) term ‘algebra‘ one usually means the domain that studies all forms of different computational systems.

Definition: In this context the expression ‘an algebra‘ will refer to a computational system involving two operations where at least one of them distributes over the other.

REMARK: In a boolean algebra both operations have this property.

The operations of an algebra will are called ‘addition’ and ‘multiplication’ and denoted by ‘plus’ ($\, + \,$) respectively ‘times’ ($\, * \,$). The distributive relation between addition and multiplication can then be verbalized as “multiplication distributes over addition” and ‘formulized’ as

$\,\,\,\,\,\,\, (a \, + \, b) * (c \, + \, d) \, = a * c \, + \, a * d \, + \, b * c \, + \, b * d \,$.

NOTATION: If $\, X \,$ is an algebra over the algebra $\, Y \,$ under the conditions $\, C \,$,
then one says that $\, "X \,$ is a $\, C$-algebra over $\, Y \, "$.

Example 1:

The (commutative) algebra of polynomials in one variable with real coefficients.

A typical element of this algebra, a typical polynomial of one variable, can be expressed as

$\, c_0 \, + \, c_1 x \, + \, c_2 x^2 \, + \, \cdots \, + \, c_n x^n$,

where $\, x \,$ denotes the variable and $\, {c_0, \, c_1, \, \cdots \, , c_n} \,$ denote the real coefficients.

Example 2: The clifford algebra $\, C_l(e_1, e_2, e_3) \,$ in three variables $\, e_1, e_2, e_3 \,$
over the (algebra of) real numbers $\, \mathbb{R}$.

Let $\, {\mathbb{E}}^3 \,$ denote the ordinary Euclidean 3-space with an orthonormal basis consisting of the vectors $\, {e_1, e_2, e_3}$. Orthogonality means that we have $\, e_1 \perp e_2 \, , \, e_2 \perp e_3 \, , \, e_3 \perp e_1 \,$ and orthonormality means that we also have $\, e_1 \cdot e_1 \, = \, e_2 \cdot e_2 \, = \, e_3 \cdot e_3 \, = 1$.

The exterior product $\, a \wedge b \,$ of two vectors $\, a \,$ and $\, b \,$ in $\, {\mathbb{E}}^3 \,$ represents an oriented, planar patch of surface, whose normal vector is parallel to the cross product $\, a \times b \,$ and whose area is equal to the length of this vector. The orientation of the planar surface patch $\, a \wedge b \,$ is seen in the anti-symmetry (= the sign-change) when we switch the order of $\, a \,$ and $\, b \,$:

$\, a \wedge b \, = \, - \, b \wedge a \,$.

Geometrically this corresponds to changing the direction of the surface normal.

Let $\, a \,$ and $\, b \,$ be two non-zero vectors in $\, {\mathbb{E}}^3$. From the properties of the cross product it follows that

$\, a \wedge b = 0 \,$ if and only if $\, a \,$ is parallel to $\, b$, a configuration that we denote by $\, a \parallel b$.

Given two vectors $\, a \, , \, b \, \in {\mathbb{E}}^3$, William Kingdon Clifford defined their geometric product $\, a \, b \,$ by

Definition: $\, a \, b \, \stackrel {\mathrm{def}}{=} \, a \cdot b \, + \, a \wedge b$.

We then have $\, e_1 \, e_1 = e_1 \cdot e_1 + e_1 \wedge e_1 = 1 + 0 = 1$, and similar for $\, e_2 \,$ and $\, e_3$.

We can write these relationships as $\; e_1^2 \, = \, e_2^2 \, = \, e_3^2 \, = \, 1$.

It is also evident from the definition of the geometric product that for two non-zero vectors $\, a \,$ and $\, b \,$ we have $\, a \, b \, = \, a \wedge b \,$ if and only if $\, a \perp b$. Hence, for the orthonormal basis vectors $\, e_i \in {\mathbb{E}}^3$, we have if $\, i \neq j \,$:

$\, e_i \, e_j \, = \, e_i \wedge e_j \, = \, - \, e_j \wedge e_i \, = \, - \, e_j \, e_i$.

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FACT: The non-commutative polynomial algebra $\, {\mathbb{R}}\{e_1, e_2, e_3\} \,$ in three variables
with real coefficients is a clifford algebra over the real numbers $\, \mathbb{R}$.

NOTATION: This algebra is called $\, C_l({\mathbb{E}}^3)$ or $\, C_l(e_1, e_2, e_3)$.

The Clifford conditions are given by

(1) $\,\,\,\,\, e_1^2 = e_2^2 = e_3^2 = 1$, and

(2) $\,\,\,\,\, e_k \, e_i = - \, e_i \, e_k \,$ if $\, k \neq i$.

From the associativity of the geometric product, and from its antisymmetry on the pairs of basis vectors $\, e_1, e_2, e_3$, it follows that:

(3) $\,\,\,\,\, (e_i e_k)^2 = e_i e_k e_i e_k = - e_k e_i e_i e_k = - e_k e_k = -1 \,$ if $\, k \neq i$, and that
(4) $\,\,\,\,\, (e_1 e_2 e_3)^2 = e_1 e_2 e_3 e_1 e_2 e_3 = (-1)^2 e_1 e_1 e_2 e_3 e_2 e_3 = - e_2 e_2 e_3 e_3 = -1$.

