{"id":{"repo_id":"washington","oai_identifier":"oai:digital.lib.washington.edu:1773/52102"},"canonical_url":"https://search.dev.ndltd.org/etd/washington/oai:digital.lib.washington.edu:1773/52102","repository":{"repo_id":"washington","name":"University of Washington","base_url":"https://digital.lib.washington.edu/server/oai/request"},"display":{"title":"The Koszul dual to n-Lie, n-Com algebras, and Young tableaux","abstract":"We study the operad $n\\text{-}Lie_d$, whose algebras are $n$-Lie algebras, which was first introduced in Nambu mechanics to extend Hamiltonian mechanics to more than one Hamiltonian. We find the Koszul dual of $n\\text{-}Lie_{-d+n-2}$ to be the operad $n\\text{-}Com_d$, whose relations come from the Specht module $S^{(n,n-1)}$. We combine the operads $n$-Lie and $m$-Com to construct the operad $(m,n)-Poiss$, where the rewriting rule that relates them is through a generalized Leibniz rule. We generalize the above Koszul duality to different types of generalization of $Lie$ and $Com$ which arise from the eigenspaces of the general Kneser graphs $\\cO_{n,s}$, where the operads $n\\text{-}Lie$ ($n\\text{-}Com)$ and $Lie_n^d$ ($Com_n^d)$ appear on different sides of the spectrum based on the parameter $s$. With the introduction of the new class of $n$-Com algebras through the Koszul duality, we take our first step in exploring these new types of algebras. Specifically, we start the work of trying to classify finite-dimensional simple $3$-Com algebras $C$ using the Peirce decomposition through semisimple idempotents $e$ through $\\chi_e=m_3(e,e,-)$ to obtain important structural properties. In particular, it decomposes the $3$-Com algebra $C=\\bigoplus C_e(t)$ for eigenvalues $t$ for $\\chi_e$ in which $C_e(1)$ is a commutative unital associative $k$-algebra acting on the other components, which is almost associative. Furthermore, we briefly construct an analog of the Killing form for $3$-Com algebras, denoted as $\\kappa$, and define non-degeneracy when the form is non-degenerate and fully degenerate when $\\kappa=0$. We use this to show that every non-degenerate $3$-Com algebra is a direct product of non-degenerate simple $3$-Com algebras. However, one very interesting aspect of this is that not every finite-dimensional $3$-Com algebra is non-degenerate, as our main example is fully degenerate. Towards some classifications of simple $3$-Com algebras, if $e$ is primitive semisimple idempotent with exactly two eigenvalues, and $C$ is simple, then its eigenvalues consist of $\\{1,-1\\}$ which help give a classification in dimension $2$ and $3$. Finally, we construct combinatorial objects, called Young $n$-trees, which are just rooted trees with a local Young tableaux structure at each edge following what is happening in the $n$-Com operad. In particular, we use these to give an upper bound to the arities of the dimension for the operad $n\\text{-}Com_d$, which gives a description for it.","abstract_html":"We study the operad <span class=\"etd-inline-math\">n\\text{-}Lie<sub>d</sub></span>, whose algebras are $n$-Lie algebras, which was first introduced in Nambu mechanics to extend Hamiltonian mechanics to more than one Hamiltonian. We find the Koszul dual of <span class=\"etd-inline-math\">n\\text{-}Lie<sub>-d+n-2</sub></span> to be the operad <span class=\"etd-inline-math\">n\\text{-}Com<sub>d</sub></span>, whose relations come from the Specht module <span class=\"etd-inline-math\">S<sup>(n,n-1)</sup></span>. We combine the operads $n$-Lie and $m$-Com to construct the operad $(m,n)-Poiss$, where the rewriting rule that relates them is through a generalized Leibniz rule. We generalize the above Koszul duality to different types of generalization of $Lie$ and $Com$ which arise from the eigenspaces of the general Kneser graphs <span class=\"etd-inline-math\">\\cO<sub>n,s</sub></span>, where the operads $n\\text{-}Lie$ ($n\\text{-}Com)$ and <span class=\"etd-inline-math\">Lie<sub>n</sub><sup>d</sup></span> (<span class=\"etd-inline-math\">Com<sub>n</sub><sup>d</sup>)</span> appear on different sides of the spectrum based on the parameter $s$. With the introduction of the new class of $n$-Com algebras through the Koszul duality, we take our first step in exploring these new types of algebras. Specifically, we start the work of trying to classify finite-dimensional simple $3$-Com algebras $C$ using the Peirce decomposition through semisimple idempotents $e$ through <span