Math 210C. A non-closed commutator subgroup

Size: px
Start display at page:

Transcription

1 Math 210C. A non-closed commutator subgroup 1. Introduction In Exercise 3(i) of HW7 we saw that every element of SU(2) is a commutator (i.e., has the form xyx 1 y 1 for x, y SU(2)), so the same holds for its quotient SO(3). We have seen in Exercise 2 of HW8 that if G is a connected compact Lie group then its commutator subgroup G is closed. This can be pushed a bit further. The arguments in Exercise 2 of HW8 show there is a finite set of subgroups G i of the form SU(2) or SO(3) such that the multiplication map of manifolds G 1 G n G is surjective on tangent spaces at the identity, and hence (by the submersion theorem) has image containing an open neighborhood U of e in G. Since U generates G algebraically (as for an open neighborhood of the identity in any connected Lie group), G is covered by open sets of the form Uu 1... u r for finite sequences {u 1,..., u r } in U. But any open cover of G has a finite subcover since G is compact, so there is a finite upper bound on the number of commutators needed to express any element of G. It is natural to wonder about the non-compact case. Suppose G is an arbitrary connected Lie group and H 1, H 2 G are normal closed connected Lie subgroups. Is the commutator subgroup (H 1, H 2 ) closed? It is an important theorem in the theory of linear algebraic groups that if G is a Zariski-closed subgroup of some GL n (C) with H 1, H 2 G also Zariskiclosed then the answer is always affirmative, with the commutator even Zariski-closed. But counterexamples occur for the non-compact G = SL 2 (R) SL 2 (R)! Before discussing such counterexamples, we discuss important cases in which the answer is affirmative. 2. The compact case and the simply connected case For compact G, we know from HW9 that G is closed in G. Let s push that a bit further before we explore the non-compact case. Proposition 2.1. If G is a compact connected Lie group and H 1, H 2 are normal closed connected Lie subgroups then (H 1, H 2 ) is a semisimple compact subgroup. In Corollary 2.6, we will see that if G is semisimple then the closedness hypothesis can be dropped: any normal connected Lie subgroup of G is automatically closed. Proof. If Z i H i is the maximal central torus in H i then it is normal in G since G-conjugation on H i is through automorphisms and clearly Z i is carried onto itself under any automorphism of H i. But a normal torus T in a connected compact Lie group is always central since its n-torsion T [n] subgroups are collectively dense and each is a finite normal subgroup of G (so central in G by discreteness). Thus, each Z i is central in G. By Exercise 4(ii) in HW9, H i = Z i H i with H i a semisimple compact connected Lie group. The central Z i in G is wiped out in commutators, so (H 1, H 2 ) = (H 1, H 2). Thus, we can replace H i with H i so that each H i is semisimple. Now H i G, so we can replace G with G to arrange that G is semisimple. If {G 1,..., G n } is the set of pairwise commuting simple factors of G in the sense of Theorem 1.1 in the handout Simple Factors then each H i is a generated by some of the G j s due to normality, so the isogeny G j G implies that (H 1, H 2 ) is generated by the G j s contained in H 1 and H 2. 1

