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Graduate Texts in Mathematics #222
Lie Groups, Lie Algebras, and Representations: An Elementary Introduction
This textbook treats Lie groups, Lie algebras and their representations in an elementary but fully rigorous fashion requiring minimal prerequisites. In particular, the theory of matrix Lie groups and their Lie algebras is developed using only linear algebra, and more motivation and intuition for proofs is provided than in most classic texts on the subject.
In addition to its accessible treatment of the basic theory of Lie groups and Lie algebras, the book is also noteworthy for including:
* a treatment of the Baker-Campbell-Hausdorff formula and its use in place of the Frobenius theorem to establish deeper results about the relationship between Lie groups and Lie algebras
* motivation for the machinery of roots, weights and the Weyl group via a concrete and detailed exposition of the representation theory of sl(3;C)
* rapidly develops the structure of semisimple Lie algebras by using a definition of semisimplicity that is unconventional but equivalent to the standard one
* gives independent constructions of the representations of semisimple Lie algebras and compact Lie groups
The second edition of Lie Groups, Lie Algebras, and Representations contains many substantial improvements and additions, among them: an entirely new part devoted to the structure and representation theory of compact Lie groups; a complete derivation of the main properties of root systems; the construction of finite-dimensional representations of semisimple Lie algebras has been elaborated; a treatment of universal enveloping algebras, including a proof of the Poincare Birkhoff Witt theorem and the existence of Verma modules; complete proofs of the Weyl character formula, the Weyl dimension formula and the Kostant multiplicity formula. The new edition also includes many additional figures.
In addition to its accessible treatment of the basic theory of Lie groups and Lie algebras, the book is also noteworthy for including:
* a treatment of the Baker-Campbell-Hausdorff formula and its use in place of the Frobenius theorem to establish deeper results about the relationship between Lie groups and Lie algebras
* motivation for the machinery of roots, weights and the Weyl group via a concrete and detailed exposition of the representation theory of sl(3;C)
* rapidly develops the structure of semisimple Lie algebras by using a definition of semisimplicity that is unconventional but equivalent to the standard one
* gives independent constructions of the representations of semisimple Lie algebras and compact Lie groups
The second edition of Lie Groups, Lie Algebras, and Representations contains many substantial improvements and additions, among them: an entirely new part devoted to the structure and representation theory of compact Lie groups; a complete derivation of the main properties of root systems; the construction of finite-dimensional representations of semisimple Lie algebras has been elaborated; a treatment of universal enveloping algebras, including a proof of the Poincare Birkhoff Witt theorem and the existence of Verma modules; complete proofs of the Weyl character formula, the Weyl dimension formula and the Kostant multiplicity formula. The new edition also includes many additional figures.
453 pages, Hardcover
First published August 7, 2003
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Displaying 1 - 3 of 3 reviews
August 8, 2024
Hall's text fills an important niche in this area, namely a more mathematically sophisticated but still practical introduction to Lie Groups and Algebras. This is due mostly to Hall's deliberate focus on matrix Lie Groups, which is how physicists and others who will actually use them in practical calculations will use them. For physicists, I still think Zee is the best show in town for making sense of this stuff, but Hall adds some rigor for those seeking more mathematical context for some of these ideas.
April 17, 2025
Lie groups are ubiquitous in theoretical physics because they arise naturally as symmetries in many of the special cases of problems that play an important role in illustrating the theory as paradigmatic solutions. As a rule, however, it is almost always a mistake to try to teach oneself mathematics from a textbook written by a physicist. Unfortunately, as a graduate student long ago this reviewer was not so clued in and studied the theory of Lie groups from the two-volume textbook by John Cornwell. As much as he can recall, Cornwell was indeed thorough in laying out the mechanics of computations with Lie groups and their representations but included nothing on formal manifold theory. Now, the converse by no means holds: for a textbook to be written by a mathematician will be no guarantee of its usefulness to one who wants to acquire a working knowledge of the relevant mathematics, especially for the beginner. For instance, Frank Warner’s old text on differentiable manifolds and John Lee’s recent entry on the subject (see our review here), while certainly unexceptionable as far as they go, leave one with the impression that Lie group theory forms an abstract domain remote from applications and one comes away from them having seen too few concrete examples to have a feeling, deep down, for what a Lie group or Lie algebra really is; moreover, they do not cover the extensive subject of representation theory at all.
