Naslov Verteks algebre pridružene reprezentacijama N=1 super Heisenberg-Virasorove i Schrödinger-Virasorove algebre Naslov (engleski) Vertex algebras associated with the representations of the N=1 super Heisenberg-Virasoro and Schrödinger-Virasoro algebras Autor Berislav Jandrić Mentor Dražen Adamović (mentor)Član povjerenstva Gordan Radobolja (predsjednik povjerenstva)Član povjerenstva Dražen Adamović (član povjerenstva)Član povjerenstva Ozren Perše (član povjerenstva)Ustanova koja je dodijelila Sveučilište u Zagrebu Datum i država obrane 2019-07-08, Hrvatska Znanstveno / umjetničko PRIRODNE ZNANOSTI Univerzalna decimalna klasifikacijaUDC ) 51 - Matematika Sažetak U ovom radu proučavali smo teoriju reprezentacija dviju važnih generalizacija Heisenberg-Virasorove Liejeve algebre \(\mathcal{H}\), a to su N = 1 super Heisenberg-Virasorova Liejeva algebra \(\mathcal{SH}\) i Schrödinger-Virasorova Liejeva algebra \(\mathfrak{sv}\). Prva predstavlja primjer beskonačnodimenzionalne Liejeve super-algebre koja do sada nije bila otkrivena, dok je druga primjer beskonačno dimenzionalne Liejeve algebre koja se dosta proučavala u fizikalnoj kao i matematičkoj literaturi jer je prva u seriji tzv. galilejevskih Liejevih konformnih algebri proširenih tzv. nerelativističkim spinom. Originalan doprinos ovog rada je eksplicitna konstrukcija izomorfizma univerzalne (resp. proste) omotačke verteks super-algebre \(V^{ \mathcal{SH}}(c_L, c_{L,\alpha}, c_\alpha \neq 0)\) (resp. \(L^{ \mathcal{SH}}(c_L, c_{L,\alpha}, c_\alpha \neq 0)\)) pridružene N = 1 super Heisenberg-Virasorovoj Liejevoj algebri \(\mathcal{SH}\) i verteks super-algebre \(SM(1) \bigotimes V^{\mathcal{NS}}_{c_L-c_\mu}\), (resp. \(SM(1) \bigotimes L^{\mathcal{NS}}_{c_L-c_\mu}\)), pri čemu smo pokazali da je na \(SM(1)\) definirana struktura reprezentacije proste N = 1 Heisenberg-Virasorove verteks-algebre \(L^{ \mathcal{SH}}(\frac{3}{2}-3\mu^2, \frac{\mu}{2}c_\alpha, c_\alpha), c_\alpha \neq 0\). Što se tiče realizacije verteks super-algebre \(V^{ \mathcal{SH}}(c_L, c_{L,\alpha}, c_\alpha \neq 0)\) u slučaju nivoa nula: \(c_\alpha=0\), to se nadovezuje se na radove Dražena Adamovića i Gordana Radobolje iz Journal of Pure and Applied Mathematics (2015) i Communications in Contemporary Mathematics (2019), u kojima su konstruirane free field realizacije za Heisenberg-Virasorovu Liejevu algebru \(\mathcal{H}\) također na nivou nula. Klasificirali smo singularne vektore na nivoima \(\frac{1}{2}, 1, \frac{3}{2}, 2\) i 4 te dali jednu konstrukciju njenih screening operatora, što je omogućilo dokaz egzistencije beskonačne serije singularnih vektora za N = 1 Heisenberg-Virasorovu Liejevu algebru \(\mathcal{SH}\) na nivoima konformnih težina \(\frac{p}{2}\) za \(p\) neparan. U tu svrhu smo promatrali realizacije njenih Vermaovih modula u verteks-algebri pridruženoj rešetki te operatore ispreplitanja i Schurove polinome, koji su nam dali i eksplicitne formule za te singularne vektore. Za Schrödinger-Virasorovu Liejevu algebru \(\mathfrak{sv}\), dali smo rigorozan tretman i poopćenje rezultata iz Unterbergerovog članka Nuclear Physics B (2009). Nadalje, ustanovili smo da verteks-algebra \(V^{\mathfrak{sv}}(c_L, c_{L,\beta}, c_\beta)\) sadrži HeisenbergVirasorovu Liejevu podalgebru i pronašli smo primjer singularnog vektora na nivou konformne težine 3 u univerzalnoj Schrödinger-Virasorovoj verteksalgebri za slučaj proizvoljnih centralnih naboja.
