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Računanje vektora reprezentacije Liejeve algebre $\esl(4,\C)$ obavlja se djelovanjem generatora Liejeve algebre na elemente triju vektorskih prostora. Djelovanje se računa prema određenim pravilima i kada se djeluje složenim generatorom na tenzorski produkt modula račun postaje vrlo opsežni i nepraktičan za računanje na papiru. Program napravljen u sklopu ovog rada, napisan u programskom jeziku Pythonu, olakšava računanje takvih opsežnih računa i smanjuje vjerojatnost nastanka pogrešaka. The calculation of the vector of the Lie algebra representation $\esl(4,\C)$ is performed by the operation of the Lie algebra generator on the elements of the three vector spaces. The operation is calculated according to certain rules. When operating with a complex generator on the tensor product of modules the calculation becomes very complicated and too impractical to be calculated by hand. A program, written in the Python programming language, which represents the main part of this thesis makes it easier to calculate such extensive calculations and reduces the likelihood of errors.

U ovom radu opisan je proces računanja komutatora unutar omotačke algebre sl(4,C). Veliki komutacijski izraz razvili smo u nekoliko manjih koristeći ekvivalentnu formulu. U manjim podizrazima komutacijske izraze smo zamijenili izrazima s generatorima i operatorima. Nakon što je svaki podizraz zapisan koristeći samo monome i matematičke izraze, prešli smo na razvoj operatora, zamjenu operatora matematičkim izrazom nakon čega smo svaki podizraz kanonizirali pretvorivši ga u sumu monoma. Svaki neporedani monom iz kanoniziranog podizraza razvili smo u skup poredanih nizom zamjena redoslijeda generatora umetanjem komutacijskih izraza. Na kraju smo monome grupirali i svim monomima izlučili zajedničke faktore koji mogu sadržavati isključivo Y-generatore i X-generatore. The process of computing commutators within enveloping algebra sl(4,C) is described in this paper. Big commutational expression was developed into a number of smaller ones using the equivalent formula. Inside the smaller subexpressions, the commutational expressions were replaced with expressions containing generators and operators. After every subexpression had got written using only monomials and mathematical expressions, replacing operators with a mathematical expression followed. Following operator conversion, each subexpression has been cannonized by converting it to a sum of monomials. Every unordered monomial from the cannonized subexpression was developed into a set of developed monomials after a series of generator replacements with injections of commutational expressions. In the end the monomials were grouped and joint factors of particular monomials which can only contain Y-generators and X-generators were extracted.

Ovim radom smo proučili Liejeve algebre i njezine reprezentacije. Pokazali smo kako računamo korijene Liejeve algebre, njezine težine i multiplicitete. Uz to smo pokazali i kako dolazimo do podmodula algebri Bn i Dn. I konačno smo izradili jednostavan program koji računa te iste podmodule kako bi olakšali budući rad i istraživanje s njima. With this paper we learned about Lie algebra and representation of it. We showed how to calculate Lie algebra roots, weights and multiplicities. We also showed how to get to submodules of Bn and Dn algebras. And finally, we made a simple to use program that calculates that submodules to ease future work and research on it.

Djelovanje Liejeve algebre na modul ili tenzorski produkt modula računa se po određenim pravilima te ukoliko se radi o produktu modula računi postaju preteški. Cilj ovoga rada je napisati program koji ubrzava i olakšava spomenuti problem. Za implementaciju programskog rješenja korišten je programski jezik Python. The effect of Lie algebra on a module is calculated according to certain rules, and if we use a product of a module, the calculations may become too difficult. That is why the aim of this paper was to implement an algorithm that facilitates the aforementioned problem. To implement the algorithm, we used the Python programming language.