2 2 Our main aim is to prove the following result, which addresses general connected Lie groups and highlights a special feature of the simply connected case. It rests on an important Theorem of Ado in the theory of Lie algebras that lies beyond the level of this course and will never be used in our study of compact Lie groups. Theorem 2.2. Suppose G is a connected Lie group, and H 1, H 2 are normal connected Lie subgroups. The subgroup (H 1, H 2 ) is a connected Lie subgroup with Lie algebra [h 1, h 2 ], and if G is simply connected then the H i and (H 1, H 2 ) are closed in G. In particular, G = G if and only if [g, g] = g. To prove this theorem, we first check that each h i is a Lie ideal in g (i.e., [g, h i ] h i ). More generally: Lemma 2.3. If n is a Lie subalgebra of g and N is the corresponding connected Lie subgroup of G then N is normal in G if and only if n is a Lie ideal in g. Proof. Normality of N is precisely the assertion that for g G, conjugation c g : G G carries N into itself. This is equivalent to Lie(c g ) = Ad G (g) carries n into itself for all g. In other words, N is normal if and only if Ad G : G GL(g) lands inside the closed subgroup of linear automorphisms preserving the subspace n. Since G is connected, it is equivalent that the analogue hold on Lie algebras. But Lie(Ad G ) = ad g, so the Lie ideal hypothesis on n is precisely this condition. Returning to our initial setup, since each h i is a Lie ideal in g, it follows from the Jacobi identity (check!) that the R-span [h 1, h 2 ] consisting of commutators [X 1, X 2 ] for X i h i is a Lie ideal in g. Thus, by the lemma, it corresponds to a normal connected Lie subgroup N in G. We want to prove that N = (H 1, H 2 ), with N closed if G is simply connected. Let G G be the universal cover (as in Exericse 3(iv) in HW9), and let H i be the connected Lie subgroup of G corresponding to the Lie ideal hi inside g = g. The map H i G factors through the connected Lie subgroup H i via a Lie algebra isomorphism since this can be checked on Lie algebras, so H i H i is a covering space. Remark 2.4. Beware that H i might not be the entire preimage of H i in G. Thus, although it follows that (H 1, H 2 ) is the image of ( H 1, H 2 ), this latter commutator subgroup of G might not be the full preimage of (H 1, H 2 ). In particular, closedness of ( H 1, H 2 ) will not imply closedness of (H 1, H 2 ) (and it really cannot, since we will exhibit counterexamples to such closedness later when G is not simply connected). Letting Ñ G be the normal connected Lie subgroup corresponding to the Lie ideal to [h 1, h 2 ] g, it likewise follows that Ñ G factors through a covering space map onto N, so if Ñ = ( H 1, H 2 ) then N = (H 1, H 2 ). Hence, it suffices to show that if G is simply connected then the H i and N are closed in G and N = (H 1, H 2 ). For the rest of the argument, we therefore may and do assume G is simply connected. Lemma 2.5. For simply connected G and any Lie ideal n in g, the corresponding normal connected Lie subgroup N in G is closed.

3 This is false without the simply connected hypothesis; consider a line with irrational angle in the Lie algebra of S 1 S 1 (contrasted with the analogue for its universal cover R 2!). Proof. There is an important Theorem of Ado (proved as the last result in Chapter I of Bourbaki s Lie Groups and Lie Algebras), according to which every finite-dimensional Lie algebra over a field F is a Lie subalgebra of gl n (F ); we only need the case when F has characteristic 0 (more specifically F = R). Thus, every finite-dimensional Lie algebra over R is a Lie subalgebra of gl n (R), and so occurs as a (not necessarily closed!) connected Lie subgroup of GL n (R). In particular, all finite-dimensional Lie algebras occur as Lie algebras of connected Lie groups! Hence, there is a connected Lie group Q such that Lie(Q) g/n. Since G is simply connected, so it has no nontrivial connected covering spaces (by HW9, Exercise 3(ii)!), the discussion near the end of 4 in the Frobenius theorem handout ensures that the Lie algebra map g g/n = Lie(Q) arises from a Lie group homomorphism ϕ : G Q. Consider ker ϕ. By Proposition 4.4 in the Frobenius Theorem handout, this is a closed Lie subgroup of G and Lie(ker ϕ) is identified with the kernel of the map Lie(ϕ) that is (by design) the quotient map g g/n. In other words, ker ϕ is a closed Lie subgroup of G whose Lie subalgebra of g is n, so the same also holds for its identity component (ker ϕ) 0. Thus, we have produced a closed connected subgroup of G whose Lie subalgebra coincides with a given Lie ideal n. It must therefore coincide with N by uniqueness, so N is closed. Corollary 2.6. If G is a semisimple compact connected Lie group and N is a normal connected Lie subgroup then N is closed. Proof. The case of simply connected G is already settled. In general, compactness ensures that the universal cover G G is a finite-degree covering space. The normal connected Lie subgroup Ñ G corresponding to the Lie ideal n g = g is therefore closed, yet Ñ G must factor through N via a covering space map, so N is the image of the compact Ñ and hence it is compact. Thus, N is closed. We have constructed a normal closed connected Lie subgroup N in G such that n = [h 1, h 2 ]. It remains to prove that N = (H 1, H 2 ). First we show that (H 1, H 2 ) N. Since n is a Lie ideal, we know by the preceding arguments that N is normal in G. Thus, the quotient map G G := G/N makes sense as a homomorphism of Lie groups, and Lie(G) = g/n =: g. Let h i g denote the image of h i, so it arises from a unique connected (perhaps not closed) Lie subgroup H i G. The map H i G factors through H i since this holds on Lie algebras, so to prove that (H 1, H 2 ) N it suffices to show that H 1 and H 2 commute inside G. By design, [h 1, h 2 ] = 0 inside g since [h 1, h 2 ] = n, so we will prove that in general a pair of connected Lie subgroups H 1 and H 2 inside a connected Lie group G commute with each other if their Lie algebras satisfy [h 1, h 2 ] = 0 inside g. For any h 1 H 1, we want to prove that c h1 on G carries H 2 into itself via the identity map. It suffices to prove that Ad G (h 1 ) on g is the identity on h 2. In other words, we claim that Ad G : H 1 GL(g) lands inside the subgroup of automorphisms that restrict to the identity on the subspace h 2. By connectedness of H 1, this holds if it does on Lie algebras. But Lie(Ad G ) = ad g, so we re reduced to checking that ad g on h 1 3

