On Frobenius Theorem and Classication of 2-Dimensional Real Division Algebras

U.U.D.M. Project Report 2020:18

Examensarbete i matematik, 15 hp Handledare: Martin Herschend

Examinator: Veronica Crispin Quinonez Juni 2020

Department of Mathematics

On Frobenius Theorem and Classication of 2-Dimensional Real Division Algebras

Mikolaj Cuszynski-Kruk

Abstract

A proof of Frobenius theorem which states that the only finite-dimensional real

associative division algebras up to isomorphism are R, C and H, is given. A com-

plete list of all 2-dimensional real division algebras based on the multiplication

table of the basis is given, based on the work of Althoen and Kugler. The list

is irredundant in all cases except the algebras that have exactly 3 idempotent

elements.

Contents

1 Introduction 1

2 Background 2

2.1 Rings . . . . 2 2.2 Linear algebra . . . . 4 2.3 Algebra . . . . 7

3 Frobenius Theorem 10

4 Classification of finite-dimensional real division algebras 12 4.1 2-dimensional division algebras . . . . 13

References 22

1 Introduction

An algebra A over a field F is vector space over the same field together with a bilinear multiplication. Moreover, an algebra is a division algebra if for all non-zero a in A the maps L

_a

: A → A defined by v 7→ av and R

_a

: A → A defined by v 7→ va are invertible. If an algebra has a unit and the multiplication is associative then the algebra has a ring structure.

In this paper we will only study algebras over the field of real numbers which are called real algebras, moreover we shall assume that the vector space is finite-dimensional. A trivial example of a real division algebra is the field of real numbers with standard multiplication.

The study of real division algebras originate form the study of number sys- tems. The construction of the field of complex numbers yielded a 2-dimensional real division algebra. Further developments were made by Sir William Rowan Hamilton. After studying the complex numbers he tried to construction a 3- dimensional real division algebra and according to a famous story, after many failed attempts, during a walk Hamilton came up with a 4-dimensional con- struction instead, defined by the property

i

²

= j

²

= k

²

= ijk = −1

which later would be called the quaternion algebra. Although Hamilton suc- ceeded in constructing a 4-dimensional division algebra there was a price to be paid, the quaternions are not commutative. For further history of Hamilton and the quaternions see [2].

In the same way complex numbers can be seen as pairs of real numbers with multiplication defined by (1), the quaternions can be see as pairs of complex numbers with multiplication also defined by (1).

(z

1

, z

2

) · (w

1

, w

2

) = (z

1

w

1

− w

2

z

2

, z

2

w

1

+ w

2

z

1

). (1) By continuing this line of thoughts, pairs of quaternions are a perfect candidate for a 8-dimensional real division algebra. Indeed, they form the so called octo- nions discovered independently by Graves [8] and Cayley [5]. Similarly to the quaternions not being commutative the octonions are in addition not associat- ive.

One could think that it is possible to construct 2

ⁿ

−dimensional real division algebras by continuing this process, but it is not the case. In 1958 Bott and Milnor [4] proved that the only possible dimensions of a real division algebra are 1,2,4 and 8.

In 1878 Frobenius proved that the only real associative division algebras up to isomorphism are R, C and H, a proof will be given in Section 3. Later in 1931, Zorn [12] proved that by weakening the assumption of associativity to instead only require that the algebra is alternative yields only one additional algebra, the octonion algebra.

When considering arbitrary real division algebras, the classification of 2-di-

mension real division algebras is known and will be given in Section 4.1. For

the 4- and 8-dimensional case only some special cases have been classified. An overview of some classifications is given in [6].

2 Background

In this section we define useful notions and establish results that will be neces- sary in order to prove Frobenius theorem.

2.1 Rings

Basic theory of rings will be needed in order to understand properties of an associative algebra. The following subsection is based on [11, Chapter 2]. A natural way to start is to define the notion of a ring.

Definition 2.1. A ring is a 5-tuple (R, +, ·, 0, 1) consisting of a set R, two binary operation + and · that are closed in R and two elements 0, 1 ∈ R such that for all a, b, c ∈ R

(a + b) + c = a + (b + c), (A1)

a + b = b + a, (A2)

a + 0 = 0 = 0 + a, (A3)

∃ −a ∈ R a + (−a) = 0 = −a + a, (A4)

(a · b) · c = a · (b · c), (M1)

a · 1 = a = 1 · a, (M2)

a · (b + c) = a · b + a · c, (D1)

(b + c) · a = b · a + c · a. (D2)

Remark 2.1.1. (R, +, ·, 0, 1) will often be denoted simply by R and a · b by ab.

Example 2.1.2. (C, +, ·, 0, 1) where the operations are interpreted as standard addition and multiplication of complex numbers, is a ring.

Definition 2.2. (S, +, ·, 0, 1) is called a subring of a ring (R, +, ·, 0, 1) if S ⊆ R and the following holds for all a, b ∈ S

0, 1 ∈ S, (S1)

a + b ∈ S, (S2)

−a ∈ S, (S3)

ab ∈ S. (S4)

Remark 2.2.1. In order to see if S satisfies 0 ∈ S, (S2) and (S3) it is enough to

check if S is non-empty and that for all x, y in S, x − y is also in S. Since then

x in S implies that 0 = x − x is in S and for all x, y in S, −x = 0 − x is also in

S and hence so is y + x = y − (−x).

Since Frobenius theorem concerns division algebras the ability to perform division is important. Division is defined as the inverse of multiplication but an ring does not required the existence of multiplicative inverses and rings that does have multiplicative inverses are called division rings.

Definition 2.3. A division ring is a ring in which every non-zero element has an inverse, i.e. a ring R such that

∀x ∈ R \ {0} ∃y ∈ R xy = 1 = yx.

Proposition 2.4. For each x ∈ R it’s inverse is unique and denoted x

⁻¹

. Proof. Assume y

1

, y

2

are inverses of x, then y

1

x = 1 = xy

2

and y

1

= y

1

(xy

2

) = (y

1

x)y

2

= y

2

.

The quaternions were first constructed by Hamilton and can be seen as an extension of complex number by two additional imaginary units, j and k, that satisfy i

²

= j

²

= k

²

= −1 = ijk. Hamilton’s construction was of a geometrical nature but there is another way of constructing quaternions, namely as a subset of M

2×2

(C), i.e the set of 2 × 2 matrices with complex entries.

Proposition 2.5. The set H =

 z w

−w z



: z, w ∈ C



is a subring of M

_2×2

(C).

Proof. For all z, w ∈ C it holds that z − w = z − w hence if Z, W ∈ H then Z − W ∈ H. The multiplicative identity in M

^2×2

(C) is I = 1 0

0 1



and since 1 = 1 and 0 = 0, I is in H. Moreover  x y

−y x

 ,

 z w

−w z



∈ H implies

 x y

−y x

 z w

−w z



=  xz − yw xw + yz

−yz − xw −yw + xz



=  xz − yw xw + yz

−xw + yz xz − yw



since z w = zw for all complex numbers. Hence H is closed under multiplication and thus a subring of M

_2×2

(C).

