Preliminaries
Classification
Conclusions
Classification of Associative Two-Dimensional
Algebras
Christopher A. Doherty, Jacob C. Hall, Patrick J. Ledwith, and
Rishi S. Mirchandani
Pennsylvania Governor’s School for the Sciences
Dr. Juan Jorge Schäffer, Advisor
Mr. Quang Sack, Teaching Assistant
August 1, 2014
Doherty, Hall, Ledwith, and Mirchandani
Classification of Associative Two-Dimensional Algebras
Preliminaries
Classification
Conclusions
Concept
Properties of Algebras
Isomorphism
Fields
Abstract Algebra
We normally work with a one-dimensional algebra. For example,
3 = 3(1)
10 = 10(1)
−5 = −5(1)
Starting with the unit vector, we can obtain any quantity we are
looking for by multiplying by some scalar in R.
Doherty, Hall, Ledwith, and Mirchandani
Classification of Associative Two-Dimensional Algebras
Preliminaries
Classification
Conclusions
Concept
Properties of Algebras
Isomorphism
Fields
Two-Dimensional Algebras
Consider a negative real number, say −1. We may want x such
that x 2 = −1.
But our one-dimensional algebra cannot handle this situation. So
we use a second dimension in the i direction.
http://pirate.shu.edu/ wachsmut/complex/numbers/graphics/plane.gif
Doherty, Hall, Ledwith, and Mirchandani
Classification of Associative Two-Dimensional Algebras
Preliminaries
Classification
Conclusions
Concept
Properties of Algebras
Isomorphism
Fields
Describing an Algebra
Multiplication tables describe an algebra visually. They are read
much like Punnett squares in biology.
Example: The complex number system
×
1
i
1
1
i
i
i
−1
→
Doherty, Hall, Ledwith, and Mirchandani
×
u
v
u
u
v
v
v
−u
Classification of Associative Two-Dimensional Algebras
Preliminaries
Classification
Conclusions
Concept
Properties of Algebras
Isomorphism
Fields
Possible Properties of Two-Dimensional Algebras
An algebra A is
I
associative if and only if (xy )z = x(yz) for all x, y , z ∈ A
I
commutative if and only if xy = yx for all x, y ∈ A
I
unital if and only if ∃u ∈ A | ux = x for all x ∈ A
I
a division algebra if and only if for every x ∈ A there exists
x 0 ∈ A | xx 0 = x 0 x = 1.
For the purposes of this investigation, we only assume associativity.
Doherty, Hall, Ledwith, and Mirchandani
Classification of Associative Two-Dimensional Algebras
Preliminaries
Classification
Conclusions
Concept
Properties of Algebras
Isomorphism
Fields
Properties of Two-Dimensional Algebras
Doherty, Hall, Ledwith, and Mirchandani
Classification of Associative Two-Dimensional Algebras
Concept
Properties of Algebras
Isomorphism
Fields
Preliminaries
Classification
Conclusions
Isomorphism
From Greek iso, meaning “equal,” and morphosis, meaning “to
form”
Example: Two isomorphic algebras
×
u
v
u
u
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v
0
0
×
u
v
u
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v
v
u
v
Doherty, Hall, Ledwith, and Mirchandani
Classification of Associative Two-Dimensional Algebras
Preliminaries
Classification
Conclusions
Concept
Properties of Algebras
Isomorphism
Fields
Isomorphism (continued)
Why are these two algebras isomorphic?
×
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v
u
u
v
×
u
w
v
0
0
u
u
w
w
u
w
Start with the first algebra. Then choose w := u + βv .
I
uu does not change
I
uw = uu + βuv = uu = u
I
wu = uu + βvu = u + βv = w
I
ww = (u + βv )(u + βv ) = uu + βvu = u + βv = w
So there exists a mapping between these two algebras.
Doherty, Hall, Ledwith, and Mirchandani
Classification of Associative Two-Dimensional Algebras
Preliminaries
Classification
Conclusions
Concept
Properties of Algebras
Isomorphism
Fields
The Field
The field F is a one-dimensional, unital, and commutative algebra
that includes all the allowable coefficients for our two-dimensional
algebras.
We assume the field to be of characteristic 0.
For non-unital algebras, there are no other restrictions on the field.
However, for division algebras, we assume the field to be R.
Doherty, Hall, Ledwith, and Mirchandani
Classification of Associative Two-Dimensional Algebras
Preliminaries
Classification
Conclusions
A Useful Lemma
Non-unital Algebras
Unital Algebras
A Useful Lemma
Lemma 1.
