Vizualizacija kombinatornih baza standardnih modula simplektičkih afinih Lievih algebri

undergraduate thesis

• Title (english): Visualization of combinatorial bases for standard modules of symplectic affine Lie algebras
• Author: Dominik Špiljak
• Mentor: Tomislav Šikić (mentor)
• Committee member: Tomislav Šikić (predsjednik povjerenstva)
• Committee member: Domagoj Kovačević (član povjerenstva)
• Committee member: Mate Puljiz (član povjerenstva)
• Granter: University of Zagreb; Faculty of Electrical Engineering and Computing; Zagreb
• Defense date and country: 2020-07-08, Croatia
• Scientific / art field, discipline and subdiscipline: TECHNICAL SCIENCES; Computing

Abstract

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.

Abstract (english)

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.

• Keywords: Liejeva algebra; simplektička Liejeva algebra; vizualizacija baze proste i afine simplektičke Liejeve algebre; izomorfizam; podalgebre; linearna Liejeva algebra; glavna linearna algebra; specijalna Liejeva algebra; bilinearna forma; antidijagonalizacijski teorem; Cartanova dekompozicija
• Keywords (english): Lie algebra; symplectic Lie algebra; visualization of the base for simple and affine symplectic Lie algebra; isomorphism; subalgebras; linear Lie algebra; general linear algebra; simple Lie algebra; bilinear form; antidiagonalization theorem; Cartan decomposition
• Language: croatian
• URN:NBN: urn:nbn:hr:168:666491
• Study programme: Title: Computing; Study programme type: university; Study level: undergraduate; Academic / professional title: sveučilišni/a prvostupnik/ prvostupnica (baccalaureus/baccalaurea) inženjer/inženjerka računarstva (sveučilišni/a prvostupnik/ prvostupnica (baccalaureus/baccalaurea) inženjer/inženjerka računarstva)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Closed access
• Created on: 2023-01-14 10:05:03