Remark 2.5.1. Note that every matrix in H is of the form

H 3  x

₁

+ x

2

√ −1 x

3

+ x

4

√ −1

−x

3

+ x

4

√ −1 x

1

− x

2

√ −1



= x

₁

1 0 0 1



+ x

₂

√−1 0

0 − √

−1



+ x

₃

 0 1

−1 0

 + x

₄

 0 √

√ −1

−1 0



where x

1

, x

2

, x

3

, x

4

∈ R, moreover if we denote the matrices 1 0 0 1

 ,

√−1 0

0 − √

−1



,  0 1

−1 0

 and

 0 √

√ −1

−1 0



by 1, i, j and k, respectively,

then it can easily be checked that i

²

= j

²

= k

²

= −1 = ijk holds.

Proposition 2.6. The ring of quaternions is a division ring.

Proof. Let

 z w

−w z



∈ H \ {0} then

1

|z|

²

+ |w|

²

 z −w

w z



= 1

|z|

²

+ |w|

²

 z −w

−−w z



∈ H and 1

|z|

²

+ |w|

²

 z w

−w z

  z −w

w z



= 1 0 0 1

 .

Remark 2.6.1. Note that the equation i

²

= j

²

= k

²

= −1 = ijk determine the multiplication in H completely.

Definition 2.7. The centre of a ring R is the subset Z(R) = {a ∈ R : ∀b ∈ R, ab = ba}.

Lemma 2.8. If r ∈ R then r1 ∈ Z(H).

Proof. Let r ∈ R then r1 = r 0 0 r



and for all z, w ∈ C,

r 0 0 r

  z w

−w z



=

 rz rw

−rw rz



=

 z w

−w z

 r 0 0 r

 .

The notion of a fields will be useful in the next subsection and the following lemma is a technical one.

Definition 2.9. A field is a non-zero division ring in which the multiplication commutes, i.e. a division ring R 6= {0} such that ∀x, y ∈ R, xy = yx.

Lemma 2.10. Let F be a field. If xy = 0 for some x, y ∈ F then then x = 0 or y = 0.

Proof. If x is non-zero and xy = 0, then since F is a field x has an inverse x

⁻¹

. 0 = xy ⇔ 0 = x

⁻¹

· 0 = x

⁻¹

xy = y.

Hence x = 0 or y = 0 must hold.

2.2 Linear algebra

Vector spaces are another important part of algebras hence an overview of some basis properties is in order. This subsection is based on [3, Chapters 1, 2 &

3.A].

Definition 2.11. A set V together with two binary operations is called a vector space over a field F if the two operations + : V × V → V and · : F × V → V satisfies the following axioms for all u, v, w ∈ V and λ, µ ∈ F

u + (v + w) = (u + v) + w, (A1)

u + v = v + u, (A2)

∃0 ∈ V, 0 + v = v = v + 0, (A3)

∃ −v ∈ V, v + (−v) = 0 = −v + v, (A4)

1

F

· v = v, (M1)

λ(µv) = (λµ)v, (M2)

λ(u + v) = λu + λv, (D1)

(λ + µ)v = λv + µv. (D2)

Remark 2.11.1. In the rest of this section all vector spaces will be over a field F if not stated otherwise.

Definition 2.12. A subset W of a vector space V is called a subspace if W satisfies

0 ∈ W , (S1)

∀u, v ∈ W, u + v ∈ W , (S2)

∀v ∈ W, λ ∈ F, λv ∈ W . (S3)

Remark 2.12.1. A subspace is a vector space and all vector spaces have {0} as a subspace.

In order to be able to easily describe elements of a vector spaces the following notions are important.

Definition 2.13. Let V be a vector space. The span of a set of vectors {v

i

∈ V : i = 1, . . . , n} is the set

span{v

1

, . . . , v

n

} = {λ

1

v

1

+ . . . + λ

n

v

n

: λ

1

, . . . , λ

n

∈ F }.

If moreover span{v

1

, . . . , v

n

} = V then v

1

, . . . , v

n

spans V . If the set consist of only one vector v then the span can also be denoted by vF .

Remark 2.13.1. The span is a subspace.

Definition 2.14. A collection of vectors {v

1

, . . . , v

n

} is called linearly inde- pendent if the equation

λ

₁

v

₁

+ · · · + λ

_n

v

_n

= 0, λ

_i

∈ F

has only the trivial solution, λ

1

= · · · = λ

n

= 0. Otherwise the vectors are called linearly dependent.

Definition 2.15. A set of vectors B is called a basis of a vector space V if B is

a linearly independent set and B spans V .

Definition 2.16. If a vector space V has a basis B then the dimension of V is defined as the cardinality of B and denoted dim V = |B|.

Remark 2.16.1. The dimension is well-defined since if a vector space V has a basis then the cardinalities of all bases of V are equal, for proof see [3, Theorem 2.35].

The notion of direct sum will be useful when constructing division algebras and in order to define a vector space structure on the quaternions.

Definition 2.17. Let U

₁

, . . . , U

_n

be subspaces of a vector space V . The sum U

1

+ · · · + U

n

= {u

1

+ · · · + u

n

: u

i

∈ U

i

, i = 1, . . . , n}

is called a direct sum of U

1

, . . . , U

n

if each element can be written as a sum of u

i

:s in exactly one way and is denoted U

1

⊕ · · · ⊕ U

n

.

Lemma 2.18. Let V be a vector space with basis B = {v

1

, . . . , v

n

} then the sum v

1

F + · · · + v

n

F is a direct sum and v

1

F ⊕ · · · ⊕ v

n

F = V .

Proof. Let u, w ∈ v

1

F + · · · + v

n

F and I = {1, 2, . . . , n} then there exists λ

i

, µ

i

∈ F for all i ∈ I such that u = λ

1

v

1

+· · ·+λ

n

v

n

and w = µ

1

v

1

+· · ·+µ

n

v

n

. Assume that u = w. Then

0 = u − w = (λ

₁

− µ

₁

)v

₁

+ · · · + (λ

_n

− µ

_n

)v

_n

and since v

₁

, . . . , v

_n

are linearly independent λ

_i

= µ

_i

must hold for all i ∈ I.

Thus every element has a unique representation and the sum is a direct sum.

For the equality let v ∈ V , then since B spans V there exists λ

i

∈ F for all i ∈ I such that v = λ

1

v

1

+ · · · + λ

n

v

n

and so v ∈ v

1

F ⊕ · · · ⊕ v

n

F . On the other hand v

i

F is a subspace of V and hence a subset, so v ∈ v

i

F implies that v ∈ V and the equality follows from the fact that V is closed under addition.