For an algebra A there exists z ∈ A× such that either zz = 0 or
zz = z.
Proof.
Consider any u ∈ A× .
I
uu ∈ Fu. Let uu = ru for r ∈ F. If r = 0, then uu = 0.
Otherwise, define v := 1r u. vv = r12 uu = 1r u = v .
I
Case 2: u and uu are not collinear. Thus (u, uu) is a basis.
Suppose uuu = αu + βuu for α, β ∈ F, and choose
y := uu − βu. Then we have yy = αy . We have now reduced
the problem to a form covered by Case 1, for which we have
already shown the conclusion holds.
Doherty, Hall, Ledwith, and Mirchandani
Classification of Associative Two-Dimensional Algebras
Preliminaries
Classification
Conclusions
A Useful Lemma
Non-unital Algebras
Unital Algebras
Classifying the Non-unital Algebras
Assume ∃u ∈ A | uu = u.
×
u
v
u
u
γu + δv
v
αu + βv
I
u(uv ) = u(αu + βv ) = αuu + βuv = αu + β(αu + βv ) =
(α + αβ)u + β 2 v
I
(uu)v = uv = αu + βv
Doherty, Hall, Ledwith, and Mirchandani
Classification of Associative Two-Dimensional Algebras
Preliminaries
Classification
Conclusions
A Useful Lemma
Non-unital Algebras
Unital Algebras
Classifying the Non-unital Algebras (continued)
By the Associative Law,
u(uv ) = (uu)v
(α + αβ)u + β 2 v = αu + βv
Matching the coefficients gives us a system of equalities:
(
(
α + αβ = α
αβ = 0
therefore
2
β =β
β(β − 1) = 0
Doherty, Hall, Ledwith, and Mirchandani
Classification of Associative Two-Dimensional Algebras
Preliminaries
Classification
Conclusions
A Useful Lemma
Non-unital Algebras
Unital Algebras
Classifying the Non-unital Algebras (continued)
The Associative Law requires that


αβ = 0



γδ = 0
β(β − 1) = 0



δ(δ − 1) = 0



α(δ − 1) = γ(β − 1)
Thus there are four cases:
1. β = δ = 0
2. β = 1, δ = 0
3. β = 0, δ = 1
4. β = δ = 1
Doherty, Hall, Ledwith, and Mirchandani
Classification of Associative Two-Dimensional Algebras
Preliminaries
Classification
Conclusions
A Useful Lemma
Non-unital Algebras
Unital Algebras
Case 1: β = δ = 0
α(δ − 1) = γ(β − 1) =⇒ γ = α
×
u
v
u
u
αu
v
αu
We can simplify this algebra by assuming α 6= 0 and choosing
w := u − α1 v .
1
uw = u u − v = u − u = 0
α
1
wu = u − v u = u − u = 0
α
Doherty, Hall, Ledwith, and Mirchandani
Classification of Associative Two-Dimensional Algebras
Preliminaries
Classification
Conclusions
A Useful Lemma
Non-unital Algebras
Unital Algebras
Case 1: β = δ = 0 (continued)
Let ww := u + ζw . By the Associative Law,
( + ζw )u = (ww )u = w (wu) = 0 =⇒ = 0
and thus ww = ζw .
I
If ζ 6= 0, choose v 0 := ζ1 w which gives v 0 v 0 = ζ12 ww = ζ1 w = v 0 .
Next, choose z := u + v 0 . If (z, v 0 ) is the basis, we obtain a unital
case. This is isomorphic to the α = 0 case.
×
z
v0
z
z
v0
0
0
v
v
v0
I
If ζ = 0, we obtain a non-unital
×
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w
Doherty, Hall, Ledwith, and Mirchandani
case.
u
u
0
w
0
0
Classification of Associative Two-Dimensional Algebras
Preliminaries
Classification
Conclusions
A Useful Lemma
Non-unital Algebras
Unital Algebras
The Five Non-unital Algebras
The algebra we just found is denoted by I. Cases 2, 3, and 4 yield
II and III, and the @u ∈ A× | uu = u scenario yields IV and V.
I
II
III
IV
V
uu
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u
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uv
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0
Some important observations:
I
I, IV, and V are commutative while II and III are not.
