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Quantum Engineering Year 0: Mathematical Foundations Month 11Day 295

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Year 0·Month 11·Week 3

Day 295: Introduction to Lie Groups — Continuous Symmetries

Day 295 of 2,016~12 min read

Learning Objectives

•Define Lie groups and understand their manifold structure
•Identify common matrix Lie groups
•Understand the dimension of a Lie group
•Connect Lie groups to physics symmetries
•Work with one-parameter subgroups
•Compute the dimension and basic properties of standard Lie groups

Today's Schedule (7 hours)

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Overview Prerequisites Learning Objectives 1 What is a Lie Group Formal Definition Matrix Lie Groups Dimension 2 The Classical Matrix Lie Groups The General Linear Group GL_n The Special Linear Group SL_n The Orthogonal Group On The Special Orthogonal Group SOn The Unitary Group Un The Special Unitary Group SUn Summary Table 3 One-Parameter Subgroups Definition Matrix Exponential Examples 4 Connectedness and Components Connected Components Simply Connected 5 Quantum Mechanics Connection Symmetries in QM Conservation Laws Noethers Theorem Generators and Observables 6 Worked Examples Example 1 Verify SO2 is a Lie Group Example 2 Dimension of SU2 Example 3 The Lorentz Group 7 Computational Lab 8 Practice Problems Problem Set A Basic Properties Problem Set B Exponentials Problem Set C Physics Applications 9 Summary Key Definitions Important Lie Groups 10 Preview Day 296

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