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{"work":{"id":20923958,"created_at":"2016-01-27T04:28:10.841-08:00","from_world_paper_id":147119554,"updated_at":"2025-02-01T18:01:38.020-08:00","_data":{"abstract":"Survey article on loop groups and their representations, following a course of three lectures held at the summer school \"algebraic groups\" at the Georg-August-Universitaet zu Goettingen, June 27--July 13, 2005. We discuss loop groups, their central extensions, and positive energy representations.","ai_title_tag":"Loop Groups: Extensions and Representations","publication_date":"2008,2,26"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Loop groups and string topology","broadcastable":true,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [35644036]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "full_page_mobile_sutd_modal"; window.loswp.useOptimizedScribd4genScript = false; window.loginModal = {}; window.loginModal.appleClientId = 'edu.academia.applesignon'; window.userInChina = "false";</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":41630568,"attachmentType":"pdf"}"><img alt="First page of “Loop groups and string topology”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/41630568/mini_magick20190218-30460-13oytyv.png?1550560029" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/images/single_work_splash/adobe_icon.svg" /><div class="ds-work-cover--hover-container"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span><p>Download Free PDF</p></div><div class="ds-work-cover--ribbon-container">Download Free PDF</div><div class="ds-work-cover--ribbon-triangle"></div></button></div></div></div><div class="ds-work-card--work-information"><h1 class="ds-work-card--work-title">Loop groups and string topology</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="35644036" href="https://uni-goettingen.academia.edu/TSchick"><img alt="Profile image of Thomas Schick" class="ds-work-card--author-avatar" src="https://0.academia-photos.com/35644036/10332640/11530375/s65_t..schick.jpg" />Thomas Schick</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2008</p><div class="ds-work-card--work-metadata"><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">visibility</span><p class="ds2-5-body-sm" id="work-metadata-view-count">…</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span><p class="ds2-5-body-sm">18 pages</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">link</span><p class="ds2-5-body-sm">1 file</p></div></div><script>(async () => { const workId = 20923958; const worksViewsPath = "/v0/works/views?subdomain_param=api&work_ids%5B%5D=20923958"; const getWorkViews = async (workId) => { const response = await fetch(worksViewsPath); if (!response.ok) { throw new Error('Failed to load work views'); } const data = await response.json(); return data.views[workId]; }; // Get the view count for the work - we send this immediately rather than waiting for // the DOM to load, so it can be available as soon as possible (but without holding up // the backend or other resource requests, because it's a bit expensive and not critical). const viewCount = await getWorkViews(workId); const updateViewCount = (viewCount) => { try { const viewCountNumber = parseInt(viewCount, 10); if (viewCountNumber === 0) { // Remove the whole views element if there are zero views. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); return; } const commaizedViewCount = viewCountNumber.toLocaleString(); const viewCountBody = document.getElementById('work-metadata-view-count'); if (!viewCountBody) { throw new Error('Failed to find work views element'); } viewCountBody.textContent = `${commaizedViewCount} views`; } catch (error) { // Remove the whole views element if there was some issue parsing. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); throw new Error(`Failed to parse view count: ${viewCount}`, error); } }; // If the DOM is still loading, wait for it to be ready before updating the view count. if (document.readyState === "loading") { document.addEventListener('DOMContentLoaded', () => { updateViewCount(viewCount); }); // Otherwise, just update it immediately. } else { updateViewCount(viewCount); } })();</script></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">Survey article on loop groups and their representations, following a course of three lectures held at the summer school "algebraic groups" at the Georg-August-Universitaet zu Goettingen, June 27--July 13, 2005. We discuss loop groups, their central extensions, and positive energy representations.</p><div class="ds-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--work-card","attachmentId":41630568,"attachmentType":"pdf","workUrl":"https://www.academia.edu/20923958/Loop_groups_and_string_topology"}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--work-card","attachmentId":41630568,"attachmentType":"pdf","workUrl":"https://www.academia.edu/20923958/Loop_groups_and_string_topology"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div><div class="ds-signup-banner-trigger-container"><div class="ds-signup-banner-trigger ds-signup-banner-trigger-control"></div></div><div class="ds-signup-banner ds-signup-banner-control"><div id="ds-signup-banner-close-button"><button class="ds2-5-button ds2-5-button--secondary ds2-5-button--inverse"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">close</span></button></div><div class="ds-signup-banner-ctas"><img src="//a.academia-assets.com/images/academia-logo-capital-white.svg" /><h4 class="ds2-5-heading-serif-sm">Sign up for access to the world's latest research</h4><button class="ds2-5-button ds2-5-button--inverse ds2-5-button--full-width js-swp-download-button" data-signup-modal="{"location":"signup-banner"}">Sign up for free<span class="material-symbols-outlined" style="font-size: 20px" translate="no">arrow_forward</span></button></div><div class="ds-signup-banner-divider"></div><div class="ds-signup-banner-reasons"><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Get notified about relevant papers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Save papers to use in your research</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Join the discussion with peers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Track your impact</span></div></div></div><script>(() => { // Set up signup banner show/hide behavior: // 1. 