Example 2.18.1. C is a vector space over R with basis {1, i} so C = R ⊕ iR.

Corollary 2.19. The ring of quaternions has a real vector space structure with basis {1, i, j, k}.

Proof. From Remark 2.5.1 it is clear that the set {1, i, j, k} spans H and is linearly independent hence it is a basis and thus H = R ⊕ iR ⊕ jR ⊕ kR.

Definition 2.20. A map L from a vector space V to a vector space W , both over a field F , is called a linear map is it satisfies

∀u, v ∈ V L(u + v) = L(u) + L(v), (L1)

∀v ∈ V, ∀λ ∈ F L(λv) = λL(v). (L2)

Definition 2.21. Two vector space V and W are called isomorphic if there

exists a bijective linear map L : V → W .

The rest of this section is based on [11, Sections 6.1-4] and contains useful tools for finding basis of a vector space.

Definition 2.22. A bilinear map from a vector space V to a vector space W is a map B : V ×V → W such that for all v ∈ V the maps R

v

: V → W, u 7→ B(u, v) are linear and for all u ∈ V the maps L

u

: V → W, v 7→ B(u, v) are linear. If moreover B(u, v) = B(v, u) holds for all u, v ∈ V then B is called symmetric and if W is the underlying field of V then B is called a bilinear form.

Definition 2.23. A symmetric bilinear form B on a real vector space V is called positive-definite if B(v, v) ≥ 0 for all v ∈ V with equality only when v = 0.

Definition 2.24. A quadratic form on a vector space V is a map Q : V → F such that,

∀v ∈ V, ∀λ ∈ F, Q(λv) = λ

²

Q(v) and (Q1)

B(u, v) = Q(u + v) − Q(u) − Q(v) is a bilinear form. (Q2) Definition 2.25. Let V be a real vector space. Two vectors u, v ∈ V are called orthogonal with respect to a positive-definite symmetric bilinear form B if B(u, v) = 0. Moreover, a set of vectors S is called orthogonal if all vectors in S are pairwise orthogonal.

Lemma 2.26. Let v

1

, . . . , v

n

be orthogonal non-zero vectors then they are lin- early independent.

Proof. Let v

1

, . . . , v

n

be non-zero and orthogonal vectors with respect to a positive-definite symmetric bilinear form B. If λ

1

v

1

+ · · · + λ

n

v

n

= 0 then,

0 = B(0, v

i

) = B(λ

1

v

1

+ · · · + λ

n

v

n

, v

i

) = λ

i

B(v

i

, v

i

)

for all i = 1, . . . , n. The positive-definiteness of B yields that λ

₁

= · · · = λ

_n

= 0.

2.3 Algebra

In this subsection we define the notion of an algebra and prove that the qua- ternions are a real division algebra. But first some technical results that are taken from [9, Section III.8].

Definition 2.27. Let F be a field. A non-zero polynomial P ∈ F [X] of degree at least 1 is called irreducible if whenever P (X) = Q(X)R(X) for some Q, R ∈ F [X] then either Q or R is constant.

Theorem 2.28 (The fundamental theorem of algebra). Every complex polyno- mial P of degree n ≥ 1 has a factorisation

P (X) = c(X − r

₁

)(X − r

₂

) . . . (X − r

_n

)

where c, r

1

, . . . , r

n

∈ C. The factorisation is unique up to permutation of the

factors.

We will not prove this theorem, for a proof see [7, pages 4 & 188].

Corollary 2.29. A real polynomial is irreducible if and only if it has degree 1 or degree 2 and no real roots.

Proof. If a polynomial P of degree 1 is equal to Q(X)R(X) for some Q, R ∈ R[X]

then 1 = deg P = deg Q + deg R hence either Q or R has degree 0, meaning that it is constants, so P is irreducible. If P has degree 2 and no real root then if P is a product of two non-constant polynomials then both must have degree 1 hence, P (X) = (c

1

X − r

1

)(c

2

X − r

2

) for some c

1

, c

2

, r

1

, r

2

∈ R but then

^r_c1¹

is real and a root of P , a contradiction, hence P is irreducible. Conversely, assume that P is irreducible and has degree n then by Theorem 2.28, P (X) = c(x − r

1

) . . . (x − r

n

) where r

1

, . . . , r

n

are roots of P , moreover since P has real coefficients P (X) = P (X) = c(X − r

1

) . . . (X − r

n

). Now the uniqueness of factorisation yields that the sets of roots are equal, {r

1

, . . . , r

n

} = {r

1

, . . . , r

n

}, hence for all i = 1, . . . , n either r

i

is real or r

i

= r

j

for some j, in which case

(X − r

_i

)(X − r

_i

) = X

²

− (r

i

+ r

_i

) + r

_i

r

_i

∈ R[X]

since r

i

+ r

i

= 2Re(r

i

), r

i

r

i

= |r

i

|

²

∈ R. Let 2m be the number of non-real roots. Then P is a product of m real polynomials of degree 2 and n − 2m real polynomials of degree 1, but since P is irreducible one of the following two cases must hold, m = 1, n − 2m = 0 or m = 0, n − 2m = 1. The first case implies that n = 2 and P has no real roots and the second implies that n = 1.

Remark 2.29.1. Note that if X

²

− 2pX + q is a real polynomial then it’s roots are X = p ± p

p

²

− q meaning that a monic degree 2 polynomial has no real roots and hence is irreducible if and only if p

²

< q.

Finally we are ready to define the notion of an algebra. The rest of this section is based on [11, Sections 7.1 & 7.7].

Definition 2.30. An algebra A over a field F is a vector space together with an operation · : (u, v) 7→ uv that is bilinear, i.e. for all u, v, w ∈ A, λ ∈ F,

(u + v)w = uw + vw, (A1)

u(v + w) = uv + uw, (A2)

λ(uv) = (λu)v = u(λv). (A3)

If moreover A together with + and · is a ring then A is called an associative algebra.

Definition 2.31. If A and B are two algebras both over F then a map ϕ : A → B is called an algebra-homomorphism if for all u, v ∈ A and λ ∈ F ,

ϕ(λv) = λϕ(v), (I1)

ϕ(u + v) = ϕ(u) + ϕ(v), (I2)

ϕ(uv) = ϕ(u)ϕ(v). (I3)

If moreover the map is bijective then ϕ is called an algebra-isomorphism.

Remark 2.31.1. If A and B are associative algebras then ϕ(1

A

) = 1

B

and ϕ is a ring-homomorphism. An algebra-isomorphism will often be called an isomorphism.

Definition 2.32. The dimension of an algebra is defined as the dimension of the underlying vector space, if moreover the dimension is finite then the algebra is called finite-dimensional.

Remark 2.32.1. In the remaining part of section we shall call an associative algebra just algebra and assume it is finite-dimensional.

Definition 2.33. An division algebra is an algebra in which the underlying ring is a division ring.

