Sumários · Grupos de Lie e Álgebras de Lie
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Sumários · Grupos de Lie e Álgebras de Lie
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25. Classification of finite dimensional irreducible representations of semisimple Lie algebras
Conclusion of the proof of the classification theorem. This lecture followed section 9.7 Brian Halls' book.
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pmr@math.tecnico.ulisboa.pt (Pedro Resende)
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Sumários
Fri, 18 Dec 2020 11:00:00 +0000

23. Representations of semisimple Lie algebras
The tensor algebra of a vector space. The enveloping associative algebra of a Lie algebra. The PoincaréBirkhoffWitt theorem. Introduction to Verma modules. Exercise 7.13 of Brian Hall's book.
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Sumários
Fri, 11 Dec 2020 11:00:00 +0000

17. Representations of Lie groups and Lie algebras
Exercises from chapter 6 of Brian Hall's book: 1, 2, 3, 6, 7, 10, 12.
https://fenix.tecnico.ulisboa.pt/disciplinas/GLAL/20202021/1semestre/verpost/17representationsofliegroupsandliealgebras
pmr@math.tecnico.ulisboa.pt (Pedro Resende)
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Sumários
Fri, 20 Nov 2020 11:00:00 +0000

16. Semisimple Lie algebras
Cartan subalgebras of semisimple Lie algebras. Basic properties and examples. Roots and root spaces relative to a Cartan subalgebra. The subalgebra isomorphic to sl(2;C) associated to each root. This lecture was based on Brian Hall, sections 7.2–7.3.
https://fenix.tecnico.ulisboa.pt/disciplinas/GLAL/20202021/1semestre/verpost/16semisimpleliealgebras
pmr@math.tecnico.ulisboa.pt (Pedro Resende)
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Sumários
Thu, 19 Nov 2020 11:00:00 +0000

15. Semisimple Lie algebras
Simple Lie algebras. Derived series and lower central series of a Lie algebra, solvable Lie algebras and nilpotent Lie algebras. Reductive Lie algebras and semisimple Lie algebras (defined to be reductive Lie algebras with trivial center, following Brian Hall's book). Decomposition of a reductive Lie algebra as the direct sum of its center and a semisimple Lie algebra. Decomposition of a semisimple Lie algebra as the direct sum of uniquely determined simple Lie subalgebras. Semisimplicity of the complexification of the Lie algebra of a simply connected matrix Lie group. This lecture followed sections 3.2 and 7.1 of Brian Hall's book.
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pmr@math.tecnico.ulisboa.pt (Pedro Resende)
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Sumários
Fri, 13 Nov 2020 11:00:00 +0000

13. Representations of Lie groups and Lie algebras
Proof of the classification of the irreducible representations of sl(3;C) (based on sections 6.4–6.5 of Brian Hall's book): highest weight cyclic representations and their identification with the irreducible representations; construction of irreducible representations as invariant subspaces of tensor powers of the standard representation and its dual. Example: construction of the irreducible representation with highest weight (1,1).
https://fenix.tecnico.ulisboa.pt/disciplinas/GLAL/20202021/1semestre/verpost/13representationsofliegroupsandliealgebras
pmr@math.tecnico.ulisboa.pt (Pedro Resende)
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Sumários
Fri, 06 Nov 2020 11:00:00 +0000

11. Representations of Lie groups and Lie algebras
Corollaries of Schur's lemma. Brief overview of Haar measures for locally compact Hausdorff groups, and of the construction of Haar measures via left/right invariant volume forms in the case of Lie groups. Proof that any representation of a compact Hausdorff group G is unitary with respect to an inner product obtained by integration on G. Corollary: every finite dimensional representation of a compact Hausdorff group (in particular a compact Lie group) is completely reducible. Representations of SU(2), su(2), and sl(2;C); classification of the irreducible representations. Representations of SO(3). This lecture is based on chapter 4 of Brian Hall (sections 4.5–4.7 plus Theorem 4.28 and Example 4.10).
https://fenix.tecnico.ulisboa.pt/disciplinas/GLAL/20202021/1semestre/verpost/11representationsofliegroupsandliealgebras
pmr@math.tecnico.ulisboa.pt (Pedro Resende)
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Sumários
Fri, 30 Oct 2020 11:00:00 +0000

10. Lie groups, Lie algebras and representations
Exercises from chapters 4 and 5 of Brian Hall's book.
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pmr@math.tecnico.ulisboa.pt (Pedro Resende)
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Sumários
Thu, 29 Oct 2020 11:00:00 +0000

