Struktura Cliffordove algebre pridružene reduktivnoj Liejevoj grupi

doctoral thesis

• Title (english): Structure of the Clifford algebra associated to a reductive Lie group
• Author: Karmen Grizelj
• Mentor: Pavle Pandžić (mentor)
• Committee member: Ana Prlić (predsjednik povjerenstva)
• Committee member: Domagoj Kovačević (član povjerenstva)
• Committee member: Pavle Pandžić (član povjerenstva)
• Committee member: Dražen Adamović (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2023-06-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

Neka je \(\mathfrak{g}\) kompleksifikacija Liejeve algebre pridružene realnoj reduktivnoj grupi \(G\). Pretpostavimo da je \((\mathfrak{g},\mathfrak{k})\) simetrični par takav da je Spin\(_{ad}\) primarna reprezentacija, odnosno takav da za Weylove grupe vrijedi \(W_{\mathfrak{k},t} = W_{\mathfrak{g},t}\), gdje je \(\mathfrak{t}\) Cartanova podalgebra od \(\mathfrak{k}\). U ovom radu pokazat će se analogoni nekih poznatih rezultata u slučajnu relativne Weilove i Cliffordove algebre. Za transgresiju na relativnoj Weilovoj algebri \(W(\mathfrak{g},\mathfrak{k}) = (S (\mathfrak{g}) \otimes \wedge \mathfrak{p})^{\mathfrak{k}} \) će se odrediti eksplicitna formula i onda dokazati teorem o transgresiji. Teorem o transgresiji opisuje sliku i jezgru transgresije. Za simetrični par \((\mathfrak{g},\mathfrak{k})\) kao ranije neka je \(\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}\) Cartanova dekompozicija. Neka je \(\mathfrak{h}\) fundamentalna Cartanova podalgebra, odnosno \(\mathfrak{h} = t \oplus \mathfrak{a}\), gdje je \(\mathfrak{a} = \mathfrak{h} \cap \mathfrak{p}\). Drugi cilj ovog rada je uvođenje Harish-Chandrinog preslikavanja na Cliffordovoj algebri \(C(\mathfrak{p})\) kao projekcije na \(C(\mathfrak{a})\) za neki izbor pozitivnih \((\mathfrak{g},\mathfrak{t})\) korijena. Pokazat će se da je restrikcija Harish-Chandrinog preslikavanja na \(\mathfrak{k}\)-invarijante u \(C(\mathfrak{p})\) izomorfizam algebri \(C(\mathfrak{p})^{\mathfrak{k}}\) i \(C(\mathfrak{a})\)

Abstract (english)

Let \(\mathfrak{g}\) be the complexification of the Lie algebra of a real reductive group \(G\). Assume that \((\mathfrak{g},\mathfrak{k})\) is a symmetric pair such that Spin\(_{ad}\) is a primary representation. Put differently, if we denote by \(\mathfrak{t}\) Cartan subalgebra of \(\mathfrak{k}\), then we have an equality of Weyl groups \((W_{\mathfrak{k},t} = W_{\mathfrak{g},t}\). Main goal of this dissertation is to show some new results in the relative Weil and Clifford algebra, analogous to famous results in the absolute case. First result will be an explicit formula for the transgression in the relative Weil algebra \(W(\mathfrak{g},\mathfrak{k}) = (S (\mathfrak{g}) \otimes \wedge \mathfrak{p})^{\mathfrak{k}} \). Further, the transgression theorem will be proved in this case; this theorem gives us the image and the kernel of the transgression map. For a symmetric pair \((\mathfrak{g},\mathfrak{k})\) as before let \(\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}\) be the Cartan decomposition. Let \(\mathfrak{h}\) be the fundamental Cartan subalgebra, so \(\mathfrak{h} = t \oplus \mathfrak{a}\), where \(\mathfrak{a} = \mathfrak{h} \cap \mathfrak{p}\). Another goal of this dissertation is to introduce Harish-Chandra map on the Clifford algebra \(C(\mathfrak{p})\) as a projection to \(C(\mathfrak{a})\) for a choice of positive \((\mathfrak{g},\mathfrak{t})\) roots. It will be shown that the restriction of the Harish-Chandra map to \(\mathfrak{k}\)-invariants in \(C(\mathfrak{p})\) is an algebra isomorphism \(C(\mathfrak{p})^{\mathfrak{k}} \rightarrow C(\mathfrak{a})\).

• Keywords: Liejeva algebra; Weylova grupa; Cliffordova algebra
• Keywords (english): Lie algebra; Weyl group; Clifford algebra
• Language: croatian
• URN:NBN: urn:nbn:hr:217:969559
• Promotion: 2023
• 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: vi, 79 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2023-07-21 11:11:36