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It is a useful reference and a source of explicit computations. Each section contains a range of exercises, and 24 figures help illustrate geometric concepts. As much as the content is great, the print quality is outrageous. This edition is a photocopy of the original printing, and it's hard to read. Surprised by the fact that Springer couldn't be bothered to even digitise the text. Good start book for Rep. This book is NEW, no mark at all!
Most lie groups books fall into one of two categories: I prefer the former. If you're perfectly happy with a more algebraic treatment, read no further. There aren't very many geometrically flavored books on lie groups; I can think of only this one and "Compact lie groups" by Sepanski. His book has a nicer treatment of harmonic analysis, but this one beats his in almost every other respect.
Sepanski is often too concise. Still, this book has a long way to go to be really great. Some special notes are in order. With respect to these bases, realize the c1 ,. In other words, once a basis is chosen and the G action is realized by matrix multiplication, the action of G on V is obtained from the action of G on V simply by taking the conjugate of the matrix. It should also be noted that few of these constructions are independent of each other. Also the actions in 4 , 5 , and 6 really only make repeated use of number 2. For this to be successful, it is necessary to examine the smallest possible building blocks.
Thus U is a representation of G in its own right. A nonzero representation is called reducible if there is a proper i. For more general representations, this approach is often impossible to carry out. In those cases, other tools are needed. One important tool is based on the next result. Similarly, the image of T is nonzero and G-invariant, so irreducibility implies T is surjective and therefore a bijection.
Noncompact groups abound with nonunitary representations Exercise 2. However, compact groups are much more nicely behaved. Every representation of a compact Lie group is unitary. Reducible but not completely reducible representations show up frequently for noncompact groups Exercise 2. Finite-dimensional representations of compact Lie groups are completely reducible.
Suppose V is a representation of a compact Lie group G that is reducible. As a result, any representation V of a compact Lie group G may be written as 2. Understanding the set of irreducible representations will take much more work. The bulk of the remaining text is, in one way or another, devoted to answering this question. For part 2 , assume V is irreducible.
It is clear that c must be in R and positive. The following result gives a uniform method of handling this ambiguity as well as giving a formula for the n i in Equation 2. Let W be a representation of a Lie group G. Use this to show that the set of irreducible representations of R is indexed by C and that the unitary ones are indexed by iR. In fact, we will later see Theorem 3. Thus only H0 Rn contains a nonzero S O n -invariant function. The desired result now follows from the previous observation and Lemma 2.
It is nonzero since f 1, 0,. Hm Rn is an irreducible S O n -module: A relatively small dose of functional analysis Exercise 3. This section shows that these representations are irreducible. For n odd, the spin representation S of Spinn R is irreducible. Thus both half-spin representation are irreducible. The 46 2 Representations function: This chapter studies a number of function spaces on G such as the set of continuous functions on G, C G , or the set of square integrable functions on G, L 2 G , with respect to the Haar measure dg.
These function spaces are examined in the light of their behavior under left and right translation by G. Integrating on each coordinate of the u,v: It turns out that character theory provides a powerful tool for studying representations. In fact, we will see in Theorem 3. Each statement of the theorem is straightforward to prove. We prove parts 1 , 4 , 5 , and 7 and leave the rest as an exercise Exercise 3. The next theorem calculates the L 2 inner product of characters corresponding to irreducible representations.
Begin with the assumption that V, W are irreducible. Since dim V G is a real number, the integrand may be conjugated with impunity and part 1 follows. This allows us to eventually focus our study on compact Lie groups that are as small as possible. It is said to be of quaternionic type if there is a quaternionic vector space on which G acts that gives rise to the action on V by restriction of scalars. It is said to be of complex type if it is neither real nor quaternionic type. Show that V is of quaternionic type if and only if V possesses an invariant nondegenerate skew-symmetric bilinear form.
Now show that the map Exercise 3. Thus when U is closed, U is a representation of G in its own right and is also called a submodule or a subrepresentation. A nonzero representation is called reducible if there is a proper closed G-invariant subspace of V.
We will soon see Lemma 3. Start with V and W irreducible. To clear this hurdle, we invoke a standard theorem from a functional analysis course. It has already been shown that nonzero elements of HomG V, V are injective. In particular, S is a multiple 56 3 Harmonic Analysis of the identity. Since S is injective, S is a nonzero multiple of the identity. For instance, vector-valued integration in this setting was already used in the proof u,v.
Obvious generalizations can be made to of Theorem 3. In any case, functional analysis provides a general framework for this type of operation which we recall now see [74] for details. Remember that G is still a compact Lie group throughout this chapter. Let V be a Hausdorff locally convex topological space and F: The linear map T is compact if the closure of the image of the unit ball under T is compact. It is a standard fact from functional analysis that the set of compact operators is a closed left and right ideal under composition within the set of bounded operators e.
The hardest part is getting started. In fact, the heart of the matter is really contained in Lemma 3. T is G-invariant since dg is u,v. Using the left invariant e. Using the fact that T0 is self-adjoint, it therefore follows that T is also self-adjoint. Partially order this collection by inclusion. By taking a union, every linearly ordered subset clearly has an upper bound.