Example 2.33.1. R is a one-dimensional vector space over R and a field, hence an associative division algebra over R of dimension 1.

Example 2.33.2. The field C and the vector space C = R ⊕ iR over R form an associative division algebra over R of dimension 2.

We have seen the quaternion ring in Proposition 2.5 and the quaternion vector space in Corollary 2.19, combining both results yields the following pro- position.

Proposition 2.34. The ring H from Proposition 2.5 is a real associative divi- sion algebra. This algebra is called the quaternion algebra.

Proof. Corollary 2.19 yields that the ring H has a real vector space structure.

Axioms (A1) and (A2) follows then from the ring axioms. For (A3), let λ ∈ R, then λ is in the centre of H so it commutes with all elements. Thus for all u, v ∈ H, λ(uv) = (λu)v = (uλ)v = u(λv) where the first and third equality follows from the associativity of H.

Remark 2.34.1. Note that the dimension of the quaternion algebra is 4.

Definition 2.35. Let A be an algebra over a field F . The minimal polynomial of a ∈ A is a monic polynomial m

a

∈ F [X] such that m

a

(a) = 0 and for every other monic non-zero polynomial P ∈ F [X] such that P (a) = 0 it holds that deg m

_a

≤ deg P .

Proposition 2.36. Every element a in an division algebra has a unique min- imal polynomial m

_a

. Moreover, m

_a

is irreducible.

Proof. For existence, let n be the dimension of the algebra, take a ∈ A and consider the set {a

ⁱ

: i = 0, . . . , n} ⊆ A, then this set consists of n + 1 vectors so they must be linearly dependent and hence there exists λ

0

, . . . λ

n

∈ F not all zero, such that λ

0

a

⁰

+ . . . + λ

n

a

ⁿ

= 0. Let m be the larges index such that λ

m

6= 0 then the polynomial P (X) =

_λ¹

m

(λ

m

X

^m

+ . . . + λ

0

) is monic and has a

as a root. Since the degree of a polynomial is bounded from below there must

exist a polynomial with smallest degree with this property.

For uniqueness, let A be an algebra over the field F and assume that m

a

, m

⁰_a

∈ F [X] are minimal polynomials of a ∈ A and denote theirs degree by n, then m

_a

= X

ⁿ

+ P

n−1

i=0

λ

_i

X

ⁱ

and m

⁰_a

= X

ⁿ

+ P

n−1

i=0

µ

_i

X

ⁱ

for some λ

_i

:s and µ

i

:s is F . Now the polynomial (m

a

− m

⁰_a

) has degree at most n − 1 and (m

a

− m

⁰_a

)(a) = m

a

(a) − m

⁰_a

(a) = 0 which implies that m

a

− m

⁰_a

is the zero polynomial hence m

a

= m

⁰_a

and thus the minimal polynomial is unique.

For irreducibility, if m

a

(X) = P (X)Q(X) for some P, Q ∈ F [X] then 0 = m

a

(a) = P (a)Q(a) so either P (a) = 0 or Q(a) = 0 meaning that at least one of P and Q must have degree greater than or equal to m

a

and hence the other must be constant.

Proposition 2.37. Let A be an division algebra over R then the minimal poly- nomial of a ∈ A is

m

a

(X) =

( X − a if a ∈ R

X

²

− 2pX + q for some p, q ∈ R with p

²

< q if a / ∈ R . Proof. The case when a ∈ R is clear and if a / ∈ R then for every polynomial P

r

of degree 1, P

_r

(X) = X − r for some r ∈ R and P

r

(a) = 0 implies a = r so the minimal polynomial can not have degree one and since the minimal polynomial is irreducible Corollary 2.29 yields the other case.

3 Frobenius Theorem

We have now the necessary background to prove Frobenius theorem. But first we shall prove a lemma that will simplify the proof.

The elements of a real division algebra can be characterised by their minimal polynomial. Proposition 2.37 has shown that the minimal polynomial of an element a is, X − a if a is real and X

²

− 2pX + q with p

²

< q otherwise which yields the following lemma.

Lemma 3.1. Let A be a finite-dimensional real associative division algebra then each a ∈ A can be written on the form a = x + y where x ∈ R and y = 0 or y

²

∈ R

<0

.

Proof. Let a ∈ A, if a is real then the condition hold trivially so assume a / ∈ R then by Proposition 2.37 the minimal polynomial of a is m

a

(X) = X

²

− 2pX + q with p

²

− q < 0, moreover let b = a − p / ∈ R then

0 = m

_a

(a) = m

_a

(b + p) = (b + p)

²

− 2p(b + p) + q = b

²

− p

²

+ q which implies that b

²

= p

²

− q < 0 and a = p + b with p ∈ R and b

²

< 0.

Theorem 3.2 (Frobenius theorem). The only finite-dimensional real associat-

ive division algebras, up to isomorphism, are (i) R, (ii) C and (iii) H.

Proof. Let A be a finite-dimensional real associative division algebra and con- sider the subset A

⁰

consisting of all elements whose square is real and non- positive. Note that a / ∈ A

⁰

for all non-zero a ∈ R. We shall now prove that A

⁰

is a subspace of A. It is closed under scalar multiplication since if v ∈ A

⁰

and λ ∈ R then λ

²

≥ 0 and (λv)

²

= λ

²

v

²

≤ 0 so λv ∈ A

⁰

. In order to see that A

⁰

is closed under addition we take non-zero u, v ∈ A

⁰

and show that u + v ∈ A

⁰

. If u = λv for some λ ∈ R then u + v = (λ + 1)u ∈ A

⁰

. Otherwise, assume that u and v are linearly independent. Note that u, v, 1 are now pair-wise linearly independent. In fact we claim that all three are linearly independent. Consider the equation νu + µv + λ1 = 0 with ν, µ, λ ∈ R. Then −νu = µv + λ and

ν

²

u

²

= (µv + λ)

²

= (µv)

²

+ 2λµv + λ

²

.

The left hand side is real and so is (µv)

²

+ λ

²

but λµv is in A

⁰

meaning that λµv is real only if λµ = 0 so either λ = 0 or µ = 0. But then since u, v, 1 are pair-wise linearly independent λ = µ = ν = 0 is the only solution, thus 1, u, v are linearly independent.

It follows that u + v, u − v ∈ A \ R. By Proposition 2.37 the minimal polynomials of both u + v and u − v have degree 2. Let m

₁

(X) = X

²

+ p

₁

X + q

₁

and m

₂

(X) = X

²

+ p

₂

X + q

₂

be the minimal polynomials of u + v and u − v, respectively. Then

0 = m

1

(u + v) = u

²

+ uv + vu + v

²

+ p

1

u + p

1

v + q

1

0 = m

₂

(u − v) = u

²

− uv − vu + v

²

+ p

₂

u − p

₂

v + q

₂

)

⇒ 0 = (p

1

+ p

2

)u + (p

1

− p

2

)v + (2u

²

+ 2v

²

+ q

1

+ q

2

)1.