U radu se proučavaju osnovni koncepti Liejeve algebre – posebno simplektičke Liejeve algebre te vizualizacije kombinatornih baza. Na početku kao motivacijsko poglavlje, dane su neke od primjena Liejeve algebre u matematici, fizici i računarskoj znanosti. Objašnjene su definicija te osnovna svojstva Liejeve algebre, definirani su pojmovi izomorfizma i podalgebre. Objašnjen je pojam linearne Liejeve algebre uz pojam glavne linearne algebre, definirana je jedna od linearnih Liejevih algebra, specijalna Liejeva algebra 𝖘𝖑(𝑛,ℂ). Objašnjen je pojam bilinearne forme i antidijagonalizacijskog teorema. Zatim je definirana i objašnjena simplektička Liejeva algebra 𝖘𝖕(2𝑙,𝔽). U drugom poglavlju, objašnjen je način vizualizacije baza simplektičke Liejeve algebre uz pomoć Cartanove dekompozicije i objašnjena je programska izvedba vizualizacije baza simplektičke Liejeve algebre. The paper studies the basic concepts of Lie algebra - especially sympletic Lie algebras and visualizations of combinatorial bases. Initially as a motivational chapter, some of the applications of Lie algebra in mathematics, physics, and computer science are given. The definition and basic properties of Lie algebra are explained, the concepts of isomorphism and subalgebra are defined. The notion of linear Lie algebra is explained along with the notion of general linear algebra, one of the linear Lie algebras is defined, the simple Lie algebra\ \mathbf{sl}(n,\ \mathbb{C}). The notion of bilinear form and antidiagonalization theorem is explained. Then, the symplectic Lie algebra \mathbf{sp}(2l,\ \mathbb{F}) is defined and explained. In the second chapter, the way of visualization of bases of symplectic Lie algebra is explained using Cartan decomposition and the program implementation of visualization of bases of symplectic Lie algebra is explained.

Uvodi se pojam Liejeve grupe i matrične Liejeve grupe te se posebna pažnja daje upravo Liejevoj grupi SU(2,C). Opisat ćemo Liejevu algebru su(2) od SU(2,C) te ćemo istražiti vezu reprezentacija od komplksikacije su(2) i izospina.

Rad ne sadrži sažetak.

Cilj ovog rada je implementacija Freudenthalove formule za sve proste Liejeve algebre. Opisane su osnovne definicije i svojstva potrebna za razumijevanje Liejevih algebri. Također je opisan i način reprezentacije Lijeve algebre te različitosti između istih. Rad je zaključen Freudenthalovom formulom te kratkim primjerom računanja multipliciteta. The main goal of this final thesis is implementation of Freudenthal’s formula for simple Lie algebra. The basic definitions and features needed to understand Lie algebra, representation of Lie algebra and differences between them are described. Thesis is concluded with Freudenthal’s formula itself and short example of multiplicity calculation.

The concept of inherited orthogonality is motivated and an optimality statement for it is derived. Basic adaptive discretizations are introduced. Various properties of difference operators which are directly related to basic adaptive discretizations are looked at. A Lie-algebraic concept for obtaining basic adaptive discretizations is explored. Some of the underlying moment problems of basic difference equations are investigated in greater detail.

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 \(\hat{A}_n\) 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. 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 \(\mathfrak{g}_0\) 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.

U radnji proučavamo strukturnu teoriju i teoriju reprezentacija verteks algebri \(W_{1+\infty}\) i \(W_\infty\) za neke centralne naboje. Dajemo eksplicitnu formulu za familiju singularnih vektora u univerzalnim verteks algebrama \(W_{1+\infty}\) i \(W_\infty\). Dokazano je da je netrivijalni kvocijent verteks algebre \(W_\infty\) za \(c =-2\) izomorfan \(W_{2,3}\) algebri iz članka W. Wanga [36]. U slučaju centralnog naboja \(c =-4\) prosti kvocijent od \(W_\infty\) je izomorfan parafermionskoj verteks algebri pridruženoj afinoj Liejevoj algebri \(A_1^{(1)}\) nivoa -1. U slučaju nekih drugih centralnih naboja određen je minimalan skup generatora kvocijenata univerzalne verteks–algebre za \(W_\infty\). Promatrana je teorija dualnih parova i realizirana je verteks algebra \(W_\infty\) kao podalgebra verteks superalgebre pridružene simplektičkim fermionima. Dokazano je da je verteks superalgebra pridružena simplektičkim fermionima potpuno reducibilna reprezentacija od \(W_\infty\). In this thesis we study structure and representation theory for vertex algebras \(W_{1+\infty}\) and \(W_\infty\) for certain central charges. We give explicit formula for a family of singular vectors in universal vertex algebras \(W_{1+\infty}\) and \(W_\infty\) It is proved that a non-trivial quotient of the vertex algebra \(W_\infty\) with central charge \(c =-2\) is isomorphic to \(W_{2,3}\) algebra from the paper of W. Wang [36]. In the case of central charge \(c =-4\) simple quotient of \(W_\infty\) is isomorphic to the parafermionic vertex algebra associated to the affine Lie algebra \(A_1^{(1)}\) of the level -1. For some other central charges, the minimal set of generators for the quotient of universal vertex algebra of \(W_\infty\) is determined. The theory of dual pairs is used for realization of the vertex algebra \(W_\infty\) as a subalgebra of the vertex superalgebra associated to the symplectic fermions. It is proved that the vertex superalgebra associated to the symplectic fermions is completely reducible representation of \(W_\infty\).

