Realizacija Liejevih algebri i diferencijalni račun na nekomutativnim prostorima

doctoral thesis

• Title (english): Realization of Lie algebras and differential calculus on noncummutative spaces
• Author: Tea Martinić Bilać
• Mentor: Saša Krešić-Jurić (mentor)
• Committee member: Dražen Adamović (predsjednik povjerenstva)
• Committee member: Ozren Perše (član povjerenstva)
• Committee member: Saša Krešić-Jurić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2016-12-16, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

U prvom dijelu disertacije se proučavaju proširenja Liejeve algebre $\mathfrak{g}_0$ generirane bazom $\{X1, X2, \dots, Xn\}$ s Abelovom familijom generatora $T_{\mu \nu}, 1 \leq \mu,\nu \leq n$, koji djeluju na omotačku algebru $U(\mathfrak{g}_0)$. Pomoću tako definiranog djelovanja možemo opisati komutacijske relacije izmedu $X_{\mu}$ i proizvoljnih monoma u $U(\mathfrak{g}_0)$. Nadalje, proučavamo realizacije Liejeve algebre $\mathfrak{g}_0$, tj. njezina ulaganja u... More $\hat{A}_n$ gdje je $\hat{A}_n$ upotpunjenje $n$-te Weylove algebre $A_n$ obzirom na stupanj diferencijalnog operatora $\partial^{k_1}_1 \partial^{k_2}_2 \dots \partial^{k_n}_n$. Posebno se proučavaju realizacije koje induciraju simetrično uređenje na omotačkoj algebri $U(\mathfrak{g}_0)$. Koristeći svojstva proširene Liejeve algebre pokazano je da je odgovarajuća simetrična realizacija izražena pomoću funkcije izvodnice za Bernoullijeve brojeve. Svakoj realizaciji Liejeve algebre $\mathfrak{g}_0$ pridružen je zvijezda-umnožak (engl. star-product) na simetričnoj algebri $X=\mathbb{C}[x_1, x_2, \dots , x_n] \subset A_n$ koji se definira pomoću kanonskog djelovanja algebre $A_n$ na podalgebru $X$. Uveden je pojam lijevo-desno dualnih zvijezda-umnožaka i njihovih pripadnih realizacija. Lijevo-desna dualnost detaljno je proučena u slučaju simetrične realizacije gdje se koristi konstruirano proširenje Liejeve algebre $\mathfrak{g}_0$. Drugi dio disertacije se bavi bikovarijantnim diferencijalnim računom na kvantnom prostoru $U(\mathfrak{g}_0)$. U tu svrhu konstruirana je Liejeve superalgebra $\mathfrak{g}=\mathfrak{g}_0 \bigoplus \mathfrak{g}_1$ gdje se elementi baze $\xi_1, \xi_2, \dots , \xi_m$ neparnog dijela $\mathfrak{g}_1$ interpretiraju kao jedan-forme na prostoru $U(\mathfrak{g}_0)$. Generalizacijom rezultata iz prvog dijela disertacije dobiveno je proširenje od $\mathfrak{g}$ s Abelovom familijom generatora $T_{\mu\nu}$ čije djelovanje na omotačku algebru $U(\mathfrak{g})$ opisuje komutacijske relacije između jedan-formi i monoma u $U(\mathfrak{g}_0)$. Također je konstruirana realizacija, tj. ulaganje superalgebre $\mathfrak{g}$ u upotpunjenje Clifford-Weylove algebre $\hat{A}_{n,m}$. U slučaju kada je $dim(\mathfrak{g}_0) = dim(\mathfrak{g}_1)$ definirana je vanjska derivacija $d: U(\mathfrak{g}_0) \to \Omega$ gdje je $\Omega=\bigoplus^n_{\mu=1} U(\mathfrak{g}_0)\xi_\mu$ bimodul nad $U(\mathfrak{g}_0)$. Diferencijalni račun prvog reda $(d, \Omega)$ je bikovarijantan obzirom na primitivnu Hopfovu strukturu od $U(\mathfrak{g}_0)$. Koristeći realizaciju Liejeve superalgebre $\mathfrak{g}$, diferencijalni račun je dobiven kao deformacija klasičnog diferencijalnog računa na Euklidskom prostoru. Less

