Algebra in Flatland Connie Dennis*, Cassie Stamper* Department of - - PowerPoint PPT Presentation
Algebra in Flatland Connie Dennis*, Cassie Stamper* Department of Mathematics Kansas State University Manhattan, KS 66506 Mentor: Dr. David Yetter* July 24, 2012 * Supported by NSF grant # DMS1004336 Motivation C. Dennis, C.Stamper (KSU)
Connie Dennis*, Cassie Stamper*
Department of Mathematics Kansas State University Manhattan, KS 66506
Mentor: Dr. David Yetter* July 24, 2012 * Supported by NSF grant # DMS1004336
SUMaR 2012 July 24, 2012 2 / 35
The starting point is to think of computations as physical things that take place in space and time, rather than abstractions.
SUMaR 2012 July 24, 2012 2 / 35
The starting point is to think of computations as physical things that take place in space and time, rather than abstractions. Secondly, we wanted to compute things by representing mathematical quantatities by quantum mechanical states.
SUMaR 2012 July 24, 2012 2 / 35
The starting point is to think of computations as physical things that take place in space and time, rather than abstractions. Secondly, we wanted to compute things by representing mathematical quantatities by quantum mechanical states. This is the idea of quantum computing.
SUMaR 2012 July 24, 2012 2 / 35
The starting point is to think of computations as physical things that take place in space and time, rather than abstractions. Secondly, we wanted to compute things by representing mathematical quantatities by quantum mechanical states. This is the idea of quantum computing. Physics in two spatial dimensions is different than physics in three and higher dimensions.
SUMaR 2012 July 24, 2012 2 / 35
The starting point is to think of computations as physical things that take place in space and time, rather than abstractions. Secondly, we wanted to compute things by representing mathematical quantatities by quantum mechanical states. This is the idea of quantum computing. Physics in two spatial dimensions is different than physics in three and higher dimensions. In three and higher dimensions there are only two types of particles: bosons and fermions.
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If two bosons in identical states are swapped, the state remains the same.
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If two bosons in identical states are swapped, the state remains the same. If two fermions in identical states are swapped, the state negates.
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If two bosons in identical states are swapped, the state remains the same. If two fermions in identical states are swapped, the state negates. In a plane, quasi particles can exhibit the fractional quantum hall effect.
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If two bosons in identical states are swapped, the state remains the same. If two fermions in identical states are swapped, the state negates. In a plane, quasi particles can exhibit the fractional quantum hall effect. When two quasi particles are swapped, the state is multiplied by a complex number with absolute value 1.
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If two bosons in identical states are swapped, the state remains the same. If two fermions in identical states are swapped, the state negates. In a plane, quasi particles can exhibit the fractional quantum hall effect. When two quasi particles are swapped, the state is multiplied by a complex number with absolute value 1. The motivation for this project is to represent math in quantum mechanical states, where when two things are swapped, the state is multiplied by a phase ζ which is a primitive Nth root of unity.
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Definition
Let Z be the set of all integers. A Z graded vector space is a vector space, V which decomposes into a direct sum of the form: V =
Vn Where each Vn is a vector space. For a given n the elements of Vn are then called homogeneous elements of degree n.
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We will use || to represent degree.
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We will use || to represent degree. The anyonic braiding associated to ζ (ζN = 1 primitive) is given by: ∀ a,b homogeneous σ(a ⊗ b) = ζ|a||b|b ⊗ a
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Question
What happens to ordinary commutativity and associativity in the anyonic setting?
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Question
What happens to ordinary commutativity and associativity in the anyonic setting? Answer 1: Commutative became a · b = ζ|a||b| b · a
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Question
What happens to ordinary commutativity and associativity in the anyonic setting? Answer 1: Commutative became a · b = ζ|a||b| b · a Answer 2: Associative stays the same unless the multiplication changes degrees.
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Question
What happens to ordinary commutativity and associativity in the anyonic setting? Answer 1: Commutative became a · b = ζ|a||b| b · a Answer 2: Associative stays the same unless the multiplication changes degrees. Answer 3: If |m| = 0 we will answer later.
SUMaR 2012 July 24, 2012 6 / 35
Question
What happens to ordinary commutativity and associativity in the anyonic setting? Answer 1: Commutative became a · b = ζ|a||b| b · a Answer 2: Associative stays the same unless the multiplication changes degrees. Answer 3: If |m| = 0 we will answer later.
Question
Do the resulting axioms give reasonable algebraic systems?
SUMaR 2012 July 24, 2012 6 / 35
Question
What happens to ordinary commutativity and associativity in the anyonic setting? Answer 1: Commutative became a · b = ζ|a||b| b · a Answer 2: Associative stays the same unless the multiplication changes degrees. Answer 3: If |m| = 0 we will answer later.
Question
Do the resulting axioms give reasonable algebraic systems? We used rewrite systems as a tool to answer this question.
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Definition
A rewrite system is a collection of directed rules for replacing parts of symbol strings with other symbol strings.