Therefore, we learners are very fortunate to have in Brian C. Hall a gifted expositor of the elementary theory in the work presently to be reviewed, his Lie groups, Lie algebras and their representations: An elementary introduction (second edition) from the distinguished series of graduate texts in mathematics published by Springer Verlag. About his restriction to matrix Lie groups and algebras: this move allows one to focus on the group-theoretical aspect without getting caught up in all the complications of the modern theory of differentiable manifolds. For every matrix Lie group can be realized as a closed submanifold of the general linear group, which, when viewed concretely as a space of matrices has the structure of a Euclidean space over the reals, and, in as much as the defining conditions will always be given by polynomials, it all but trivially inherits a smooth manifold structure in the subspace topology. Now, the Lie groups expressible in this form cover all the classical Lie groups and virtually every case of relevance to physics. Moreover, the exponential permits easy definition in such terms as a convergent power series and one can then go back and forth readily between the Lie group and its Lie algebra. So the gain is considerable and very little is lost after all.
The first three chapters are devoted to preliminaries: basic definitions of the relevant terms, the matrix exponential and matrix logarithm, polar decomposition, simplicity, solvability, nilpotency, irreducibility and the adjoint map. Very clear on what it means to complexify a real Lie algebra. Only in chapter four does Hall begin a presentation of representation theory itself. Schur’s lemma receives a nice and concise proof. The general theory is illustrated through the all-important special case of representations of sl(2,C) and their classification into irreducibles (which students of physics will recognize as angular momentum or spin, basic to atomic theory as it yields good quantum numbers). Lastly, Hall gives an example of a non-matrix Lie group.
So far, everything has remained fairly elementary. Hall is prepared now, however, to get into the substance of Lie theory. Chapter five explains and proves the so-called Baker-Campbell-Hausdorff formula, which surprisingly expresses the product of the exponentials of two matrices in terms of the exponential of a series of nested commutators, starting with the special case of the Heisenberg group where the formula simplifies due to the fact that all the commutators after the first vanish identically and the indicated series terminates. The way Halls breaks the problem down into lemmas involving the derivative of an exponential makes for an easy-to-follow exposition.
But the heart of the text lies in Part II, chapters six to nine. Here, Hall’s pedagogical decision is in favor of clarity at the expense of maximal logical condensation. Hence, he will first show how directly to handle weights and roots in the rank-2 case of representations of sl(3,C) in chapter six before presenting the general case of semisimple Lie algebras in chapter seven – which necessitates a certain amount of repetition in that one must now formally define again the concepts of Cartan algebras, roots and root spaces, the Weyl group etc. which one has already met with in a concrete special case in the previous chapter. This reviewer doesn’t mind, as neither should any beginner more interested in getting acquainted with the ideas than in logical economy. (By the way, Hall uses a non-standard but equivalent definition of semisimplicity in order to streamline some of his arguments; for one who is not an expert it’s impossible to tell whether this constitutes an improvement.) Chapter eight recapitulates everything once more by working with abstract root systems, defined by a set of axioms designed to incorporate the essential features one has already encountered in the semisimple case. Nevertheless, this procedure does lead to an efficient account of bases, Weyl chambers, dominant integral elements, Dynkin diagrams etc., all of which are very geometrical and generously illustrated with figures. The classification of irreducible root systems, a fundamental and satisfying result of Lie theory, is presented without proof. A glance at Humphreys, say, who gives all the details, suggests that the proof is neither all that straightforward nor all that illuminating and it would have taken Hall too far afield to include in his elementary text. In any case, a full set of homework exercises allows one to acquire some familiarity with the concepts.
Chapter nine goes on to derive the theorem of highest weight for irreducible finite-dimensional representations of complex semisimple Lie algebras, via the universal enveloping algebra and construction of the Verma module. The key structural result is the Poincaré-Birkhoff-Witt theorem, proved in full. Roughly speaking, it says that a representation of a semisimple Lie algebra can be obtained by tying together a number of copies of sl(2,C), each equipped with its raising and lowering operators. Naturally, these are not independent of one another: one starts from a highest-weight vector and repeatedly applies all possible combinations of lowering operators; whatever non-zero one is left with will span the carrying space of the representation corresponding to the weights of the highest-weight vector with which one starts and, in the semisimple case, all possible finite-dimensional irreducible representations will be obtained in this manner.