4 4 restricts to 0 on h 2, and that is precisely the vanishing hypothesis on Lie brackets between H 1 and H 2. This completes the proof that (H 1, H 2 ) N. To prove that the subgroup (H 1, H 2 ) exhausts the connected Lie group N, it suffices to prove that it contains an open neighborhood of the identity in N. By design, n = [h 1, h 2 ] is spanned by finitely many elements of the form [X i, Y i ] with X i h 1 and Y i h 2 (1 i n). Consider the C map f : R 2n G given by n n f(s 1, t 1,..., s n, t n ) = (exp G (s i X i ), exp G (t i Y i )) = (α Xi (s i ), α Yi (t i )). i=1 This map is valued inside (H 1, H 2 ) N and so by the Frobenius theorem (applied to the integral manifold N in G to a subbundle of the tangent bundle of G) it factors smoothly through N. Rather generally, for any X, Y g, consider the C map F : R 2 G defined by F (s, t) = (exp G (sx), exp G (ty )) = exp G (Ad G (α X (s))(ty )) exp G ( ty ). For fixed s 0, the parametric curve F (s 0, t) passes through e at t = 0 with velocity vector Ad G (α X (s 0 ))(Y ) Y. Hence, if s 0 0 then t F (s 0, t/s 0 ) sends 0 to e with velocity vector at t = 0 given by (1/s 0 )(Ad G (α X (s 0 ))(Y ) Y ). But Ad G (α X (s 0 )) = Ad G (exp G (s 0 X)) = e adg(s 0X) as linear automorphisms of g, and since ad g (s 0 X) = s 0 ad g (X), its exponential is 1+s 0 [X, ]+ O(s 2 0) (with implicit constant depending only on X), so i=1 Ad G (α X (s 0 ))(Y ) Y = s 0 [X, Y ] + O(s 2 0). Hence, F (s 0, t/s 0 ) has velocity [X, Y ] + O(s 0 ) at t = 0 (where O(s 0 ) has implied constant depending just on X and Y ). For fixed s i 0, we conclude that the map R n (H 1, H 2 ) N given by (t 1,..., t n ) f(s 1, t 1 /s 1,..., s n, t n /s n ) carries (0,..., 0) to e with derivative there carrying ti 0 to [X i, Y i ] + O(s i ) n with implied constant depending just on X i and Y i. The elements [X i, Y i ] n form a spanning set, so likewise for elements [X i, Y i ] + O(s i ) if we take the s i s sufficiently close to (but distinct from) 0. Hence, we have produced a map R n N valued in (H 1, H 2 ) carrying 0 to e with derivative that is surjective. Thus, by the submersion theorem this map is open image near e, which gives exactly that (H 1, H 2 ) contains an open neighborhood of e inside N, as desired. This completes the proof of Theorem Non-closed examples We give two examples of non-closed commutators. One is an example with (G, G) not closed in G, but in this example G is not built as a matrix group over R (i.e., a subgroup of some GL n (R) that is a zero locus of polynomials over R in the matrix entries). The second example will use the matrix group G = SL 2 (R) SL 2 (R) and provide closed connected Lie subgroups H 1 and H 2 for which (H 1, H 2 ) is not closed.