class=\"etd-inline-math\">\\chi<sub>e</sub>=m<sub>3</sub>(e,e,-)</span> to obtain important structural properties. In particular, it decomposes the $3$-Com algebra <span class=\"etd-inline-math\">C=\\bigoplus C<sub>e</sub>(t)</span> for eigenvalues $t$ for <span class=\"etd-inline-math\">\\chi<sub>e</sub></span> in which <span class=\"etd-inline-math\">C<sub>e</sub>(1)</span> is a commutative unital associative $k$-algebra acting on the other components, which is almost associative. Furthermore, we briefly construct an analog of the Killing form for $3$-Com algebras, denoted as $\\kappa$, and define non-degeneracy when the form is non-degenerate and fully degenerate when $\\kappa=0$. We use this to show that every non-degenerate $3$-Com algebra is a direct product of non-degenerate simple $3$-Com algebras. However, one very interesting aspect of this is that not every finite-dimensional $3$-Com algebra is non-degenerate, as our main example is fully degenerate. Towards some classifications of simple $3$-Com algebras, if $e$ is primitive semisimple idempotent with exactly two eigenvalues, and $C$ is simple, then its eigenvalues consist of $\\{1,-1\\}$ which help give a classification in dimension $2$ and $3$. Finally, we construct combinatorial objects, called Young $n$-trees, which are just rooted trees with a local Young tableaux structure at each edge following what is happening in the $n$-Com operad. In particular, we use these to give an upper bound to the arities of the dimension for the operad <span class=\"etd-inline-math\">n\\text{-}Com<sub>d</sub></span>, which gives a description for it.","abstract_has_math":true,"creators":["Tipton, Cody Allen"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Zhang, James J"],"committee_chairs":[],"committee_members":[],"year":2024,"date_issued":"2024-09-09","date_published":"2024-09-09","updated_at":"2026-07-24T05:58:05Z","subjects":["Commutative Algebra","n-Lie Algebras","Non-Associative Algebras","Operad Theory","Young Tableaux","Mathematics"],"languages":["en_US"],"rights":["CC BY"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/1773/52102","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Zhang, James J"]},{"key":"dc:creator","label":"Author","values":["Tipton, Cody Allen"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2024-09-09T23:12:41Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2024-09-09T23:12:41Z"]},{"key":"dc:date.issued","label":"Date","values":["2024-09-09"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Commutative Algebra","n-Lie Algebras","Non-Associative Algebras","Operad Theory","Young Tableaux","Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]},{"key":"dc:rights","label":"Dc Rights","values":["CC BY"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["Tipton_washington_0250E_26919.pdf"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/1773/52102"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Thesis (Ph.D.)--University of Washington, 2024"]},{"key":"dc:description.abstract","label":"Abstract","values":["We study the operad $n\\text{-}Lie_d$, whose algebras are $n$-Lie algebras, which was first introduced in Nambu mechanics to extend Hamiltonian mechanics to more than one Hamiltonian. We find the Koszul dual of $n\\text{-}Lie_{-d+n-2}$ to be the operad $n\\text{-}Com_d$, whose relations come from the Specht module $S^{(n,n-1)}$. We combine the operads $n$-Lie and $m$-Com to construct the operad $(m,n)-Poiss$, where the rewriting rule that relates them is through a generalized Leibniz rule. We generalize the above Koszul duality to different types of generalization of $Lie$ and $Com$ which arise from the eigenspaces of the general Kneser graphs $\\cO_{n,s}$, where the operads $n\\text{-}Lie$ ($n\\text{-}Com)$ and $Lie_n^d$ ($Com_n^d)$ appear on different sides of the spectrum based on the parameter $s$. With the introduction of the new class of $n$-Com algebras through the Koszul duality, we take our first step in exploring these new types of algebras. Specifically, we start the work of trying to classify finite-dimensional simple $3$-Com algebras $C$ using the Peirce decomposition through semisimple idempotents $e$ through $\\chi_e=m_3(e,e,-)$ to obtain important structural properties. In particular, it decomposes the $3$-Com algebra $C=\\bigoplus C_e(t)$ for eigenvalues $t$ for $\\chi_e$ in which $C_e(1)$ is a commutative unital associative $k$-algebra acting on the other components, which is almost associative. Furthermore, we briefly construct