I
None of these five algebras are isomorphic. Hence, we have a
complete classification of the non-unital algebras.
Doherty, Hall, Ledwith, and Mirchandani
Classification of Associative Two-Dimensional Algebras
Preliminaries
Classification
Conclusions
A Useful Lemma
Non-unital Algebras
Unital Algebras
Unital Algebras
An algebra is unital if there exists a unity u such that uv = v = vu
for all v . A unital algebra is divisional if and only if there exists y
for each x such that yx = u.
Lemma 2.
For all unital algebras A where u is the unity,
∃w ∈ A \ Fu | ww = ru for suitable r ∈ F.
Proof.
If (u, v ) is the basis, choose
I
I
I
r := α + 14 β 2
w := v − 21 βv
v v := αu + βv
Then verify that ww = ru.
Doherty, Hall, Ledwith, and Mirchandani
Classification of Associative Two-Dimensional Algebras
Preliminaries
Classification
Conclusions
A Useful Lemma
Non-unital Algebras
Unital Algebras
Unital Algebras (continued)
Theorem 3.
Consider a unital algebra A with unity u and a vector v | vv = ru.
A is a division algebra if and only if r 6= s 2 ∀s ∈ F.
Proof.
We choose x := αu + βv ∈ A× .
1
1. If r 6= s 2 , then α2 − r β 2 6= 0. Choose y := α2 −r
(αu − βv ).
β2
Then xy = yx = u and so the algebra is a division algebra.
2. If A is divisional, y (the inverse of x) exists for all α and β,
and thus α2 − r β 2 6= 0 =⇒ r 6= (α/β)2 .
Doherty, Hall, Ledwith, and Mirchandani
Classification of Associative Two-Dimensional Algebras
Preliminaries
Classification
Conclusions
A Useful Lemma
Non-unital Algebras
Unital Algebras
For the division algebras, we specify that the field is R.
1. r = 0: not a division algebra
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−u
2. r = 1: not a division algebra
×
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v
3. r = −1: a division algebra
×
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v
Doherty, Hall, Ledwith, and Mirchandani
Classification of Associative Two-Dimensional Algebras
Preliminaries
Classification
Conclusions
Summary of Findings
Further Research
Thanks
The Eight Two-Dimensional Associative Algebras
Theorem 4 (Main result).
Unital
Non-unital
Every two-dimensional associative algebra is either equivalent or
isomorphic to one of the eight algebras specified below.
I
II
III
IV
V
VI
VII
VIII
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u
0
0
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u
u
Doherty, Hall, Ledwith, and Mirchandani
uv
0
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vu
0
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−u
Classification of Associative Two-Dimensional Algebras
Preliminaries
Classification
Conclusions
Summary of Findings
Further Research
Thanks
Further Research
Retracting any one of the fundamental assumptions made in this
investigation changes the problem significantly.
Future research could entail classification of algebras that are
I
of three or more dimensions
I
nonassociative
I
over a field not of characteristic 0
Additionally, there is more work to be done with the
two-dimensional associative division algebras.
Doherty, Hall, Ledwith, and Mirchandani
Classification of Associative Two-Dimensional Algebras
Preliminaries
Classification
Conclusions
Summary of Findings
Further Research
Thanks
Acknowledgments
We extend our sincere gratitude to
I Dr. Barry Luokkala, Director of the PGSS
I Our host, Carnegie Mellon University
I All PGSS donors, including:
I
I
I
I
I
I
I
I
I
I
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Pennsylvania Department of Education
PGSS Campaign, Inc.
EQT
AT&T
Alcoa
PPG
Teva Pharmaceuticals
Westinghouse
Dr. Russ Walker, Discrete Math Professor
Quang Sack, Teaching Assistant
Dr. Juan Jorge Schäffer, Project Advisor
Doherty, Hall, Ledwith, and Mirchandani
Classification of Associative Two-Dimensional Algebras
Preliminaries
Classification
Conclusions
Summary of Findings
Further Research
Thanks
Thank you. Any questions?
Doherty, Hall, Ledwith, and Mirchandani
Classification of Associative Two-Dimensional Algebras
Preliminaries
Classification
Conclusions
Summary of Findings
Further Research
Thanks
Sources
Images
http://pirate.shu.edu/ wachsmut/complex/numbers/graphics/plane.gif
http://i.imgur.com/F6N5EYC.png
Doherty, Hall, Ledwith, and Mirchandani
Classification of Associative Two-Dimensional Algebras