Chapter ten sets forth in full detail several further properties of representations, such as Casimir elements, the Weyl character formula and the Kostant multiplicity formula. Here, one appreciates what an elegant subject Lie theory can be; from a rather straightforward starting point, it permits the elaboration of a number of deep structural results without demanding very much at all in terms of hard analysis.
The last three chapters sketch the representation theory of connected compact matrix Lie groups which turns out to be quite analogous to what one can do in the case of semisimple Lie algebras. There is again a theorem of highest weight, arrived at this time by employing completely different methods: in place of the Verma module, one uses the Haar integral to decompose the space of functions on the compact group under the left and right action. This, in point of fact, corresponds to how Hermann Weyl originally approached the subject. While Hall does everything in these chapters with his customary clarity, one ends up actually doing little with the objects one has obtained – perhaps a text such as that by Theodor Bröcker and Tammo tom Dieck would be a better resource for someone looking for a fuller exposition. But the final chapter wraps things up nicely by showing how all the apparatus one has so patiently developed can be applied to determine the fundamental groups of the classical Lie groups, viewed as topological spaces. The proofs of these results turn out to be admirably efficient.
Problems for the most part not very hard and more often than not are outfitted with hints, even the easy ones, which could suggest a plan of solution or just refer the reader to Lemma A or to the method of proof of Theorem B. If one heeds the hints, it does speed things up but one suspects most of the time it would not take very long to realize what is needed in any case, and one will thereby be deprived of the pleasure of having figured it out for oneself. In a few problems more computational in nature, on the other hand, the author declines to spell out anything about how to proceed, for instance, to construct the isomorphism between su(2) and the cross product in Euclidean 3d space or between complexified so(1,3) and sl(2,C) ⊕ sl(2,C). These occasional instances turn out to be rather harder than the general run of Hall’s exercises and it gives a sense of mounting excitement to wind one’s way to the right solution in the end.
Therefore, we learners are very fortunate to have in Brian C. Hall a gifted expositor of the elementary theory in the work presently to be reviewed, his Lie groups, Lie algebras and their representations: An elementary introduction (second edition) from the distinguished series of graduate texts in mathematics published by Springer Verlag. About his restriction to matrix Lie groups and algebras: this move allows one to focus on the group-theoretical aspect without getting caught up in all the complications of the modern theory of differentiable manifolds. For every matrix Lie group can be realized as a closed submanifold of the general linear group, which, when viewed concretely as a space of matrices has the structure of a Euclidean space over the reals, and, in as much as the defining conditions will always be given by polynomials, it all but trivially inherits a smooth manifold structure in the subspace topology. Now, the Lie groups expressible in this form cover all the classical Lie groups and virtually every case of relevance to physics. Moreover, the exponential permits easy definition in such terms as a convergent power series and one can then go back and forth readily between the Lie group and its Lie algebra. So the gain is considerable and very little is lost after all.
The first three chapters are devoted to preliminaries: basic definitions of the relevant terms, the matrix exponential and matrix logarithm, polar decomposition, simplicity, solvability, nilpotency, irreducibility and the adjoint map. Very clear on what it means to complexify a real Lie algebra. Only in chapter four does Hall begin a presentation of representation theory itself. Schur’s lemma receives a nice and concise proof. The general theory is illustrated through the all-important special case of representations of sl(2,C) and their classification into irreducibles (which students of physics will recognize as angular momentum or spin, basic to atomic theory as it yields good quantum numbers). Lastly, Hall gives an example of a non-matrix Lie group.
So far, everything has remained fairly elementary. Hall is prepared now, however, to get into the substance of Lie theory. Chapter five explains and proves the so-called Baker-Campbell-Hausdorff formula, which surprisingly expresses the product of the exponentials of two matrices in terms of the exponential of a series of nested commutators, starting with the special case of the Heisenberg group where the formula simplifies due to the fact that all the commutators after the first vanish identically and the indicated series terminates. The way Halls breaks the problem down into lemmas involving the derivative of an exponential makes for an easy-to-follow exposition.