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Definition
A rewrite system is a collection of directed rules for replacing parts of symbol strings with other symbol strings.
Definition
Descending Chain Condition- given a word there are no infinite strings of successive rule applications which can be made starting at the word.
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Definition
A rewrite system is a collection of directed rules for replacing parts of symbol strings with other symbol strings.
Definition
Descending Chain Condition- given a word there are no infinite strings of successive rule applications which can be made starting at the word.
Definition
Local Confluence- If two instances apply to a word, then there are sequences of rule applications to each of the results which give equal results.
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Theorem
Knuth - Bendix Theorem: If a rewrite system satisfies two properties, the Descending Chain Condition (DCC) and Local Confluence, then: (a) Given any word, there is a unique reduced word. The reduced word is the canonical representative of the equivalence class. (b) If two words are equivalent, the reduced word reachable from each is the same.
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C[x(d1)
1
....x(dn)
n
]ζ
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C[x(d1)
1
....x(dn)
n
]ζ freely generated by x1......xn
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C[x(d1)
1
....x(dn)
n
]ζ freely generated by x1......xn
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C[x(d1)
1
....x(dn)
n
]ζ freely generated by x1......xn
ζN = 1
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C[x(d1)
1
....x(dn)
n
]ζ freely generated by x1......xn
ζN = 1 ∀a, b homogenous :
SUMaR 2012 July 24, 2012 9 / 35
C[x(d1)
1
....x(dn)
n
]ζ freely generated by x1......xn
ζN = 1 ∀a, b homogenous : a · b = ζ|a||b|b · a
SUMaR 2012 July 24, 2012 9 / 35
C[x(d1)
1
....x(dn)
n
]ζ freely generated by x1......xn
ζN = 1 ∀a, b homogenous : a · b = ζ|a||b|b · a b · a = ζ|b||a|a · b
SUMaR 2012 July 24, 2012 9 / 35
C[x(d1)
1
....x(dn)
n
]ζ freely generated by x1......xn
ζN = 1 ∀a, b homogenous : a · b = ζ|a||b|b · a b · a = ζ|b||a|a · b So by solving:
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C[x(d1)
1
....x(dn)
n
]ζ freely generated by x1......xn
ζN = 1 ∀a, b homogenous : a · b = ζ|a||b|b · a b · a = ζ|b||a|a · b So by solving: a · b = ζ−|a||b|b · a
SUMaR 2012 July 24, 2012 9 / 35
C[x(d1)
1
....x(dn)
n
]ζ freely generated by x1......xn
ζN = 1 ∀a, b homogenous : a · b = ζ|a||b|b · a b · a = ζ|b||a|a · b So by solving: a · b = ζ−|a||b|b · a a · b = 0 ⇒ N | 2|a||b|
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N ∤ |a|2 ⇒ a2 = 0
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N ∤ |a|2 ⇒ a2 = 0 N ∤ 2|a||b| ⇒ ab = 0
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N ∤ |a|2 ⇒ a2 = 0 N ∤ 2|a||b| ⇒ ab = 0 N|2|a||b|and N ∤ |a||b| ⇒ ab = −ba
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N ∤ |a|2 ⇒ a2 = 0 N ∤ 2|a||b| ⇒ ab = 0 N|2|a||b|and N ∤ |a||b| ⇒ ab = −ba N||a||b| ⇒ ab = ba
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n 1 2 C C C 1 C 2 C
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n 1 2 C C C 1 C 2 C All phase orders that are odd primes behave this way
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n Must square to 0 ? No 1 Yes 2 Yes
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n 1 2 3 C C C C 1 C A 2 C A C A 3 C A
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n Must square to 0 ? No 1 Yes 2 No 3 Yes
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n 1 2 3 4 5 6 7 C C C C C C C C 1 C A 2 C A C A 3 C A 4 C A C A C A C A 5 C A 6 C A C A 7 C A
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n 1 2 3 4 5 6 7 C C C C C C C C 1 C A 2 C A C A 3 C A 4 C A C A C A C A 5 C A 6 C A C A 7 C A All phase orders that are a power of 2 behave this way
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n Must square to 0 ? No 1 Yes 2 Yes 3 Yes 4 No 5 Yes 6 Yes 7 Yes
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k > 2µ2(|a|) ⇒ a2 = 0
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k > 2µ2(|a|) ⇒ a2 = 0 k − 1 > µ2(|a|) + µ2(|b|) ⇒ ab = 0
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k > 2µ2(|a|) ⇒ a2 = 0 k − 1 > µ2(|a|) + µ2(|b|) ⇒ ab = 0 k − 1 = µ2(|a|) + µ2(|b|) ⇒ ab = −ba
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k > 2µ2(|a|) ⇒ a2 = 0 k − 1 > µ2(|a|) + µ2(|b|) ⇒ ab = 0 k − 1 = µ2(|a|) + µ2(|b|) ⇒ ab = −ba k ≦ µ2(|a|) + µ2(|b|) ⇒ ab = ba
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n 1 2 3 4 5 C C C C C C 1 C A 2 C C 3 C A C A C A 4 C C 5 C A
SUMaR 2012 July 24, 2012 18 / 35
n 1 2 3 4 5 C C C C C C 1 C A 2 C C 3 C A C A C A 4 C C 5 C A All phase orders that are a power of 2 multiplied with an odd prime behave this way