Abstract (english)

In the first part of the thesis we study extensions of a Lie algebra $\mathfrak{g}_0$ with basis $\{X1, X2, \dots, Xn\}$ by an Abelian family of generators $T_{\mu \nu}, 1 \leq \mu,\nu \leq n$, which act on the enveloping algebra $U(\mathfrak{g}_0)$. This action describes the commutation relations between $X_{\mu}$ and arbitrary monomials in $U(\mathfrak{g}_0)$. Furthermore, we study realizations of the Lie algebra $\mathfrak{g}_0$, i.e. embedding of... More $\mathfrak{g}_0$ into $\hat{A}_n$ where $\hat{A}_n$ is the completion of the $n$-th Weyl algebra $A_n$ by the degree of the differential operator $\partial^{k_1}_1 \partial^{k_2}_2 \dots \partial^{k_n}_n$. In particular, we study realizations which induce the symmetric ordering on the enveloping algebra $U(\mathfrak{g}_0)$. Using properties of the extended Lie algebra it is shown that the corresponding symmetric realization is given in terms of the generating function for the Bernoulli numbers. To each realization of the Lie algebra $\mathfrak{g}_0$ one can associate a star–product on the symmetric algebra $X=\mathbb{C}[x_1, x_2, \dots , x_n] \subset A_n$ which is defined using the canonical action of the algebra $A_n$ on the subalgebra $X$. We introduce the notion of left–right dual star–products and the corresponding realizations. The left–right duality is studied in detail in case of the symmetric realization using the aforementioned extension of the Lie algebra $\mathfrak{g}_0$. The second part of the thesis deals with bicovariant differential calculus on the quantum space $U(\mathfrak{g}_0)$. For this purpose we construct a Lie superalgebra $\mathfrak{g}=\mathfrak{g}_0 \bigoplus \mathfrak{g}_1$ where the basis elements $\xi_1, \xi_2, \dots , \xi_m$ of the odd part $\mathfrak{g}_1$ are interpreted as one–forms on the space $U(\mathfrak{g}_0)$. By generalizing the result from the first part of the thesis we obtain an extension of $\mathfrak{g}$ by an Abelian family of generators $T_{\mu\nu}$ whose action on the enveloping algebra $U(\mathfrak{g})$ describes the commutation relations between one–forms and monomials in $U(\mathfrak{g}_0)$. We also construct a realization, i.e. an embedding of the superalgebra $\mathfrak{g}$ into a completion of the Clifford–Weyl algebra $\hat{A}_{n,m}$. In case when $dim(\mathfrak{g}_0) = dim(\mathfrak{g}_1)$ we define an exterior derivative $d: U(\mathfrak{g}_0) \to \Omega$ where $\Omega=\bigoplus^n_{\mu=1} U(\mathfrak{g}_0)\xi_\mu$ is an $U(\mathfrak{g}_0)$–bimodule. The first order differential calculus $(d, \Omega)$ is bicovariant with respect to the primitive Hopf structure of $U(\mathfrak{g}_0)$. Using the realization of the Lie superalgebra $\mathfrak{g}$, the differential calculus is obtained as a deformation of the classical differential calculus on the Euclidean space. Less

• Keywords: Liejeva algebra; Abelova familija generatora; komutacijske relacije; Weylova algebra; Bernoullijevi brojevi
• Keywords (english): Lie algebra; Abelian family of generators; commutation relations; Weyl algebra; Bernoulli numbers
• Language: croatian
• URN:NBN: urn:nbn:hr:217:128591
• Study programme: Title: Mathematics; Study programme type: university; Study level: postgraduate; Academic / professional title: doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika (doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika)
• Type of resource: Text
• Extent: vii, 97 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-03-18 12:57:54