But the heart of the text lies in Part II, chapters six to nine. Here, Hall’s pedagogical decision is in favor of clarity at the expense of maximal logical condensation. Hence, he will first show how directly to handle weights and roots in the rank-2 case of representations of sl(3,C) in chapter six before presenting the general case of semisimple Lie algebras in chapter seven – which necessitates a certain amount of repetition in that one must now formally define again the concepts of Cartan algebras, roots and root spaces, the Weyl group etc. which one has already met with in a concrete special case in the previous chapter. This reviewer doesn’t mind, as neither should any beginner more interested in getting acquainted with the ideas than in logical economy. (By the way, Hall uses a non-standard but equivalent definition of semisimplicity in order to streamline some of his arguments; for one who is not an expert it’s impossible to tell whether this constitutes an improvement.) Chapter eight recapitulates everything once more by working with abstract root systems, defined by a set of axioms designed to incorporate the essential features one has already encountered in the semisimple case. Nevertheless, this procedure does lead to an efficient account of bases, Weyl chambers, dominant integral elements, Dynkin diagrams etc., all of which are very geometrical and generously illustrated with figures. The classification of irreducible root systems, a fundamental and satisfying result of Lie theory, is presented without proof. A glance at Humphreys, say, who gives all the details, suggests that the proof is neither all that straightforward nor all that illuminating and it would have taken Hall too far afield to include in his elementary text. In any case, a full set of homework exercises allows one to acquire some familiarity with the concepts.
Chapter nine goes on to derive the theorem of highest weight for irreducible finite-dimensional representations of complex semisimple Lie algebras, via the universal enveloping algebra and construction of the Verma module. The key structural result is the Poincaré-Birkhoff-Witt theorem, proved in full. Roughly speaking, it says that a representation of a semisimple Lie algebra can be obtained by tying together a number of copies of sl(2,C), each equipped with its raising and lowering operators. Naturally, these are not independent of one another: one starts from a highest-weight vector and repeatedly applies all possible combinations of lowering operators; whatever non-zero one is left with will span the carrying space of the representation corresponding to the weights of the highest-weight vector with which one starts and, in the semisimple case, all possible finite-dimensional irreducible representations will be obtained in this manner.
Chapter ten sets forth in full detail several further properties of representations, such as Casimir elements, the Weyl character formula and the Kostant multiplicity formula. Here, one appreciates what an elegant subject Lie theory can be; from a rather straightforward starting point, it permits the elaboration of a number of deep structural results without demanding very much at all in terms of hard analysis.
The last three chapters sketch the representation theory of connected compact matrix Lie groups which turns out to be quite analogous to what one can do in the case of semisimple Lie algebras. There is again a theorem of highest weight, arrived at this time by employing completely different methods: in place of the Verma module, one uses the Haar integral to decompose the space of functions on the compact group under the left and right action. This, in point of fact, corresponds to how Hermann Weyl originally approached the subject. While Hall does everything in these chapters with his customary clarity, one ends up actually doing little with the objects one has obtained – perhaps a text such as that by Theodor Bröcker and Tammo tom Dieck would be a better resource for someone looking for a fuller exposition. But the final chapter wraps things up nicely by showing how all the apparatus one has so patiently developed can be applied to determine the fundamental groups of the classical Lie groups, viewed as topological spaces. The proofs of these results turn out to be admirably efficient.
Problems for the most part not very hard and more often than not are outfitted with hints, even the easy ones, which could suggest a plan of solution or just refer the reader to Lemma A or to the method of proof of Theorem B. If one heeds the hints, it does speed things up but one suspects most of the time it would not take very long to realize what is needed in any case, and one will thereby be deprived of the pleasure of having figured it out for oneself. In a few problems more computational in nature, on the other hand, the author declines to spell out anything about how to proceed, for instance, to construct the isomorphism between su(2) and the cross product in Euclidean 3d space or between complexified so(1,3) and sl(2,C) ⊕ sl(2,C). These occasional instances turn out to be rather harder than the general run of Hall’s exercises and it gives a sense of mounting excitement to wind one’s way to the right solution in the end.
November 20, 2018
Excellent transition between a theoretical linear algebra course and a full-on Lie groups course.
Most everything is done in a computable, low-dimensional setting.
Successfully taught to both physics and math students at the Senior and First Year Graduate level.
Most everything is done in a computable, low-dimensional setting.
Successfully taught to both physics and math students at the Senior and First Year Graduate level.
Displaying 1 - 3 of 3 reviews