7. Matrix Lie groups and Lie algebras
Global extension of local homomorphisms on simply connected matrix Lie groups. Lifting of Lie algebra homomorphisms for simply connected matrix Lie groups. Some functoriality properties of the liftings. Polar decomposition of matrices in SL(2;R) and SL(2;C). Simple connectedness of SL(2;C) and nonsimple connectedness of SL(2;R). Lifting of Lie algebra homomorphisms from sl(2;R).
<div>
Lecture based on section 5.7 of Brian Hall's book (and also briefly section 2.5).
</div>
https://fenix.tecnico.ulisboa.pt/disciplinas/GLAL/20202021/1semestre/verpost/7matrixliegroupsandliealgebras
pmr@math.tecnico.ulisboa.pt (Pedro Resende)
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Sumários
Fri, 16 Oct 2020 11:00:00 +0100

6. Matrix Lie Groups and Lie Algebras
Exercises from Brian Hall's book: Chapter 3, ex. 6, 7, 10, 11, 15, 16, 17, 18, 19.
https://fenix.tecnico.ulisboa.pt/disciplinas/GLAL/20202021/1semestre/verpost/6matrixliegroupsandliealgebras
pmr@math.tecnico.ulisboa.pt (Pedro Resende)
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Sumários
Thu, 15 Oct 2020 11:00:00 +0100

5. Matrix Lie groups and Lie algebras
Sections 3.7 and 3.8 of Brian Hall's book — Smooth structure on a matrix Lie group, exponential coordinates. The Lie algebra of a matrix Lie group as the tangent space at the identity. Smoothness of the continuous homomorphisms between matrix Lie groups. Representation of the elements of a connected matrix Lie group by products of exponentials. Corollary: connected matrix lie groups whose Lie algebras are abelian are themselves abelian. Section 3.5 — Functoriality of the Lie algebra of a matrix Lie group: the differential of a homomorphism of matrix Lie groups; the differential of F as the unique Lie algebra homomorphism f that commutes with the exponentials:
<div>
<br>
</div>
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exp o f = F o exp.
</div>
<div>
<br>
</div>
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Properties of the functor Lie : LieGrp > LieAlg: faithfulness (in the case of connected matrix Lie groups); preservation of kernels. The adjoint representation
</div>
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<br>
</div>
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Ad : G > GL(g).
</div>
<div>
<br>
</div>
<div>
The adjoint representation of Lie algebras ad as the differential of Ad. Section 3.6 — Complexification of real Lie algebras. Section 5.3 — The BakerCampbellHausdorff (BCH) formula. Section 5.7 — Introduction to the problem of integrating homomorphisms of Lie algebras: obtaining local homomorphisms of matrix Lie groups from homomorphisms of their Lie algebras.
</div>
https://fenix.tecnico.ulisboa.pt/disciplinas/GLAL/20202021/1semestre/verpost/summary1690348803926797
pmr@math.tecnico.ulisboa.pt (Pedro Resende)
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Sumários
Fri, 09 Oct 2020 11:00:00 +0100

4. Lie algebras
Exercises from Brian Hall's book: Chapter 2, ex. 6, 7, 8, 9, 10; Chapter 3, ex. 2, 5. Existence of a locally defined homeomorphism between a matrix Lie group (around the identity) and its Lie algebra (around 0): Theorem 3.42 (section 3.7).
https://fenix.tecnico.ulisboa.pt/disciplinas/GLAL/20202021/1semestre/verpost/4liealgebras
pmr@math.tecnico.ulisboa.pt (Pedro Resende)
https://fenix.tecnico.ulisboa.pt/disciplinas/GLAL/20202021/1semestre/verpost/4liealgebras#1689399616181975
Sumários
Thu, 08 Oct 2020 11:00:00 +0100

1. Lie groups and matrix Lie groups
Definition of Lie group, homomorphism, isomorphism, Lie subgroup, matrix Lie group. Examples, including the general and special linear groups, orthogonal, special orthogonal, unitary and special unitary groups, generalized orthogonal groups, Lorenz group, symplectic groups, Euclidian groups, Poincaré groups, compact symplectic groups. Topological properties: compactness, connectedness, simple connectedness. The unit component of a Lie group (respectively matrix Lie group). Examples. SU(2) as the 3sphere, SO(3) as projective 3space. The double cover SU(2)>SO(3).
<div>
This lecture corresponds closely to chapter 1 of Brian Hall's book minus section 1.2.8.
</div>
<div>
Exercises for next week (from that chapter): 4, 5, 9, 10, 13, 14, 16, 17, 18.
</div>
https://fenix.tecnico.ulisboa.pt/disciplinas/GLAL/20202021/1semestre/verpost/summary1690348803925417
pmr@math.tecnico.ulisboa.pt (Pedro Resende)
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Sumários
Fri, 25 Sep 2020 11:00:00 +0100