an analog of the Killing form for $3$-Com algebras, denoted as $\\kappa$, and define non-degeneracy when the form is non-degenerate and fully degenerate when $\\kappa=0$. We use this to show that every non-degenerate $3$-Com algebra is a direct product of non-degenerate simple $3$-Com algebras. However, one very interesting aspect of this is that not every finite-dimensional $3$-Com algebra is non-degenerate, as our main example is fully degenerate. Towards some classifications of simple $3$-Com algebras, if $e$ is primitive semisimple idempotent with exactly two eigenvalues, and $C$ is simple, then its eigenvalues consist of $\\{1,-1\\}$ which help give a classification in dimension $2$ and $3$. Finally, we construct combinatorial objects, called Young $n$-trees, which are just rooted trees with a local Young tableaux structure at each edge following what is happening in the $n$-Com operad. In particular, we use these to give an upper bound to the arities of the dimension for the operad $n\\text{-}Com_d$, which gives a description for it."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["The Koszul dual to n-Lie, n-Com algebras, and Young tableaux"]}]}],"canonical_facts":{"dc:contributor.advisor":["Zhang, James J"],"dc:creator":["Tipton, Cody Allen"],"dc:date.accessioned":["2024-09-09T23:12:41Z"],"dc:date.available":["2024-09-09T23:12:41Z"],"dc:date.issued":["2024-09-09"],"dc:description":["Thesis (Ph.D.)--University of Washington, 2024"],"dc:description.abstract":["We study the operad $n\\text{-}Lie_d$, whose algebras are $n$-Lie algebras, which was first introduced in Nambu mechanics to extend Hamiltonian mechanics to more than one Hamiltonian. We find the Koszul dual of $n\\text{-}Lie_{-d+n-2}$ to be the operad $n\\text{-}Com_d$, whose relations come from the Specht module $S^{(n,n-1)}$. We combine the operads $n$-Lie and $m$-Com to construct the operad $(m,n)-Poiss$, where the rewriting rule that relates them is through a generalized Leibniz rule. We generalize the above Koszul duality to different types of generalization of $Lie$ and $Com$ which arise from the eigenspaces of the general Kneser graphs $\\cO_{n,s}$, where the operads $n\\text{-}Lie$ ($n\\text{-}Com)$ and $Lie_n^d$ ($Com_n^d)$ appear on different sides of the spectrum based on the parameter $s$. With the introduction of the new class of $n$-Com algebras through the Koszul duality, we take our first step in exploring these new types of algebras. Specifically, we start the work of trying to classify finite-dimensional simple $3$-Com algebras $C$ using the Peirce decomposition through semisimple idempotents $e$ through $\\chi_e=m_3(e,e,-)$ to obtain important structural properties. In particular, it decomposes the $3$-Com algebra $C=\\bigoplus C_e(t)$ for eigenvalues $t$ for $\\chi_e$ in which $C_e(1)$ is a commutative unital associative $k$-algebra acting on the other components, which is almost associative. Furthermore, we briefly construct an analog of the Killing form for $3$-Com algebras, denoted as $\\kappa$, and define non-degeneracy when the form is non-degenerate and fully degenerate when $\\kappa=0$. We use this to show that every non-degenerate $3$-Com algebra is a direct product of non-degenerate simple $3$-Com algebras. However, one very interesting aspect of this is that not every finite-dimensional $3$-Com algebra is non-degenerate, as our main example is fully degenerate. Towards some classifications of simple $3$-Com algebras, if $e$ is primitive semisimple idempotent with exactly two eigenvalues, and $C$ is simple, then its eigenvalues consist of $\\{1,-1\\}$ which help give a classification in dimension $2$ and $3$. Finally, we construct combinatorial objects, called Young $n$-trees, which are just rooted trees with a local Young tableaux structure at each edge following what is happening in the $n$-Com operad. In particular, we use these to give an upper bound to the arities of the dimension for the operad $n\\text{-}Com_d$, which gives a description for it."],"dc:format.mimetype":["application/pdf"],"dc:identifier.other":["Tipton_washington_0250E_26919.pdf"],"dc:identifier.uri":["https://hdl.handle.net/1773/52102"],"dc:language.iso":["en_US"],"dc:rights":["CC BY"],"dc:subject":["Commutative Algebra","n-Lie Algebras","Non-Associative Algebras","Operad Theory","Young Tableaux","Mathematics"],"dc:title":["The Koszul dual to n-Lie, n-Com algebras, and Young tableaux"],"dc:type":["Thesis"]},"updated_at":"2026-07-24T05:58:05Z"}