Sažetak (engleski) In this thesis we have investigated representation theory of two important generalizations of the Hesienberg-Virasoro Lie algebra \(\mathcal{H}\), which are the N = 1 super Heisenberg-Virasoro Lie algebra \(\mathcal{SH}\) and the Schrödinger-Virasoro Lie algebra \(\mathfrak{sv}\). The former provides an example of an infinite dimensional Lie super-algebra which has not been yet discovered, while the latter is also an example of an infinite dimensional Lie algebra that has been widely studied in the physical as well as mathematical literature because it is the first of an infinite series of the so called Galilean Lie conformal algebras extended by the non-relativistic spin parametre. The important innovation in this thesis is the explicit construction of an isomorphism between the universal (resp. simple) enveloping vertex superalgebra \(V^{ \mathcal{SH}}(c_L, c_{L,\alpha}, c_\alpha \neq 0)\) (resp. \(L^{ \mathcal{SH}}(c_L, c_{L,\alpha}, c_\alpha \neq 0)\)) associated to the N = 1 super Heisenberg-Virasoro Lie algebra \(\mathcal{SH}\) and the vertex algebra \(SM(1) \bigotimes V^{\mathcal{NS}}_{c_L-c_\mu}\), (resp. \(SM(1) \bigotimes L^{\mathcal{NS}}_{c_L-c_\mu}\)), where we have shown that there is a representation of the simple N = 1 Heisenberg-Virasoro vertex-algebra \(L^{ \mathcal{SH}}(\frac{3}{2}-3\mu^2, \frac{\mu}{2}c_\alpha, c_\alpha), c_\alpha \neq 0\) on \(SM(1)\). As far as the realization of this vertex algebra at level zero case \(c_\alpha=0\), this is a continuation of the papers by Dražen Adamović and Gordan Radobolja from Journal of Pure and Applied Mathematics (2015) and Communications in Contemporary Mathematics (2019), where free field realizations of the Heisenberg-Virasoro Lie algebra at level zero had been constructed. In this thesis, we have classified singular vectors on levels \(\frac{1}{2}, 1, \frac{3}{2}, 2\) and 4 and we have also given a construction of the screening operators for this algebra, which has enabled us to prove the existence of an infinite series of singular vectors for the N = 1 Heisenberg-Virasoro Lie algebra \(\mathcal{SH}\) at the levels of conformal weight \(\frac{p}{2}\); for \(p\) odd. In order to do so, we have studied realizations of the Verma modules for this algebra in the vertex algebra associated to a certain lattice as well as intertwining operators and the Schur polynomials, which led us to the explicit formulae for this singular vectors. In the case of the Schrödinger-Virasoro Lie algebra \(\mathfrak{sv}\), we have provided a rigorous treatment as well as a generalization of the result of Unterberger from his paper Nuclear Physics B (2009). Furthermore, we have proved that the vertex algebra \(V^{\mathfrak{sv}}(c_L, c_{L,\beta}, c_\beta)\) contains a Heisenberg-Virasoro subalgebra and we have found an example of a singular vector at the level three conformal weight of the universal Schrödinger-Virasorove vertex algebra for the case of general central charges.
Ključne riječi
verteks-algebra
N=1 Heisenberg-Virasorova verteksalgebra
N=1 Neveu-Schwarz Liejeva algebra
singularni vektori
screening operatori
Schrödinger-Virasorova verteks-algebra
Ključne riječi (engleski)
vertex algebra
N=1 Heisenberg-Virasoro vertex algebra
N=1 Neveu-Schwarz Lie algebra
singular vector
screening operator
SchrödingerVirasoro vertex algebra
Jezik hrvatski URN:NBN urn:nbn:hr:217:925217 Studijski program Naziv: Matematika Vrsta resursa Tekst Opseg vii, 88 str. Način izrade datoteke Izvorno digitalna Prava pristupa Otvoreni pristup Uvjeti korištenja Datum i vrijeme pohrane 2019-07-16 11:08:28