Now the linear independence of u, v, 1 yields that

(p

₁

+ p

₂

) = (p

₁

− p

₂

) = (2u

²

+ 2v

²

+ q

₁

+ q

₂

) = 0

and thus p

1

= p

2

= 0. Hence m

1

(X) = X

²

+ q

1

, with q

1

> 0 by Proposition 2.37. It follows that (u + v)

²

= −q

₁

< 0 and thus u + v ∈ A

⁰

which completes the proof of the claim that A

⁰

is a subspace of A.

Lemma 3.1 showed that every u ∈ A can be written on the form u = a + b with a ∈ R and b ∈ A

⁰

hence A = R ⊕ A

⁰

. Let Q : A

⁰

→ R be defined by Q(v) = −v

²

then Q(0) = 0 and Q(λv) = −λ

²

v

²

> 0 for all non-zero scalars λ and non-zero v in A

⁰

. Moreover B(u, v) := Q(u + v) − Q(u) − Q(v) = −(uv + vu) is symmetric and

B((λu + µv), w) = −(λu + µv)w − −w(λu + µv)

= −λ(uw + wu) − µ(vw + wv) = λB(u, w) + µB(v, w), shows that B is bilinear, thus Q is a quadratic form and B is positive-definite.

In order to complete the proof we shall now show that the only possible

dimensions of A

⁰

are 0, 1 and 3. If A

⁰

= {0} then A = R ⊕ {0} = R which

corresponds to the case (i). Now assume A

⁰

is non-trivial. Then there exists

an i ∈ A

⁰

with 1 = Q(i) = −i

²

and iR ⊆ A

⁰

is a subspace. If A

⁰

= iR then

A = R ⊕ iR = C, which corresponds to the case (ii). Finally, if iR 6= A

⁰

then there exists a vector j, orthogonal to i with respect to B such that j

²

= −1.

Taking the bilinear form of i and j gives 0 = B(i, j) = −(ij + ji) which implies that ij = −ji hence there exists a k = ij ∈ A and k

²

= ijij = −jiij = j

²

= −1 thus k ∈ A

⁰

but

−B(i, k) = ik + ki = iij − jii = −j + j = 0 and

−B(j, k) = jk + kj = jji − ijj = −i + i = 0,

meaning that k is orthogonal to both i and j and thus A

⁰

can not have dimension 2. It is left to show that A

⁰

must now be equal to span{i, j, k}. Let v ∈ A

⁰

and consider

` = v − B(v, i)

B(i, i) i − B(v, j)

B(j, j) j − B(v, k) B(k, k) k.

Then ` is orthogonal to i, j and k, since B(`, i) = B(v, i) − B(v, i)

B(i, i) B(i, i) = 0

and similarly for j and k. Moreover, the orthogonality yields i` = −`i, j` = −`j, k` = −`k and so

k` = −`k = −`ij = i`j = −ij` = −k`, which implies that 0 = ` so

v = B(v, i)

B(i, i) i + B(v, j)

B(j, j) j + B(v, k)

B(k, k) k ∈ span{i, j, k} .

Thus A

⁰

= span{i, j, k}. Moreover, it holds that i

²

= j

²

= k

²

= −1 = ijk hence the last case is given by A = R ⊕ iR ⊕ jR ⊕ kR = H by Remark 2.6.1.

4 Classification of finite-dimensional real divi- sion algebras

Before a classification can be given the definition of an arbitrary division algebra is in order.

Definition 4.1. An algebra A is called a division algebra if for all non-zero a ∈ A the maps L

_a

: A × A → A defined by v 7→ av and R

_a

: A × A → A defined by v 7→ va are invertible.

This generalises the notion of an associative division algebra.

Frobenius theorem yields that the only possible dimensions of a finite-dimen-

sional real associative algebras are 1, 2 or 4, by weakening the assumption of

associativity we get a new type of algebras.

Definition 4.2. An alternative algebra is an algebra A such that (uu)v − u(uv) = 0 = (uv)v − u(vv)

holds for all u, v ∈ A.

By Zorn’s theorem [12] requiring that the algebra is alternative instead of associative yields only one additional algebra, the octonions which are 8- dimensional. A natural question to ask is which dimension an finite-dimensional real division algebra can have? The answer turn out to by 1, 2, 4 or 8 and has been proved by Bott and Milnor in [4, Corollary 1].

The classification of all finite-dimensional real division algebras is still an open problem. The 1-dimensional case has only the trivial solution R [6]. In this section a solution of the 2-dimensional case, when the number of idempotent elements in not three, will be given.

4.1 2-dimensional division algebras

Another approach to classification of real division algebras is rather than fixing the properties of the algebra instead fix it’s dimension. In this subsection a almost complete classification of 2-dimensional real division algebras will be given based on the work of Althoen and Kugler [1].

In this subsection every algebra will be assumed to be a 2-dimensional real algebra if not stated otherwise.

An 2-dimensional real algebra if fully determined by the multiplication of the basis, since then the properties (A1), (A2) and (A3) from Definition 2.30 yields the multiplication of arbitrary elements. Thus such algebras can be described by a multiplication table as in Table 1 where {u, v} is some basis and λ

ij

, µ

ij

∈ R for i, j ∈ {1, 2}.

· u v

u λ

₁₁

u + µ

₁₁

v λ

₁₂

u + µ

₁₂

v v λ

₂₁

u + µ

₂₁

v λ

₂₂

u + µ

₂₂

v

Table 1

Remark 4.2.1. Although a basis is defined as a set of vectors in this subsection the order will of the essence and the basis {u, v} and {v, u} will be considered to be different.

Definition 4.3. Two multiplication tables as in Table 2 are called identical if λ

_ij

= λ

⁰_ij

and µ

_ij

= µ

⁰_ij

for all i, j ∈ {1, 2}.

Lemma 4.4. Let A and A

⁰

be two algebras. If there exist bases such that the multiplication tables of A and A

⁰

are identical then A and A

⁰

are isomorphic.

Conversely, if ϕ : A → A

⁰

is an isomorphism and {u, v} a basis for A then the

multiplication tables with basis {u, v} of A and {ϕ(u), ϕ(v)} of A

⁰

are identical.