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Then the space LG of all maps from the circle S to G becomes a group by pointwise multiplication. Actually, there are different variants of LG, depending on the classes of maps one considers, and the topology to be put on the mapping space. In these lectures, we will always look at the space of smooth (i.e. C∞) maps, with the topology of uniform convergence of all derivatives. These groups certainly are not algebraic groups in the usual sense of the word. Nevertheless, they share many properties of algebraic groups (concerning e.g. their representation theory). There are actually analogous objects which are very algebraic (compare e.g. [1]), and it turns out that those have properties remarkably close to those of the smooth loop groups. The lectures are organized as follows.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Loop groups and string topology Lectures for the summer school algebraic groups Gottingen, July 2005","attachmentId":82368015,"attachmentType":"pdf","work_url":"https://www.academia.edu/74090537/Loop_groups_and_string_topology_Lectures_for_the_summer_school_algebraic_groups_Gottingen_July_2005","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/74090537/Loop_groups_and_string_topology_Lectures_for_the_summer_school_algebraic_groups_Gottingen_July_2005"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="1" data-entity-id="25334084" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/25334084/Notes_on_string_topology">Notes on string topology</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="48707945" href="https://spanalumni.academia.edu/AlexanderVoronov">Alexander A Voronov</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Arxiv preprint math/0503625, 2005</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Notes on string topology","attachmentId":45632546,"attachmentType":"pdf","work_url":"https://www.academia.edu/25334084/Notes_on_string_topology","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/25334084/Notes_on_string_topology"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="2" data-entity-id="98935547" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/98935547/The_extended_loop_group_An_infinite_dimensional_manifold_associated_with_the_loop_space">The extended loop group: An infinite dimensional manifold associated with the loop space</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="3120684" href="https://psico.academia.edu/RodolfoGambini">Rodolfo Gambini</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Communications in Mathematical Physics, 1993</p><p class="ds-related-work--abstract ds2-5-body-sm">A set of coordinates in the non parametric loop-space is introduced. We show that these coordinates transform under infinite dimensional linear representations of the diffeomorphism group. An extension of the group of loops in terms of these objects is proposed. The enlarged group behaves locally as an infinite dimensional Lie group. Ordinary loops form a subgroup of this group. The algebraic properties of this new mathematical structure are analized in detail. Applications of the formalism to field theory, quantum gravity and knot theory are considered.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"The extended loop group: An infinite dimensional manifold associated with the loop space","attachmentId":100154317,"attachmentType":"pdf","work_url":"https://www.academia.edu/98935547/The_extended_loop_group_An_infinite_dimensional_manifold_associated_with_the_loop_space","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/98935547/The_extended_loop_group_An_infinite_dimensional_manifold_associated_with_the_loop_space"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="3" data-entity-id="47985555" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/47985555/String_structures_on_loop_bundles">String structures on loop bundles</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="9472496" href="https://independent.academia.edu/RobertCoquereaux">Robert Coquereaux</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Communications in Mathematical Physics, 1989</p><p class="ds-related-work--abstract ds2-5-body-sm">Differential geometry and topology of principal loop bundles (bundles of loop groups over loop spaces) are investigated. String structures, defined as bundle extensions corresponding to the central extension of the structure group, do not always exist. Various methods for deriving the obstruction to the existence of string structures are discussed.