SUMaR 2012 July 24, 2012 18 / 35
n Must square to 0 ? No 1 Yes 2 Yes 3 Yes 4 Yes 5 Yes
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n 1 2 3 4 5 6 7 8 C C C C C C C C C 1 C 2 C 3 C C C 4 C 5 C 6 C C C 7 C 8 C
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n 1 2 3 4 5 6 7 8 C C C C C C C C C 1 C 2 C 3 C C C 4 C 5 C 6 C C C 7 C 8 C All phase orders that are a product of two odd primes behave this way
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n Must square to 0 ? No 1 Yes 2 Yes 3 No 4 Yes 5 Yes 6 No 7 Yes 8 Yes
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n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 C C C C C C C C C C C C C C C 1 C 2 C 3 C C C 4 C 5 C C C C C 6 C C C 7 C 8 C 9 C C C 10 C C C C C 11 C 12 C C C 13 C 14 C
SUMaR 2012 July 24, 2012 22 / 35
n Must square to 0 ? No 1 Yes 2 Yes 3 Yes 4 Yes 5 Yes 6 Yes 7 Yes 8 Yes 9 Yes 10 Yes 11 Yes 12 Yes 13 Yes 14 Yes
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C{x(d1)
1
....x(dk)
k
}ζ,dm
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C{x(d1)
1
....x(dk)
k
}ζ,dm freely generated by x1......xk
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C{x(d1)
1
....x(dk)
k
}ζ,dm freely generated by x1......xk
SUMaR 2012 July 24, 2012 24 / 35
C{x(d1)
1
....x(dk)
k
}ζ,dm freely generated by x1......xk
Convention: All braidings are positive.
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C{x(d1)
1
....x(dk)
k
}ζ,dm freely generated by x1......xk
Convention: All braidings are positive. The opposite convention would be the same except with respect to ζ−1
SUMaR 2012 July 24, 2012 24 / 35
C{x(d1)
1
....x(dk)
k
}ζ,dm freely generated by x1......xk
Convention: All braidings are positive. The opposite convention would be the same except with respect to ζ−1 Weakly associative with respect to the anyonic braiding for ζ
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Elements are complex linear combinations of binary trees.
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Elements are complex linear combinations of binary trees. The final nodes are labeled from x1....xk
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Elements are complex linear combinations of binary trees. The final nodes are labeled from x1....xk A final node is a node whose subtrees are both empty.
SUMaR 2012 July 24, 2012 25 / 35
Elements are complex linear combinations of binary trees. The final nodes are labeled from x1....xk A final node is a node whose subtrees are both empty. There is a height function on the nodes with range {0, 1, ..., M} or ∅
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All final nodes are at the same height.
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All final nodes are at the same height. No two non-final nodes are at the same height.
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All final nodes are at the same height. No two non-final nodes are at the same height. The children of a node are higher than the parent.
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All final nodes are at the same height. No two non-final nodes are at the same height. The children of a node are higher than the parent. The empty tree names the identity.
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ζ|m||a|
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rewrites.pdf
T
T
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a b c d e
((ab)1(cd)2e)3
a b c d e
((ab)1(c(de)2)3 ζ (|m||c|)
a b c d e
((ab)2 (c(de)1)3 ζ
(|m||c|) – (|m||m|)
a b c d e
ζ
(|m||c|) – 2(|m||m|)
(ab)3(c(de)1)2
a b c d e
(ab)3((cd)1e)2
ζ
a b c d e
((ab)2(cd))1e)3 ζ
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Proof 2.pdf
a b c d e
a b c d e ζ(|m||a|)
a b c d e
ζ
(|m||a|) - |m||m|
a b c d e
ζ
(|m||a|) – 2(|m||m|)
a b c d e
ζ
– 2(|m||m|)
a b c d e
ζ - (|m||m|)
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Proof 3.pdf
a b c d e f a b c d e f
ζ
a b c d e f
ζ
a b c d e f
ζ
a b c d e f
ζ
a b c d e f
ζ
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Proof 4.pdf
a b c d
a b c d a b c d
a b c d
a b c d a b c d
ζ
|m|(|m|+|a|+|b|)
ζ
|m|(|a|+|b|)
ζ
2(|m|+|a|)+|b|
ζ
|m|(|m|+|a|)
ζ
(|m|+|a|)
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Every element in a free-weakly associative algebra can be uniquely represented as a linear combination of reduced trees.
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Every element in a free-weakly associative algebra can be uniquely represented as a linear combination of reduced trees. Reduced means all left subtrees are empty or have exactly one node, and all non-final nodes have two children.
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Thank You!
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