In particular, Lemma 3. The only real change replaces direct sums with Hilbert space direct sums. Let V be a unitary representation of a compact Lie group G on a Hilbert space. As in the proof of Theorem 2. In this section we decompose L 2 G under left and right translation of functions. Instead of attacking this problem directly, it turns out to be easy Lemma 3. Using the Stone— Weierstrass Theorem Theorem 3.
Both spaces carry a left and right action l g and r g of G given by 3. They are called the left and right regular representations. Since f may be chosen arbitrarily close to f 2 in the L 2 norm and since G already acts continuously on C G , the result follows. Even though there are two actions of G on C G , i. In light of Theorem 3. This map 2 1 even intertwines with the right regular action of L S.
In order to generalize to groups that are not Abelian and to accommodate both the left and right regular actions, we will phrase the result a bit differently. Thus the results of Fourier analysis on S 1 can be thought of as arising directly from the representation theory of S 1. This result will generalize to all compact Lie groups. Let G be a compact Lie group. The proof of this theorem is really not much more than the proof of Theorem 2.
To see that the map is surjective, Lemma 3. It remains to see that the map is injective. Coupling this density result with the canonical decomposition and the version of Frobenius reciprocity contained in Lemma 3. Since the two results are so linked, the following corollary is also often referred to as the Peter—Weyl Theorem.
With respect to the same conventions as in Lemma 3. The calculation given in the proof of Theorem 3. This follows immediately from Lemma 3. A compact Lie group G possesses a faithful representation, i. Even better, since compact, each is isomorphic to a closed subgroup of U n by Theorem 2. For part 1 , recall from Theorem 3. In light of Lemma 3. For part 2 , let f be a continuous class function. For this, use Theorem 3. For part 3 , let f be an L 2 class function. By dimension, each is obviously inequivalent to the others. In fact, they are the only irreducible representations up to isomorphism c.
Part 3 of Theorem 3. Since a representation is determined by its character, Theorem 3. Use the theorems of this chapter to recover the standard results of Fourier analysis on S 1. Use the fact that G acts trivially on 1-dimensional representations to show that all irreducible representations of a compact Lie group are one-dimensional if and only if G is Abelian c.
However continuity and any positive Lipschitz condition will guarantee uniform convergence. In fact, the scalar valued Fourier transform in Theorem 3. In terms of proofs, most of the work needed for the general case is already done in Corollary 3. G G For part 2 , calculate the following: Some comments are in order.
The inverse operator valued Fourier transform, I: These details will be checked in the proof below. To check that the algebra structures are preserved, simply use Lemma 3. Part 1 follows immediately from the Plancherel Theorem. For part 3 , the Plancherel Theorem and Lemma 3. However, using Lie algebra techniques and the Plancherel Theorem, it is possible to show that the Scalar Fourier Inversion Theorem holds when f is continuously differentiable. In particular, the Scalar Fourier Inversion Theorem holds for smooth f. This result is implicitly embedded in the proof of Theorem 3.
However it is trivial to check directly. In fact, the main part of this result is true in a much more general setting than Hilbert space representations. Now only assume V is a Hausdorff complete locally convex topological space. Let V be a representation of a compact Lie group G on a Hausdorff complete locally convex topological space.
For part 2 , Lemma 3. However, the existence of the projections in Theorem 3. Looking to use Corollary 3. As an example, S could be the set of smooth functions on G or the set of real analytic functions on G. One interpretation of Corollary 3. Let f be a smooth class function on SU 2. Show that G is Abelian if and only if the convolution on C G is commutative c. However, it turns out there is a way to linearize their study by looking at the tangent space to the identity.
The resulting object is called a Lie algebra. Simply by virtue of the fact that vector spaces are simpler than groups, the Lie algebra provides a powerful tool for studying Lie groups and their representations. Since we are interested in compact groups, there is a way to bypass much of this differential geometry. Recall from Theorem 3. In the setting of Lie subgroups of G L n, C , the Lie algebra has an explicit matrix realization which we develop in this chapter.
It should be remarked, however, that the theorems in this chapter easily generalize to any Lie group. Given a compact group G, Theorem 3. The statements regarding the basic properties of the Lie bracket in part b are elementary and left as an exercise Exercise 4. Since left multiplication is a diffeomorphism, Equation 4. It is obviously complete. In fact, we will see that the differential of exp at I is the identity map on all of G L n, C.
Part c follows from Theorem 1. Note from the proof of Theorem 4. Also in general, exp need not be onto Exercise 4. To check this, use Theorem 4. It remains to calculate the Lie algebras for the compact classical Lie groups. Again, this follows 4. Suppose X is in the Lie algebra of U n. It is handled as in the case of S L n, F. Using the fact that the tangent space has the same dimension as the manifold, we now have a simple way to calculate the dimension of U n and SU n.
We give the corresponding Lie algebra for each.
Mathematics Algebra · Graduate Texts in Mathematics This book is intended for a one year graduate course on Lie groups and Lie algebras. The author. This textbook treats Lie groups, Lie algebras and their representations in an elementary but fully rigorous fashion requiring Graduate Texts in Mathematics.