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"String structures on loop bundles","attachmentId":66832794,"attachmentType":"pdf","work_url":"https://www.academia.edu/47985555/String_structures_on_loop_bundles","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/47985555/String_structures_on_loop_bundles"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="4" data-entity-id="22613179" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/22613179/A_homotopy_theoretic_realization_of_string_topology">A homotopy theoretic realization of string topology</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="44175622" href="https://stanford.academia.edu/RalphCohen">Ralph Cohen</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Mathematische Annalen, 2002</p><p class="ds-related-work--abstract ds2-5-body-sm">Let M be a closed, oriented manifold of dimension d. Let LM be the space of smooth loops in M . In [2] Chas and Sullivan defined a product on the homology H * (LM ) of degree −d. They then investigated other structure that this product induces, including a Lie algebra structure on H * (LM ), and an induced product on the S 1 equivariant homology,</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A homotopy theoretic realization of string topology","attachmentId":43212159,"attachmentType":"pdf","work_url":"https://www.academia.edu/22613179/A_homotopy_theoretic_realization_of_string_topology","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/22613179/A_homotopy_theoretic_realization_of_string_topology"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="5" data-entity-id="60034192" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/60034192/Geometry_of_the_analytic_loop_group">Geometry of the analytic loop group</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="47048435" href="https://independent.academia.edu/CorradoDeConcini">Corrado De Concini</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Advances in Mathematics, 2013</p><p class="ds-related-work--abstract ds2-5-body-sm">We introduce and study a notion of analytic loop group with a Riemann-Hilbert factorization relevant for the representation theory of quantum affine algebras at roots of unity Uǫ(ĝ) with non trivial central charge. We introduce a Poisson structure and study properties of its Poisson dual group. We prove that the Hopf-Poisson structure is isomorphic to the semi-classical limit of the center of Uǫ(ĝ) (it is a geometric realization of the center). Then the symplectic leaves, and corresponding equivalence classes of central characters of Uǫ(ĝ), are parameterized by certain G-bundles on an elliptic curve.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Geometry of the analytic loop group","attachmentId":73657116,"attachmentType":"pdf","work_url":"https://www.academia.edu/60034192/Geometry_of_the_analytic_loop_group","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/60034192/Geometry_of_the_analytic_loop_group"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="6" data-entity-id="22613235" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/22613235/A_polarized_view_of_string_topology">A polarized view of string topology</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="44175622" href="https://stanford.academia.edu/RalphCohen">Ralph Cohen</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Proceedings of the 2002 Oxford Symposium in Honour of the 60th Birthday of Graeme Segal, 2004</p><p class="ds-related-work--abstract ds2-5-body-sm">Let M be a closed, connected manifold, and LM its loop space. In this paper we describe closed string topology operations in h * (LM ), where h * is a generalized homology theory that supports an orientation of M . We will show that these operations give h * (LM ) the structure of a unital, commutative Frobenius algebra without a counit. Equivalently they describe a positive boundary, two dimensional topological quantum field theory associated to h * (LM ). This implies that there are operations corresponding to any surface with p incoming and q outgoing boundary components, so long as q ≥ 1. The absence of a counit follows from the nonexistence of an operation associated to the disk, D 2 , viewed as a cobordism from the circle to the empty set.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A polarized view of string topology","attachmentId":43212192,"attachmentType":"pdf","work_url":"https://www.academia.edu/22613235/A_polarized_view_of_string_topology","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/22613235/A_polarized_view_of_string_topology"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="7" data-entity-id="22613249" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/22613249/The_homotopy_invariance_of_the_string_topology_loop_product_and_string_bracket">The homotopy invariance of the string topology loop product and string bracket</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="44175622" href="https://stanford.academia.edu/RalphCohen">Ralph Cohen</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Topology, 2008</p><p class="ds-related-work--abstract ds2-5-body-sm">Let f : M_1 \to M_2 be a homotopy equivalence of closed, oriented n -manifolds. Then the induced equivalence, Lf : LM_1 \to LM_2 induces a ring isomorphism in homology, and an isomorphism of Lie algebras in equivariant homology. The analogous statement also holds true for any generalized homology theory h_* that supports an orientation of the M_i 's.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"The homotopy invariance of the string topology loop product and string bracket","attachmentId":43212154,"attachmentType":"pdf","work_url":"https://www.academia.edu/22613249/The_homotopy_invariance_of_the_string_topology_loop_product_and_string_bracket","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/22613249/The_homotopy_invariance_of_the_string_topology_loop_product_and_string_bracket"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="8" data-entity-id="25334117" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/25334117/String_Topology_and_Cyclic_Homology">String Topology and Cyclic Homology</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="44175622" href="https://stanford.academia.edu/RalphCohen">Ralph Cohen</a><span>, </span><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="48707945" href="https://spanalumni.academia.edu/AlexanderVoronov">Alexander A Voronov</a></div><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline 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