· u v u λ

11

u + µ

11

v λ

12

u + µ

12

v v λ

21

u + µ

21

v λ

22

u + µ

22

v

(a)

· u

⁰

v

⁰

u

⁰

λ

⁰₁₁

u

⁰

+ µ

⁰₁₁

v

⁰

λ

⁰₁₂

u

⁰

+ µ

⁰₁₂

v

⁰

v

⁰

λ

⁰₂₁

u

⁰

+ µ

⁰₂₁

v

⁰

λ

⁰₂₂

u

⁰

+ µ

⁰₂₂

v

⁰

(b)

Table 2

Proof. Let A be an algebra given by Table 2a and A

⁰

be an algebra given by Table 2b. Assume the tables are identical and let ϕ : A → A

⁰

be defined by ϕ(λu + µv) = λu

⁰

+ µv

⁰

for all λ, µ ∈ R then clearly ϕ(λu+µv) = λϕ(u)+µϕ(v),

ϕ(uv) = ϕ(λ

12

u + µ

12

v) = λ

12

u

⁰

+ µ

12

v

⁰

= u

⁰

v

⁰

= ϕ(u)ϕ(v)

and the other cases can be calculated in a similar way. Thus ϕ is an isomorphism.

Conversely, if ϕ : A → A

⁰

is an isomorphism then

ϕ(u)

²

= ϕ(u

²

) = ϕ(λ

₁₁

u + µ

₁₁

v) = λ

₁₁

ϕ(u) + µ

₁₁

ϕ(v).

and the other cases follows in a similar way.

The classification of 2-dimensional real division algebras is based on the number of idempotent element hence the following definition is in order.

Definition 4.5. A non-zero element u of an algebra is called idempotent if u

²

= u.

Lemma 4.6. An isomorphism maps idempotent elements to idempotent ele- ments.

Proof. Let u be idempotent and ϕ an isomorphism then ϕ(u) = ϕ(u

²

) = ϕ(u)

²

.

Proposition 4.7. A 2-dimensional real division algebra has at least one idem- potent element.

For proof see [1, Theorem 2]. Thus by choosing an appropriate basis the multiplication table of every algebra can be written on the form of Table 3.

· u v

u u λ

₁₂

u + µ

₁₂

v v λ

21

u + µ

21

v λ

22

u + µ

22

v

Table 3

Definition 4.8. Let A(λ

₁₂

, µ

₁₂

, λ

₂₁

, µ

₂₁

, λ

₂₂

, µ

₂₂

) denote an algebra with mul-

tiplication table on the form of Table 3.

Proposition 4.9. An algebra given by Table 3 is a division algebra if and only if (µ

22

− A

1

)

²

< 4µ

12

A

2

where A

1

=

   

λ

12

µ

12

λ

21

µ

21

   

and A

2

=    

λ

21

µ

21

λ

22

µ

22

    .

This lemma can be proved by applying Cramer’s Rule, for details see [1, Theorem 3]. Another useful fact in the following proposition [1, Proposition on p. 629].

Proposition 4.10. An algebra has exactly one idempotent element if and only if (λ

₁₂

+ λ

₂₁

− µ

22

)

²

< 4λ

₂₂

(1 − µ

₁₂

− µ

21

).

Assume there is an algebra that has at least two idempotent elements u and v. Then the elements must be linearly independent since v = λu implies that v = v

²

= λ

²

u

²

= λ

²

u = λv so λ = 1 and u = v. Thus {u, v} is a basis and the multiplication table is

· u v

u u λ

12

u + µ

12

v v λ

21

u + µ

21

v v

.

Table 4

Lemma 4.11. In a 2-dimensional real division algebra given by the Table 4, (λ

12

+ λ

21

)(µ

12

+ µ

21

) 6= 1.

Proof. An algebra given by Table 4 must satisfy

1 + (λ

12

µ

21

)

²

+ (λ

21

µ

12

)

²

− 2(λ

12

µ

21

) − 2(λ

21

µ

12

) − 2(λ

12

µ

21

)(λ

21

µ

12

) < 0, (2) by Proposition 4.9. Hence λ

12

, λ

21

, µ

12

, µ

21

are all non-zero. Let A = λ

12

µ

21

and B = λ

21

µ

12

. Then (2) simplifies to (A + B − 1)

²

− 4AB < 0. If (λ

12

+ λ

21

)(µ

12

+ µ

21

) = 1, then substitution µ

12

=

_λ^B

21

and µ

21

=

_λ^A

12

yields B λ

₁₂

λ

21

+ A + B + A λ

₂₁

λ

12

= 1 ⇔ B  λ

₁₂

λ

21



²

+ (A + B − 1)  λ

₁₂

λ

21



+ A = 0

so 

λ₁₂ λ₂₁



is a root of a quadratic equation. Thus the discriminant (A + B − 1)

²

− 4AB is non-negative which contradicts (2).

The first step in classification of 2-dimensional algebras is the following the- orem.

Theorem 4.12. A 2-dimensional real division algebra has exactly one, two or three idempotent elements.

Proof. By Proposition 4.7 all such algebras have at least one idempotent ele-

ment, hence let A be such algebra with basis {u, v} and at least two idempotent

elements. Then the multiplication in A can be written on the form of Table 4.

An arbitrary element xu + yv in A is idempotent if

xu + yv = (xu + yv)

²

= x

²

u + (λ

12

+ λ

21

)xyu + (µ

12

+ µ

21

)xyv + y

²

holds, which yields the system

( x = x

²

+ (λ

12

+ λ

21

)yx

y = y

²

+ (µ

₁₂

+ µ

₂₁

)xy (3)

and the solutions are x = ( 0

1 − (λ

12

+ λ

21

)y and y = ( 0

1 − (µ

12

+ µ

21

)x . If x = 0 or y = 0 then the only possible solutions are (x, y) = (0, 0), (0, 1) and (1, 0). Otherwise, let Λ = λ

12

+ λ

21

and M = µ

12

+ µ

21

then x =

_1−ΛM^1−Λ

and y =

_1−ΛM^1−M

is the only possible solution by Lemma 4.11. Thus the idempotent elements are u, v and

_1−ΛM^1−Λ

u +

_1−ΛM^1−M

v. Note that they are not necessarily distinct and

_1−ΛM^1−Λ

u +

_1−ΛM^1−M

v is non-zero by Lemma 4.11.

By Lemma 4.6 two algebras can only be isomorphic if they have the same number of idempotent elements, which together with Theorem 4.12 yields three classes of algebras, 1-, 2- and 3-idempotent algebras.

Let A be an algebra with at least two idempotent elements u, v. Then the multiplication can be written on the form Table 4 with basis {u, v}. From proof of Theorem 4.12 we know that A to has exactly two idempotent elements if and only if

_1−ΛM^1−Λ

u +

_1−ΛM^1−M

v is equal to u or v. Thus exactly one of Λ and M must be equal to 1 in order for A to have exactly two idempotent elements.

If A is a 2-idempotent algebra with M = 1 then a change of basis to {v, u}

yields a multiplication table of the form of Table 4 with Λ = 1. Thus a multiplic- ation table with Λ = 1 can be considered to be the canonical multiplication table for a 2-idempotent algebra. The following theorem classifies all 2-idempotent algebras.

Theorem 4.13. Two 2-idempotent 2-dimensional real division algebras are iso- morphic if and only if theirs multiplication tables are identical when written on the form of Table 4 with λ

12

+ λ

21

= 1.

