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Reformulate, complete and turn this file into a po...

Creato il: 28 settembre 2026

Risposto usando GPT-5.6 Thinking da Chat01

Domanda

Liegroups.pdf

Reformulate, complete and turn this file into a polished standalone tex file of interesting exercises in Lie groups

Risposta

I’ll turn the sheet into a self-contained LaTeX source, keeping its mathematical core while fixing notation, numbering, incomplete prompts, and presentation. I’ll also make the exercises more coherent as a standalone problem set rather than a transcription.

Pensiero

Used pdfs skill

Pensiero

Checking representation dimensions

I’m noticing a likely inconsistency: the three-dimensional Heisenberg Lie algebra cannot embed faithfully into gl2(R)\mathfrak{gl}_2(\mathbb R), although a faithful representation into gl3(R)\mathfrak{gl}_3(\mathbb R) exists. I’m clarifying the wording before constructing examples.

Checked representation dimensions

Organizing exercises

I’m restructuring the material into a coherent standalone set, renumbering continuously, clarifying incomplete prompts, and preserving key topics such as exponentials, connectedness, quotients, actions, matrix groups, and the Heisenberg example.

Pensiero

Created and visually verified a polished Lie groups exercise handout PDF

Worked for 4m 33s

Risposta

Done. I turned the original sheet into a polished, self-contained problem set rather than a literal transcription. The source topics—from invariant vector fields and one-parameter subgroups through SU(2)SU(2), SO(3)SO(3), and matrix exponentials—are retained and reorganized coherently. (Liegroups.pdf) (Liegroups.pdf)

I also repaired a couple of mathematical issues in the original: the inconsistent left/right-invariant pushforward statement (Liegroups.pdf) and the requested injection of the Heisenberg algebra into gl2(R)\mathfrak{gl}_2(\mathbb R), which is replaced by the more meaningful task of proving nonexistence and constructing the standard faithful gl3\mathfrak{gl}_3-representation. (Liegroups.pdf)

The result has 18 continuously numbered exercises, standardized notation and conventions, completed missing examples, several natural extensions/challenges, and five thematic sections.

Download the standalone LaTeX source

Download the compiled PDF preview

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