Skip to main content
MathOverflow

Return to Answer

Fixed the link.
Source Link
Brian Hall
Brian Hall
  • 376
  • 2
  • 5

As mentioned above, my book, "Lie groups, Lie algebras, and representations," discusses this question in detail. See http://www.amazon.com/Lie-Groups-Algebras-Representations-Introduction/dp/0387401229http://www.amazon.com/Lie-Groups-Algebras-Representations-Introduction/dp/3319134663/ref=sr_1_3?s=books&ie=UTF8&qid=1457839028&sr=1-3&keywords=brian+hall. The book is now in its second edition; in this version, the relevant part is Chapter 5. I do things from the matrix group perspective. In addition, I use the Baker-Campbell-Hausdorff formula rather than appealing to the Frobenius theorem. I think the BCH formula shows most clearly "why the result is really true." As already pointed out, compactness is not needed in the general statement.

One can, however, look at the question specifically from the perspective of the compact case. The classification of the representations of the group and of the Lie algebra both take the form of a "theorem of the highest weight". (These are discussed in Part II of my book for the Lie algebra case and Part III for the group case.) In the simply connected case, we expect that the classifications will match up. To verify this directly (without appealing to the results of Chapter 5), we have to show that the possible highest weights in the group result are the same as in the Lie algebra result. I show this in Chapter 13 of the book; see Corollary 13.20. Since the argument is a bit involved, it is probably simplest just to look at the material in Chapter 5.

As mentioned above, my book, "Lie groups, Lie algebras, and representations," discusses this question in detail. See http://www.amazon.com/Lie-Groups-Algebras-Representations-Introduction/dp/0387401229. The book is now in its second edition; in this version, the relevant part is Chapter 5. I do things from the matrix group perspective. In addition, I use the Baker-Campbell-Hausdorff formula rather than appealing to the Frobenius theorem. I think the BCH formula shows most clearly "why the result is really true." As already pointed out, compactness is not needed in the general statement.

One can, however, look at the question specifically from the perspective of the compact case. The classification of the representations of the group and of the Lie algebra both take the form of a "theorem of the highest weight". (These are discussed in Part II of my book for the Lie algebra case and Part III for the group case.) In the simply connected case, we expect that the classifications will match up. To verify this directly (without appealing to the results of Chapter 5), we have to show that the possible highest weights in the group result are the same as in the Lie algebra result. I show this in Chapter 13 of the book; see Corollary 13.20. Since the argument is a bit involved, it is probably simplest just to look at the material in Chapter 5.

As mentioned above, my book, "Lie groups, Lie algebras, and representations," discusses this question in detail. See http://www.amazon.com/Lie-Groups-Algebras-Representations-Introduction/dp/3319134663/ref=sr_1_3?s=books&ie=UTF8&qid=1457839028&sr=1-3&keywords=brian+hall. The book is now in its second edition; in this version, the relevant part is Chapter 5. I do things from the matrix group perspective. In addition, I use the Baker-Campbell-Hausdorff formula rather than appealing to the Frobenius theorem. I think the BCH formula shows most clearly "why the result is really true." As already pointed out, compactness is not needed in the general statement.

One can, however, look at the question specifically from the perspective of the compact case. The classification of the representations of the group and of the Lie algebra both take the form of a "theorem of the highest weight". (These are discussed in Part II of my book for the Lie algebra case and Part III for the group case.) In the simply connected case, we expect that the classifications will match up. To verify this directly (without appealing to the results of Chapter 5), we have to show that the possible highest weights in the group result are the same as in the Lie algebra result. I show this in Chapter 13 of the book; see Corollary 13.20. Since the argument is a bit involved, it is probably simplest just to look at the material in Chapter 5.

Source Link
Brian Hall
Brian Hall
  • 376
  • 2
  • 5

As mentioned above, my book, "Lie groups, Lie algebras, and representations," discusses this question in detail. See http://www.amazon.com/Lie-Groups-Algebras-Representations-Introduction/dp/0387401229. The book is now in its second edition; in this version, the relevant part is Chapter 5. I do things from the matrix group perspective. In addition, I use the Baker-Campbell-Hausdorff formula rather than appealing to the Frobenius theorem. I think the BCH formula shows most clearly "why the result is really true." As already pointed out, compactness is not needed in the general statement.

One can, however, look at the question specifically from the perspective of the compact case. The classification of the representations of the group and of the Lie algebra both take the form of a "theorem of the highest weight". (These are discussed in Part II of my book for the Lie algebra case and Part III for the group case.) In the simply connected case, we expect that the classifications will match up. To verify this directly (without appealing to the results of Chapter 5), we have to show that the possible highest weights in the group result are the same as in the Lie algebra result. I show this in Chapter 13 of the book; see Corollary 13.20. Since the argument is a bit involved, it is probably simplest just to look at the material in Chapter 5.