Proof. Let A and A

⁰

be such algebras. By the discussion above every 2-idem-

potent algebra can be written on the form of Table 4 with Λ = 1, let {u, v} and

{u

⁰

, v

⁰

} be the basis of A and A

⁰

in which the multiplication is on the desired

form. If the algebras are isomorphic then there exists an isomorphism ϕ which

must map u to either u

⁰

or v

⁰

. In the first case v is then mapped to v

⁰

and

Lemma 4.4 yields that the tables with basis {u, v} and {u

⁰

, v

⁰

} are identical. If

ϕ(u) = v

⁰

then by the same lemma the tables {u, v} and {v

⁰

, u

⁰

} are identical,

but then λ

12

= µ

⁰₂₁

, λ

21

= µ

⁰₁₂

so µ

⁰₁₂

+ µ

⁰₂₁

= 1 but by the assumption λ

⁰₁₂

+ λ

⁰₂₁

is also 1 which contradicts Lemma 4.11 so ϕ is not an isomorphism in this

case. Conversely, if the tables are identical then by Lemma 4.4, A and A

⁰

are

isomorphic.

Corollary 4.14. The set

D

2

= {A(a, c, 1 − a, d, 0, 1) : c + d 6= 1, (1 − ad + (1 − a)c)

²

< 4c(1 − a)}

is a complete and irredundant list of 2-idempotent 2-dimensional real division algebras.

The case of three idempotent elements is similar and hence shall be con- sidered next. Let A be a 3-idempotent algebra with idempotent elements u, v and basis {u, v}. Then as before the multiplication can be written on the form of Table 4 and ΛM 6= 1. As seen in the proof of Theorem 4.12 the third idem- potent element is then w =

_1−ΛM^1−Λ

u+

_1−ΛM^1−M

v, meaning that neither Λ nor M can be equal to one. But now {u, w} is also a basis with multiplication of the form of Table 4. In fact a 3-idempotent algebra has 6 bases with such multiplication.

A change of basis to {u, w} yields that

uw = (1 − Λ)u + (1 − M )(λ

₁₂

u + µ

₁₂

v)

1 − ΛM = ν

₁₂

u + µ

₁₂

w, where ν

12

=

(1−Λ)+(1−M )λ12−(1−Λ)µ12

1−ΛM

=

^{λ12µ21−(µ}_{ΛM −1}^{12−1)(λ21−1)}

and similarly ν

21

is ν

12

with all indices changed from 12 to 21 and vice versa. Which yield the three multiplication tables in Table 5, the other three are given by simply reversing the order of the basis vectors in these tables.

· u v

u u λ

₁₂

u + µ

₁₂

v v λ

₂₁

u + µ

₂₁

v v

(a)

· u w

u u ν

₁₂

u + µ

₁₂

w

w ν

₂₁

u + µ

₂₁

w w

(b)

· v w

v v ν

₁₂

u + λ

₂₁

w

w ν

₂₁

u + λ

₁₂

w w

(c)

Table 5

Proposition 4.15. If A(λ

12

, µ

12

, λ

21

, µ

21

, 0, 1) is a 3-idempotent division al- gebra then it is isomorphic to A(λ

21

, µ

21

, λ

12

, µ

12

, 0, 1), A(ν

12

, µ

12

, ν

21

, µ

21

, 0, 1), A(ν

21

, µ

₂₁

, ν

₁₂

, µ

₁₂

, 0, 1), A(ν

₁₂

, λ

₂₁

, ν

₂₁

, λ

₁₂

, 0, 1) and A(ν

₂₁

, λ

₁₂

, ν

₁₂

, λ

₂₁

, 0, 1) where ν

12

=

^{λ12µ21−(µ}_{ΛM −1}^{12−1)(λ21−1)}

and ν

21

is ν

12

with all indices changed from 12 to 21 and vice versa.

Proof. If A(λ

12

, µ

12

, λ

21

, µ

21

, 0, 1) has basis {u, v} and the third idempotent element is w then a change of basis to {v, u}, {u, w}, {w, u}, {v, w} and {w, v}, respectively together with Lemma 4.4 yields the proposition.

A complete list of 3-idempotent algebras is given by the following theorem.

Theorem 4.16. Two 3-idempotent 2-dimensional real division algebras are iso- morphic if and only there is a correspondence between the idempotent elements and the corresponding multiplication tables are identical.

Proof. Let A and A

⁰

be such algebras the first one with idempotent elements u, v and w. If they are isomorphic and ϕ is an isomorphism then ϕ(u), ϕ(v) and ϕ(w) are the idempotent elements in A

⁰

. Now, the tables with basis {u, v}

and {ϕ(u), ϕ(v)} are identical, {u, w} and {ϕ(u), ϕ(w)} are identical and also {v, w} and {ϕ(v), ϕ(w)} are identical by Lemma 4.4. Conversely, if the tables are identical then by Lemma 4.4, A and A

⁰

are isomorphic.

Corollary 4.17. The set

D

3

= {A(a, c, b, d, 0, 1) : a + b 6= 1 6= c + d, (a + b)(c + d) 6= 1, (1 − ad + bc)

²

< 4cb}

is a complete list of 3-idempotent 2-dimensional real division algebras.

Note that an algebra in D

3

can be isomorphic to at most 5 other algebras in D

3

.

The last case to consider is when the algebra has exactly one idempotent element, call it u. As before there is a multiplication table on the form of Table 3 but the 1-idempotent case requires λ

₂₂

6= 0 since otherwise

_µ^v

22

would be idempotent.

Proposition 4.18. Given a 2-dimensional real division algebra A with one idempotent element u and multiplication given by Table 3 if µ

₁₂

+ µ

₂₁

6= 0 then there exists a unique up to sign w such that {u, w} is a basis in which w

²

= ±u.

It can be proven by letting w = xu + yv with y 6= 0 and solving the equation (xu + yv)

²

= ±u for details see [1, Theorem 7].

Lemma 4.19. In an algebra given by Table 3 the coefficients µ

12

and µ

21

are invariant under the change of basis to {u, w} for all w linearly independent to u.

Proof. Consider such algebra with basis {u, v} and let w = xu + yv. In order for w to be linearly independent to u, y must be non-zero. Then

uw = u(xu + yv) = xu + yλ

₁₂

u + yµ

₁₂

v = (x + yλ

₁₂

− xµ

12

)u + µ

₁₂

w, wu = (xu + yv)u = xu + yλ

21

u + yµ

21

v = (x + yλ

21

− xµ

21

)u + µ

21

w.

This lemma makes the following definition well-defined.

Definition 4.20. An algebra given by Table 3 is quasi-complex if µ

₁₂

+µ

₂₁

6= 0.

Hence a quasi-complex algebra can be written on the form of Table 6.

The following theorem classifies all quasi-complex algebras.