0. Introduction
Presentation of the course and discussion of practical matters.
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pmr@math.tecnico.ulisboa.pt (Pedro Resende)
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Sumários
Thu, 24 Sep 2020 11:00:00 +0100

24. Verma modules
Verma modules. Irreducible representations obtained from Verma modules. This material followed sections 9.5 and 9.6 of Brian Hall's book.
https://fenix.tecnico.ulisboa.pt/disciplinas/GLAL/20202021/1semestre/verpost/24vermamodules
pmr@math.tecnico.ulisboa.pt (Pedro Resende)
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Sumários
Thu, 17 Dec 2020 11:00:00 +0000

22. Representations of semisimple Lie algebras
Integral elements and dominant integral elements. Partial order on the set of integral elements. Basic properties. Integrality of the weights of the representations of semisimple Lie algebras, and invariance of the set of weights under the action of the Weyl group. Introduction to the theorem of the highest weight. This lecture was based on sections 8.7–8.8 and 9.1–9.2 of Brian Hall's book.
https://fenix.tecnico.ulisboa.pt/disciplinas/GLAL/20202021/1semestre/verpost/22representationsofsemisimpleliealgebras
pmr@math.tecnico.ulisboa.pt (Pedro Resende)
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Sumários
Thu, 10 Dec 2020 11:00:00 +0000

21. Semisimple Lie algebras and root systems
Exercises from chapter 7 of Brian Hall's book: 1, 2, 4, 6, 9, 12.
https://fenix.tecnico.ulisboa.pt/disciplinas/GLAL/20202021/1semestre/verpost/21semisimpleliealgebrasandrootsystems
pmr@math.tecnico.ulisboa.pt (Pedro Resende)
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Sumários
Fri, 04 Dec 2020 11:00:00 +0000

20. Semisimple Lie algebras and root systems
Duals of root systems. Bases and Weyl chambers. Action of the Weyl group on the set of open Weyl chambers: basic properties, transitivity and freeness. Reflections associated to the positive simple roots as generators of the Weyl group. Dynkin diagrams. Classification of the irreducible root systems and of the complex simple Lie algebras. (Cf. sections 8.3–8.6 and 8.9–8.11 of Brian Hall's book.)
https://fenix.tecnico.ulisboa.pt/disciplinas/GLAL/20202021/1semestre/verpost/20semisimpleliealgebrasandrootsystems
pmr@math.tecnico.ulisboa.pt (Pedro Resende)
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Sumários
Thu, 03 Dec 2020 11:00:00 +0000

19. Semisimple Lie algebras and root systems
The root systems of the classical Lie algebras. Simple Lie algebras versus complexification of Lie algebras. Theorem: a nonabelian Lie algebra of a compact matrix Lie group is simple if and only if its complexification is simple as a complex Lie algebra. A semisimple Lie algebra is simple if and only if its root system is irreducible. This lecture followed Brian Hall's book, sections 7.6 and 7.7.
https://fenix.tecnico.ulisboa.pt/disciplinas/GLAL/20202021/1semestre/verpost/19semisimpleliealgebrasandrootsystems
pmr@math.tecnico.ulisboa.pt (Pedro Resende)
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Sumários
Fri, 27 Nov 2020 11:00:00 +0000

18. Semisimple Lie algebras and root systems
More properties of the roots of semisimple Lie algebras: the only multiples of a root
<i>a</i> that are roots are
<i>a</i> and –
<i>a</i>; the root spaces have dimension one. Weyl group of a semisimple algebra. Invariance of the set of roots under the action of the Weyl group. Root systems. The Weyl group of a root system. Direct sum of roots systems, irreducible root systems. General facts about angles between roots and root norms. Classification of the ranktwo root systems. This lecture was based on sections 7.3–7.5 and 8.1–8.2 of Brian Hall's book.
https://fenix.tecnico.ulisboa.pt/disciplinas/GLAL/20202021/1semestre/verpost/18semisimpleliealgebrasandrootsystems
pmr@math.tecnico.ulisboa.pt (Pedro Resende)
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Sumários
Thu, 26 Nov 2020 11:00:00 +0000

14. Representations of Lie groups and Lie algebras
Matrix representation of weights and roots with respect to abelian Lie subalgebras equipped with the trace inner product. Weyl group of an abelian Lie subalgebra. Example — sl(3;C): symmetries of the set of weights and roots; weight diagrams for several irreducible representations; proof that the weights of an irreducible representation are the integral elements in the convex hull of the orbit of the highest weight under the action of the Weyl group which can be obtained from the highest weight by subtracting an integer combination of roots. This lecture followed sections 6.6–6.8 of Brian Hall's book.
https://fenix.tecnico.ulisboa.pt/disciplinas/GLAL/20202021/1semestre/verpost/14representationsofliegroupsandliealgebras
pmr@math.tecnico.ulisboa.pt (Pedro Resende)
https://fenix.tecnico.ulisboa.pt/disciplinas/GLAL/20202021/1semestre/verpost/14representationsofliegroupsandliealgebras#1126449662739989
Sumários
Thu, 12 Nov 2020 11:00:00 +0000