V doktorski disertaciji so obravnavane algebre, določene z ničelnim produktom. Ta pojem je nov. Zato bo veˇcji del disertacije namenjen ugotavljanju določenosti z ničelnim produktom standardnih primerov asociativnih, Liejevih in jordanskih algeber. V prvem delu se osredotočimo na asociativne algebre in pokažemo, da je vsaka matrična algebra nad algebro z enoto določena z ničelnim produktom. Nato sledi obravnava multiaditivnih preslikav, ki zadoščajo določenemu pogoju ohranjanja ničelnih produktov. Opisano je obnašanje teh preslikav na podkolobarju, generiranem z vsemi idempotenti danega kolobarja. Poseben primer tega rezultata je v pomoč pri dokazu, da je vsaka enotska algebra, ki je generirana s svojimi idempotenti, določena z ničelnim produktom. Prav tako je vsaka končno razsežna enostavna algebra, ki ni obseg, določena z ničelnim produktom. Drugi del je namenjen Liejevim algebram. Dokažemo, da je z ničelnim Liejevim produktom določena vsaka matrična algebra nad enotsko asociativno algebro B, določeno z ničelnim Liejevim produktom. Podan je primer matrične algebre, ki pove, da je res treba dodati določene predpostavke na algebro B. V nadaljevanju tega poglavja je dokazano še, da sta z ničelnim produktom določeni tudi Liejevi algebri poševno simetričnih matrik glede na transponiranje in simplektično involucijo. V tretjem so obravnavane najbolj znane jordanske algebre. Dokazano je, da so z ničelnim jordanskim produktom določene: algebra matrik nad poljubno enotsko algebro, algebra simetričnih matrik glede na transponiranje in simplektično involucijo, Albertova algebra ter jordanska algebra, določena z nedegenerirano simetrično bilinearno formo. Zadnji del je namenjen obravnavi določenih aditivnih preslikav na prakolobarjih. The central object in this thesis is a zero product determined algebra. This concept is new. The bulk of the thesis is therefore devoted to the question whether standard examples of associative, Lie, and Jordan algebras are zero product determined. In the first part we treat associative algebras. We show that the matrix algebra over any unital algebra is zero product determined. Next we deal with multiadditive maps satisfying a certain zero product preserver condition. Their form is described on the subring generated with all idempotents of the ring in question. This result implies that every unital algebra generated with idempotents is zero product determined. We also show that every simple finite dimensional algebra, which is not a division algebra, is zero product determined. The second chapter is devoted to Lie algebras. We show that if an associative algebra B is zero Lie product determined, then so is the matrix algebra over B. By an example we justify the assumption that B must be zero Lie product determined. We conclude this part by showing that Lie algebras of skew symmetric matrices with respect to either transpose or symplectic involution are zero product determined. Next we turn our attention to Jordan algebras. We prove that the algebra of matrices over a unital algebra endowed with the Jordan product, the Jordan algebra of symmetric matrices with respect to either transpose or symplectic involution, the Albert algebra, and the Jordan algebra determined with a nondegenerate symmetric bilinear form are all zero product determined. At the end we deal with additive maps between prime rings.