5 Such examples are best possible in the sense that (by basic results in the theory of linear algebraic groups over general fields) if G is a matrix group over R then relative to any chosen realization of G as a zero locus of polynomials in some GL n (R), if H 1, H 2 are any two subgroups Zariski-closed in G relative to such a matrix realization of G then (H 1, H 2 ) is closed in G, in fact closed of finite index in a possibly disconnected Zariski-closed subgroup of G (the Zariski topology may be too coarse to detect distinct connected components of a Lie group, as for R >0 R via the matrix realization as GL 1 (R)). In particular, for any such matrix group G, (G, G) is always closed in G. Example 3.1. Let H SL 2 (R) be the universal cover, so it fits into an exact sequence 1 Z ι H SL 2 (R) 1. Since h = sl 2 (R) is its own commutator subalgebra, we have (H, H) = H. Choose an injective homomorphism j : Z S 1, so j(n) = z n for z S 1 that is not a root of unity. Then Z is naturally a closed discrete subgroup of S 1 H via n (j( n), ι(n)), so we can form the quotient G = (S 1 H)/Z. There is an evident connected Lie subgroup inclusion H G. The commutator subgroup (G, G) is equal to (H, H) = H, and we claim that this is not closed in G. Suppose to the contrary that H were closed in G. By normality we could then form the Lie group quotient G/H that is a quotient of S 1 in which the image of z is trivial. But the kernel of f : S 1 G/H would have to be a closed subgroup, so containment of z would imply that it contains the closure of the subgroup generated by z, which is dense in S 1. In other words, ker f = S 1, which forces G/H = 1, contradicting that obviously H G. Example 3.2. The following example is an explicit version of a suggestion made by Tom Goodwillie on Math Overflow (as I discovered via Google search; try Commutator of closed subgroups to find it). Let G = SL 2 (R) SL 2 (R). Let X = ( ), Y = ( inside the space sl 2 (R) of traceless 2 2 matrices. Obviously α X (R) is the closed subgroup of diagonal elements in SL 2 (R) with positive entries (a copy of R >0), and α Y (R) is a conjugate of this since Y = gy g 1 for g = ( 1 1/ /2 ). But [X, Y ] = 2( 1 0 ) has pure imaginary eigenvalues, so its associated 1-parameter subgroup is a circle. Note that (X, 0) and (0, X) commute inside g, so they span of subspace of g that exponentiates to a closed subgroup isomorphic to R 2. In particular, (X, sx) exponentiates to a closed subgroup of G isomorphic to R for any s R. We choose s = 1 to define H 1. Likewise,any (Y, cy ) exponentiates to a closed subgroup of G isomorphic to R, and we choose c R to be irrational to define H 2. Letting Z = [X, Y ], we likewise have that (Z, 0) and (0, Z) span a subspace of g that exponentiates to a closed subgroup isomorphic to S 1 S 1. The commutator [h 1, h 2 ] is (Z, cz), so by the irrationality of c this exponentiates to a densely wrapped line L in the torus S 1 S 1. Hence, (H 1, H 2 ) = L is not closed in G. ) 5

Lemma 1.3. The element [X, X] is nonzero.