As usual, this follows from Theorem 4. Suppose X is in the Lie algebra of Sp n, C. By calculating the differential on each standard basis vector, show that the map t1 ,. The coordinates t1 ,. A smooth homomorphism of the additive group R into a Lie group G is called a one-parameter subgroup. Some notes are in order. When that is done, ad will, in fact, be a representation of g on itself. For the sake of clarity of exposition, we will verify this lemma for Mn R and leave the general case of Mn C to the reader.
Moreover, on G L n, C , their value at each point determines a smooth rank k subbundle of the tangent bundle. Finally, it is an important fact that integral submanifolds such as H , as was the case for regular submanifolds, satisfy the property that when f: There is a bijection between the set of connected Lie subgroups of G and the set of subalgebras of g.
Suppose h is a subalgebra of g. By the remark above the statement of this theorem, the multiplication and inverse operations are smooth as maps on H , so that H is a Lie subgroup of G. Hence the correspondence is surjective. Using the exponential map and Theorem 4. Let H and G be connected Lie subgroups of general linear groups with H simply connected.
Uniqueness follows from Theorem 4. Such a chart is called cubical. This result will have far-reaching consequences, including various structure theorems. Since part b follows from part a and Theorems 1. It is a familiar fact Exercise 4. In particular, the most general compact connected Abelian Lie group is a torus. Let G be a compact Abelian group. A maximal torus of G is a maximal connected Abelian subgroup of G.
Then T is a maximal torus if and only if t is a Cartan subalgebra. In particular, maximal tori and Cartan subalgebras exist. Since maximal Abelian subalgebras clearly exist, this also shows that maximal tori exist. Since it is easy to see t is a maximal Abelian subalgebra of u n Exercise 5. It is straightforward to verify that t is a Cartan subalgebra Exercise 5. Then T is a maximal torus and t is its corresponding Cartan subalgebra Exercise 5. For any such Ad-invariant inner product on g, ad is skew-symmetric, i.
To prove part a , recall that Theorem 2. Let G be a compact Lie group and t a Cartan subalgebra of g. It follows that r X is an ad X -invariant subspace. If zg X is a Cartan subalgebra, then X is called a regular element of g. Let G be a compact Lie group and t a Cartan subalgebra. Using the skewsymmetry of ad, Lemma 5. Then Ad G acts transitively on the set of Cartan subalgebras of G. Using the fact that Ad g is a Lie algebra homomorphism, Theorem 4. Since Ad g is a Lie algebra homomorphism, Ad g t1 is still Abelian.
For part b , let Ti be the maximal torus of G corresponding to ti. Let G be a compact connected Lie group. The kernel of the Adjoint map is the center of G, i. Thus cg is the identity on a neighborhood of I in G. Since G is connected and cg is a homomorphism, Theorem 1. In other words parts a and b are equivalent. We will prove part b by induction on dim g. Thus it remains to show that exp g is open. It is necessary to show that there is a neighborhood of g0 contained in exp g.
If dim a0 Corollary 5. Let G be a compact connected Lie group with maximal torus T.
In particular, T is maximal Abelian. Part b clearly follows from part a. Note that there exist maximal Abelian subgroups that are not maximal tori Exercise 5. Show that the following statements are equivalent.
It is a straightforward exercise Exercise 1. The Borel—Weil Theorem repairs this gap. Linear Representations of Finite Groups. G and so central by Lemma 1. Series by cover 1—8 of next show all. Theory of Bergman Spaces by Hakan Hedenmalm.
Show that T is a maximal torus c. Show that the center of G, Z G , is the intersection of all maximal tori in G. Suppose S is a connected Abelian Lie subgroup of G. Suppose that G is also a complex manifold whose group operations are holomorphic. We already know from Theorem 4. In fact, more is true. As is common in mathematics, we instead resort to a trick. Although now known as the Campbell—Baker—Hausdorff Series [21], [5], and [49] , the following explicit formula is actually due to Dynkin [35].
The approach of this proof follows [34]. Springer Shop Bolero Ozon. Lie Groups, Lie Algebras, and Representations: Brian Hall , Brian C.. This book provides an introduction to Lie groups, Lie algebras, and repre sentation theory, aimed at graduate students in mathematics and physics. Although there are already several excellent books that cover many of the same topics, this book has two distinctive features that I hope will make it a useful addition to the literature. First, it treats Lie groups not just Lie alge bras in a way that minimizes the amount of manifold theory needed.
Thus, I neither assume a prior course on differentiable manifolds nor provide a con densed such course in the beginning chapters. Second, this book provides a gentle introduction to the machinery of semi simple groups and Lie algebras by treating the representation theory of SU 2 and SU 3 in detail before going to the general case.
This allows the reader to see roots, weights, and the Weyl group "in action" in simple cases before confronting the general theory. The standard books on Lie theory begin immediately with the general case: The Lie algebra is then defined as the space of left-invariant vector fields and the exponential mapping is defined in terms of the flow along such vector fields.
This approach is undoubtedly the right one in the long run, but it is rather abstract for a reader encountering such things for the first time.