Definitions:

1): The polynomials of vectors that appear in any clifford algebra
are called multivectors.

Example: The polynomial $\, e_1 - 2 e_2 + \pi e_1 e_2 -3 e_2 e_3 + e_3 e_1 + e_1 e_2 e_3 \,$
is a multivector in the clifford algebra $\, C_l(e_1, e_2, e_3)$. It has six terms.

2): The number of vectors that appear in each term of a multivector
is called the degree of the term.

Example: The degrees of the respective terms
of the polynomial $\, e_1 - 2 e_2 + \pi e_1 e_2 -3 e_2 e_3 + e_3 e_1 + e_1 e_2 e_3 \,$
are (from left to right): $\, 1, 1, 2, 2, 2, 3$.

3): If each term of a multivector has the same degree
the multivector is called homogeneous.

Example: The multivector $\, 2 e_1 e_2 -3 e_2 e_3 + e_3 e_1 \,$ is homogeneous with degree 2.

4): A multivector that can be written as an exterior product of $\, k \,$ vectors
is called a $\, k$-blade or a blade of degree $\, k$.

Example: In the clifford algebra $\, C_l(e_1, e_2, e_3) \,$, $\, e_1 e_2 \equiv e_1 \wedge e_2 \,$ is a $\, 2$-blade
and $\, e_1 e_2 e_3 \equiv e_1 \wedge e_2 \wedge e_3 \,$ is a $\, 3$-blade.

5): The $\, 3$-blade $\, e_1 e_2 e_3 \,$ is called a unit pseudoscalar for $\, C_l(e_1, e_2, e_3)$.

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FACT 1: In any clifford algebra, the geometric product of any set of basis vectors
produces a pseudoscalar. Moreover, any two pseudoscalars for the algebra
differ only by a scalar factor.

FACT 2: A pseudoscalar dualizes every multivector that it operates on by multiplication.

/////// This diagram shows the 1-blades (unbroken black arrows) and the 2-blades (red, dotted, broken arrows) among the blades in the canonical basis for $\, C_l(e_1, e_2, e_3)$.
The negatives of the 1-blades are shown as dotted black arrows.

The 2-blades $\, \textcolor {red} {e_1 e_2} \textcolor {black} {,} \textcolor {red} {e_2 e_3} \textcolor {black} {,} \textcolor {red} {e_1 e_3} \,$
represent the directed area within the corresponding squares.

Example 3: The complex numbers represented as the even subalgebra
of the clifford algebra $\, C_l(e_1, e_2) \,$ over the real numbers $\, \mathbb{R}$:

$\, e_1, e_2 \,$

$\, e_1^2 = e_2^2 = 1 \,$

$\, e_2 e_1 = - e_1 e_2 \,$

Hence we have: $\, (e_1 e_2)^2 = e_1 e_2 e_1 e_2 = - e_2 e_1 e_1 e_2 = - e_2 e_2 = -1$-

$\, \alpha_1 e_1 + \alpha_2 e_2 \,$

$\, \alpha'_1 e_1 + \alpha'_2 e_2 \,$

$\, (\alpha_1 e_1 + \alpha_2 e_2) (\alpha'_1 e_1 + \alpha'_2 e_2) =$

$\, = \alpha_1 e_1 \alpha'_1 e_1 + \alpha_1 e_1 \alpha'_2 e_2 + \alpha_2 e_2 \alpha'_1 e_1 + \alpha_2 e_2 \alpha'_2 e_2 \, =$

$= \, \alpha_1 \alpha'_1 e_1 e_1 + \alpha_2 \alpha'_2 e_2 e_2 + \alpha_1 \alpha'_2 e_1 e_2 + \alpha_2 \alpha'_1 e_2 e_1 \, =$

$= \, \alpha_1 \alpha'_1 + \alpha_2 \alpha'_2 + \alpha_1 \alpha'_2 e_1 e_2 - \alpha_2 \alpha'_1 e_1 e_2 \, =$

$= \, \alpha_1 \alpha'_1 + \alpha_2 \alpha'_2 + (\alpha_1 \alpha'_2 - \alpha_2 \alpha'_1) e_1 e_2$.

/////// The even part of the clifford algebra $\, C_l(e_1, e_2) \,$:

$\, (x + y e_1 e_2) (x' + y' e_1 e_2) \, =$

$= \, x x' + x y' e_1 e_2 + y e_1 e_2 x' + y e_1 e_2 y' e_1 e_2 \, =$

$= \, x x' + x y' e_1 e_2 + y x' e_1 e_2 + y y' e_1 e_2 e_1 e_2 \, =$

$= \, x x' + x y' e_1 e_2 + y x' e_1 e_2 - y y' e_2 e_1 e_1 e_2 \, =$

$= \, x x' - y y' + (x y' + y x') e_1 e_2$.

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