12. Representations of Lie groups and Lie algebras
The reordering lemma for representations of Lie algebras. Cyclic generators versus irreducible representations. Representations of SU(3) (introduction). Basis of sl(3;C) and commutation relations of the basis elements. Irreducibility of the standard representation. Weights, weight vectors, weight spaces, weight multiplicity. Roots and root vectors. Positive simple roots. Partial order on the set of weights; highest weights. Statement of the classification theorem for the irreducible representations of sl(3;C) and SU(3). Dominant integral elements. Beginning of the proof of the classification theorem: direct sum decomposition of irreducible representations in terms of weight spaces. The material for this lecture is chapter 6 of Brian Hall's book up to and including Proposition 6.9, plus Proposition 6.12.
https://fenix.tecnico.ulisboa.pt/disciplinas/GLAL/20202021/1semestre/verpost/summary1127398850481747
pmr@math.tecnico.ulisboa.pt (Pedro Resende)
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Sumários
Thu, 05 Nov 2020 11:00:00 +0000

9. Representations of Lie groups and Lie algebras
Real and complex finite dimensional representations. Basic definitions and examples: trivial, standard, and adjoint representations. Representations versus actions. Invariant subspaces, irreducible representations. Intertwiners (equivariant maps), isomorphic representations. The differential of a representation of matrix Lie groups. Functorial properties of the differential. Complexification versus irreducibility. Direct sums and tensor products of representations. Complete reducibility. Complete reducibility of invariant subspaces of completely reducible representations. Complete reducibility of unitary representations on Euclidean spaces. Schur's Lemma. This lecture followed the material in chapter 4 of Brian Hall's book.
https://fenix.tecnico.ulisboa.pt/disciplinas/GLAL/20202021/1semestre/verpost/9representationsofliegroupsandliealgebras
pmr@math.tecnico.ulisboa.pt (Pedro Resende)
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Sumários
Fri, 23 Oct 2020 11:00:00 +0100

8. Matrix Lie groups and Lie algebras
<b>Lie's second theorem</b> for matrix groups (revision from previous lecture). An equivalence of categories between the category of simply connected matrix Lie groups and the category of their Lie algebras. Lie subgroups: connected Lie subgroups of matrix groups;
<b>Lie's third theorem</b>. Universal covers. SL(2,R) as an example without a universal cover in the class of matrix Lie groups. General universal covers.
<b>Lie's first theorem</b>. The equivalence of categories between the category of finite dimensional real Lie groups and the category of finite dimensional real Lie algebras. This lecture followed sections 5.8–5.10 of Brian Hall's book. See also these
<a href="https://fenix.tecnico.ulisboa.pt/downloadFile/1126518382258831/notas.pdf">notes</a>.
https://fenix.tecnico.ulisboa.pt/disciplinas/GLAL/20202021/1semestre/verpost/summary1127398850479338
pmr@math.tecnico.ulisboa.pt (Pedro Resende)
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Sumários
Thu, 22 Oct 2020 11:00:00 +0100

3. Lie algebras
(Roughly sections 2.1–2.4, 3.1, 3.3–3.4 of Brian Hall's book.) Introduction to Lie algebras. Homomorphisms, subalgebras, ideals, and quotients. Lie algebras of derivations. Representations of Lie algebras. The adjoint representation of a Lie algebra on itself. Faithful representations and Lie algebras of matrices. "Easy" faithful representations: abelian Lie algebras; Lie algebras with trivial center. Statement of Ado's theorem. Matrix exponentials and logarithms. Oneparameter subgroups of a matrix Lie group. The Lie algebra of a matrix Lie group. Examples.
https://fenix.tecnico.ulisboa.pt/disciplinas/GLAL/20202021/1semestre/verpost/3liealgebras
pmr@math.tecnico.ulisboa.pt (Pedro Resende)
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Sumários
Fri, 02 Oct 2020 11:00:00 +0100

2. Lie groups and matrix Lie groups
Exercises from chapter 1 of Brian Hall's book: 5, 9, 10, 13, 14, 17, 18.
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pmr@math.tecnico.ulisboa.pt (Pedro Resende)
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Sumários
Thu, 01 Oct 2020 11:00:00 +0100