Temperley–lieb algebra: from knot theory to logic and computation via quantum mechanics
<p style="text-align:justify;"> Abstract We study the Temperley-Lieb algebra, central to the Jones polynomial invariant of knots and ensuing developments, from a novel point of view. We relate the Temperley-Lieb category to the categorical formulation of quantum mechanics introduced...
| Autor principal: | |
|---|---|
| Format: | Book section |
| Publicat: |
Taylor and Francis
2007
|
| _version_ | 1826262091222220800 |
|---|---|
| author | Abramsky, S |
| author_facet | Abramsky, S |
| author_sort | Abramsky, S |
| collection | OXFORD |
| description | <p style="text-align:justify;"> Abstract We study the Temperley-Lieb algebra, central to the Jones polynomial invariant of knots and ensuing developments, from a novel point of view. We relate the Temperley-Lieb category to the categorical formulation of quantum mechanics introduced by Abramsky and Coecke as the basis for the development of high-level methods for quantum information and computation. We develop some structural properties of the Temperley-Lieb category, giving a simple diagrammatic description of epi-monic factorization, and hence of splitting idempotents. We then relate the Temperley-Lieb category to some topics in proof theory and computation. We give a direct, “fully abstract” description of the Temperley-Lieb category, in which arrows are just relations on discrete finite sets, with planarity being characterized by simple order-theoretic properties. The composition is described in terms of the “Geometry of Interaction” construction, originally introduced to analyze cut elimination in Linear Logic. Thus we obtain a planar version of Geometry of Interaction. Moreover, we get an explicit description of the free pivotal category on one self-dual object, which is easily generalized to an arbitrary generating category. Moreover, we show that the construction naturally lifts a dagger structure on the underlying category, thus exhibiting a key feature of the Abramsky-Coecke axiomatization. The dagger or “adjoint”, and the “complex conjugate”, acquire natural diagrammatic readings in this context. Finally, we interpret a non-commutative lambda calculus (a variant of the Lambek calculus, widely used in computational linguistics) in the Temperley-Lieb category, and thus show how diagrammatic simplification can be viewed as functional computation. </p> |
| first_indexed | 2024-03-06T19:30:53Z |
| format | Book section |
| id | oxford-uuid:1d6daec2-72f3-4176-b61d-e5fd4f93bb11 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T19:30:53Z |
| publishDate | 2007 |
| publisher | Taylor and Francis |
| record_format | dspace |
| spelling | oxford-uuid:1d6daec2-72f3-4176-b61d-e5fd4f93bb112022-03-26T11:10:43ZTemperley–lieb algebra: from knot theory to logic and computation via quantum mechanicsBook sectionhttp://purl.org/coar/resource_type/c_3248uuid:1d6daec2-72f3-4176-b61d-e5fd4f93bb11Symplectic Elements at OxfordTaylor and Francis2007Abramsky, S <p style="text-align:justify;"> Abstract We study the Temperley-Lieb algebra, central to the Jones polynomial invariant of knots and ensuing developments, from a novel point of view. We relate the Temperley-Lieb category to the categorical formulation of quantum mechanics introduced by Abramsky and Coecke as the basis for the development of high-level methods for quantum information and computation. We develop some structural properties of the Temperley-Lieb category, giving a simple diagrammatic description of epi-monic factorization, and hence of splitting idempotents. We then relate the Temperley-Lieb category to some topics in proof theory and computation. We give a direct, “fully abstract” description of the Temperley-Lieb category, in which arrows are just relations on discrete finite sets, with planarity being characterized by simple order-theoretic properties. The composition is described in terms of the “Geometry of Interaction” construction, originally introduced to analyze cut elimination in Linear Logic. Thus we obtain a planar version of Geometry of Interaction. Moreover, we get an explicit description of the free pivotal category on one self-dual object, which is easily generalized to an arbitrary generating category. Moreover, we show that the construction naturally lifts a dagger structure on the underlying category, thus exhibiting a key feature of the Abramsky-Coecke axiomatization. The dagger or “adjoint”, and the “complex conjugate”, acquire natural diagrammatic readings in this context. Finally, we interpret a non-commutative lambda calculus (a variant of the Lambek calculus, widely used in computational linguistics) in the Temperley-Lieb category, and thus show how diagrammatic simplification can be viewed as functional computation. </p> |
| spellingShingle | Abramsky, S Temperley–lieb algebra: from knot theory to logic and computation via quantum mechanics |
| title | Temperley–lieb algebra: from knot theory to logic and computation via quantum mechanics |
| title_full | Temperley–lieb algebra: from knot theory to logic and computation via quantum mechanics |
| title_fullStr | Temperley–lieb algebra: from knot theory to logic and computation via quantum mechanics |
| title_full_unstemmed | Temperley–lieb algebra: from knot theory to logic and computation via quantum mechanics |
| title_short | Temperley–lieb algebra: from knot theory to logic and computation via quantum mechanics |
| title_sort | temperley lieb algebra from knot theory to logic and computation via quantum mechanics |
| work_keys_str_mv | AT abramskys temperleyliebalgebrafromknottheorytologicandcomputationviaquantummechanics |