Diploma je sestavljena iz devetih poglavij. Začetek diplomskega dela vsebuje osnovne pojme in lastnosti matrik, vektorskih prostorov in osnovne lastnosti grup. V naslednjem poglavju je bolj podrobno predstavljena posebna unitarna grupa, kjer opišemo zemljepisne širine, zemljepisne dolžine ter severni in južni pol grupe. Pokažemo tudi, da so konjugirani razredi v unitarni grupi dvodimenzionalne sfere. V poglavju Ortogonalna upodobitev unitarne grupe vpeljemo orbite in pojem vlakna. Tu pokažemo, da je unitarne grupe dvojno pokritje grupe ortogonalne grupe. V nadaljevanju si pogledamo primer nekompaktne grupe. Nato sledi poglavje Enoparametričnih grup, ki so homomorfizmi, ki slikajo iz aditivne grupe v linearno grupo odvedljivih funkcij spremenljivke t ∈ ℝ. Tu omenimo pojem parcialnega odvoda in izrek o inverznih funkcijah. V nadaljevanju se ukvarjamo z Liejevo algebro, ki je prostor vektorjev tangent na G pri identiteti I. S pomočjo pojma gradient in verižnega ulomka podamo potrebne pogoje, da vektor postane tangenta za realno algebrsko množico S. V tem poglavju so definirani pojmi infinitizimalna tangenta, vektor tangent in prostor tangent. Ukvarjamo se z izračunom infinitizimalne spremembe posebne linearne grupe in ortogonalne grupe. Za konec tega poglavja zapišemo definicijo Liejeve algebre bolj abstraktno s pomočjo uporabe operacije komutator. V zadnjem poglavju z naslovom Primeri enostavnih grup navedemo nekaj primerov teh grup ter dokažemo pomemben izrek. The thesis consists from nine chapters. Begining of thesis contains the concepts and properties of matrices, vector spaces and the basic properties of groups, which are often used in thesis. In the next chapter we focus on special unitary group where we present latitude and longitude of unitary group. In this section we show that conjugate classes in unitary groups are two-dimensional spheres. In chapter The ortogonal representation of unitary group are introduced concepts of orbits and fiber. Here we show that unitary group is double covering of ortogonal group. In the continuation we meet special ortogonal group, which is example of noncompact group. Then follows chapter of Oneparametric groups, where is shown that oneparametric groups are homomorphisms from the additive group of real numbers to the general linear group, which are differentiable functions of the variable t ∈ ℝ. Here we mention concept of partial derivatives and the inverse function theorem, which is shown on ortogonal group and special linear group. In the continuation we speak about Lie algebra. Using the concepts of gradient and the chain rule we give necessary conditions for a vector to be tangent to a real algebraic set S. In this chapter we define concepts like infinitizimal tangent, vector tangent and tangent space. We are also dealing with computation of infinitizimal change of and ortogonal groups. In conclusion of this chapter we write the definition od Lie algebra more abstract, where we use the operation bracket. The final chapter, Examples of simple groups, contains some examples of these groups and we prove important theorem.

V diplomskem delu definiramo Liejevo in asociativno algebro ter opišemo povezavo med njima. Vpeljemo linearne Liejeve algebre in strukturne konstante. Nato definiramo odvajanja, (pol)direktne vsote, avtomorfizme Liejevih algeber in upodobitve Liejevih algeber. V nadaljevanju spoznamo omejene Liejeve algebre in definiramo posplošeno Jacobson-Wittovo algebro, ki se imenuje tudi algebra posebnih odvajanj. In the diploma work we define a Lie algebra and we describe the relation between them. We construct linear Lie algebras and strukture constants. After that we define derivations, (semi)direct sums, automorphisms of Lie algebras and representations of Lie algebras. In the sequel we become familiar with restricted Lie algebras and we define the generalized Jacobson-Witt algebra, which is called also the algebra of special derivations.