· u v u u λ

12

u + µ

12

v v λ

21

u + µ

21

v ±u

Table 6

Theorem 4.21. Two 1-idempotent quasi-complex 2-dimensional real division algebras given by Table 6 with basis {u, v} and {u

⁰

, v

⁰

} are isomorphic if and only if the multiplication table are identical or the multiplication tables with basis {u, v} and {u

⁰

, −v

⁰

} are identical.

Proof. Let A and A

⁰

be such algebras where A has basis {u, v} and let ϕ be an isomorphism then ϕ(u) = u

⁰

and ϕ(v)

²

= ϕ(v

²

) = ϕ(±u) = ±u

⁰

so ϕ(v) = ±v

⁰

by Proposition 4.18 and so {u

⁰

, ϕ(v)} is equal to {u

⁰

, v

⁰

} or {u

⁰

, −v

⁰

} and then by Lemma 4.4 the multiplication tables with basis {u, v} and {u, ϕ(v)} are identical. Conversely, note that the algebra given by the multiplication table with basis {u

⁰

, −v

⁰

} is just a change of basis of the algebra with basis {u

⁰

, v

⁰

}.

Thus by Lemma 4.4, A and A

⁰

are isomorphic.

Corollary 4.22. The set

D

_1a

= {A(a, c, b, d, e, 0) : c + d 6= 0, e = ±1, (bc − ad)

²

< −4cde, (a + c)

²

< 4e(1 − c − d)}

is a complete and irredundant list of 1-idempotent quasi-complex 2-dimensional real division algebras.

Another type of 1-idempotent algebras are algebras with a basis such that λ

12

= λ

21

= 0. In order to find such basis consider w = xu + yv with y 6= 0.

uw = xu + y(λ

₁₂

u + µ

₁₂

v) = ((1 − µ

₁₂

)x + yλ

₁₂

)u + µ

₁₂

w and similarly for wu which yields the following system of equations,

( (1 − µ

12

)x + yλ

12

= 0

(1 − µ

₂₁

)x + yλ

₂₁

= 0 (4)

This system has always the solution x = y = 0. Moreover, the solution is unique whenever

   

(1 − µ

12

) λ

12

(1 − µ

21

) λ

21

   

= (λ

12

µ

21

− λ

21

µ

12

) − λ

12

+ λ

21

6= 0. A non-trivial solution exists if and only if (λ

12

µ

21

− λ

21

µ

12

) = λ

12

− λ

21

. For a commutative algebra the condition is always true which explains the following definition.

Definition 4.23. A 2-dimensional real division algebra is called quasi-com- mutative if λ

₁₂

µ

₂₁

− λ

₂₁

µ

₁₂

= λ

₁₂

− λ

₂₁

.

Proposition 4.24. If a 1-idempotent 2-dimensional real division algebra is

quasi-commutative but not quasi-complex then the algebra has basis {u, v} in

which the multiplication is as in Table 7.

· u v

u u µ

12

v

v −µ

12

v u + µ

22

v Table 7

Proof. Let u be the idempotent element and the multiplication be given by Table 3. Since the algebra is non-quasi-complex, µ

12

+ µ

21

= 0 and the condition for quasi-commutativeness simplifies to µ

12

(λ

12

+ λ

21

) = λ

21

− λ

12

. If λ

12

+ λ

21

= 0 then λ

12

= λ

21

= 0. Letting w = v yields multiplication as in Table 8.

· u w

u u µ

12

w

w −µ

12

w λ

22

u + µ

22

w Table 8

Otherwise, adding both equations in the system (4) yields 2x+(λ

₁₂

+λ

₂₁

)y = 0. Then x = 1 and y =

_λ ⁻²

12+λ₂₁

is a solution. Let w = xu + yv. The change of basis to {u, w} yields multiplication on the form of Table 8.

Moreover, in a division algebra with multiplication as in Table 8, (−µ

22

)

²

< 4λ

22

must hold by Proposition 4.10. Hence λ

22

is positive. The change of basis v = w/ √

λ

22

yields that v

²

= w

²

λ

₂₂

= u + µ

22

w

λ

₂₂

= u + µ

22

√ λ

₂₂

v, while preserving the other multiplications.

Theorem 4.25. Two 1-idempotent non-quasi-complex quasi-commutative 2- dimensional real division algebras are isomorphic if and only if when the multi- plication table are written on the form of Table 7 the tables are identical.

Proof. The theorem follows from Proposition 4.24 and Lemma 4.4.

Corollary 4.26. The set

D

_1b

= A(0, c, 0, −c, 1, f ) : f

²

< 4, f

²

< 2c

²

is a complete and irredundant list of 1-idempotent non-quasi-complex quasi-com- mutative 2-dimensional real division algebras.

The last case is algebras that are neither quasi-complex nor quasi-commu- tative and they are classified by the following theorem.

Theorem 4.27. A 1-idempotent non-quasi-complex non-quasi-commutative 2-

dimensional division algebra has a basis in which the multiplication table is of

the form of Table 9. Moreover two such algebras are isomorphic if and only if

theirs multiplication tables are identical.

· u v

u u u + µ

12

v

v u − µ

12

v λ

22

u + µ

22

v Table 9

Proof. Consider such algebra with multiplication as in Table 3 then since it is not quasi-complex µ

12

= −µ

21

. The change of basis is given by the system

( (1 − µ

₁₂

)x + yλ

₁₂

= 1

(1 + µ

12

)x + yλ

21

= 1 (5)

which has a unique solution since the non-quasi-commutativeness yields that 

  

(1 − µ

12

) λ

12

(1 + µ

12

) λ

21

   

6= 0. In order for the solution to be a proper change of basis y must be non-zero. If y = 0 then

_1+µ¹

12

= x =

_1−µ¹

12

, hence µ

₁₂

must be 0.

Conversely, if µ

12

= 0 then the system of equations yields y(λ

12

− λ

21

) = 0 and non-quasi-commutativeness yields y = 0. So y = 0 if and only if µ

12

= 0. But if µ

12

= 0 then u(λ

12

u − v) = λ

12

u − λ

12

u = 0 so λ

12

u = v and hence {u, v} is not a basis. Thus there always exists a change of basis in which the multiplication is as in Table 9. Moreover, the uniqueness implies that two such algebras are isomorphic if and only if the multiplication tables are identical.

Corollary 4.28. The set

D

1c

= {A(1, c, 1, −c, e, f ) : c 6= 0 6= e, f

²

< 4c(f + ce), (2 − f )

²

< 4e}

is a complete and irredundant list of 1-idempotent non-quasi-complex quasi-com- mutative 2-dimensional real division algebras.

Thus the set D = D

_1a

∪ D

_1b

∪ D

_1c

∪ D

₂

∪ D

₃

is a complete list of all 2- dimensional real division algebras where the only redundant algebras have ex- actly three idempotent elements. A complete and irredundant classification is given in [10] although it uses a more general theory of division algebras not discussed in this paper.

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