Math 210C. The remarkable SU(2) Let G be a non-commutative connected compact Lie group, and assume that its rank (i.e., dimension of maximal tori) is 1; equivalently, G is a compact connected Lie group

Math 145. Codimension

Math 145. Codimension 1. Main result and some interesting examples In class we have seen that the dimension theory of an affine variety (irreducible!) is linked to the structure of the function field in

Math 210C. The representation ring

Math 210C. The representation ring 1. Introduction Let G be a nontrivial connected compact Lie group that is semisimple and simply connected (e.g., SU(n) for n 2, Sp(n) for n 1, or Spin(n) for n 3). Let

Math 249B. Nilpotence of connected solvable groups

Math 249B. Nilpotence of connected solvable groups 1. Motivation and examples In abstract group theory, the descending central series {C i (G)} of a group G is defined recursively by C 0 (G) = G and C

Notation. For any Lie group G, we set G 0 to be the connected component of the identity.

Notation. For any Lie group G, we set G 0 to be the connected component of the identity. Problem 1 Prove that GL(n, R) is homotopic to O(n, R). (Hint: Gram-Schmidt Orthogonalization.) Here is a sequence

(dim Z j dim Z j 1 ) 1 j i

Math 210B. Codimension 1. Main result and some interesting examples Let k be a field, and A a domain finitely generated k-algebra. In class we have seen that the dimension theory of A is linked to the

Math 249B. Applications of Borel s theorem on Borel subgroups

Math 249B. Applications of Borel s theorem on Borel subgroups 1. Motivation In class we proved the important theorem of Borel that if G is a connected linear algebraic group over an algebraically closed

Math 210C. Size of fundamental group

Math 210C. Size of fundamental group 1. Introduction Let (V, Φ) be a nonzero root system. Let G be a connected compact Lie group that is semisimple (equivalently, Z G is finite, or G = G ; see Exercise

MATRIX LIE GROUPS AND LIE GROUPS

MATRIX LIE GROUPS AND LIE GROUPS Steven Sy December 7, 2005 I MATRIX LIE GROUPS Definition: A matrix Lie group is a closed subgroup of Thus if is any sequence of matrices in, and for some, then either

10. The subgroup subalgebra correspondence. Homogeneous spaces.

10. The subgroup subalgebra correspondence. Homogeneous spaces. 10.1. The concept of a Lie subgroup of a Lie group. We have seen that if G is a Lie group and H G a subgroup which is at the same time a

Math 210C. Weyl groups and character lattices

Math 210C. Weyl groups and character lattices 1. Introduction Let G be a connected compact Lie group, and S a torus in G (not necessarily maximal). In class we saw that the centralizer Z G (S) of S in

Math 676. A compactness theorem for the idele group. and by the product formula it lies in the kernel (A K )1 of the continuous idelic norm

Math 676. A compactness theorem for the idele group 1. Introduction Let K be a global field, so K is naturally a discrete subgroup of the idele group A K and by the product formula it lies in the kernel

INVERSE LIMITS AND PROFINITE GROUPS

INVERSE LIMITS AND PROFINITE GROUPS BRIAN OSSERMAN We discuss the inverse limit construction, and consider the special case of inverse limits of finite groups, which should best be considered as topological

ALGEBRAIC GROUPS J. WARNER

ALGEBRAIC GROUPS J. WARNER Let k be an algebraically closed field. varieties unless otherwise stated. 1. Definitions and Examples For simplicity we will work strictly with affine Definition 1.1. An algebraic

Math 249B. Geometric Bruhat decomposition

Math 249B. Geometric Bruhat decomposition 1. Introduction Let (G, T ) be a split connected reductive group over a field k, and Φ = Φ(G, T ). Fix a positive system of roots Φ Φ, and let B be the unique

Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III

Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III Lie algebras. Let K be again an algebraically closed field. For the moment let G be an arbitrary algebraic group

ELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS

ELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS J. WARNER SUMMARY OF A PAPER BY J. CARLSON, E. FRIEDLANDER, AND J. PEVTSOVA, AND FURTHER OBSERVATIONS 1. The Nullcone and Restricted Nullcone We will need

where Σ is a finite discrete Gal(K sep /K)-set unramified along U and F s is a finite Gal(k(s) sep /k(s))-subset

Classification of quasi-finite étale separated schemes As we saw in lecture, Zariski s Main Theorem provides a very visual picture of quasi-finite étale separated schemes X over a henselian local ring

FINITE GROUP THEORY: SOLUTIONS FALL MORNING 5. Stab G (l) =.

FINITE GROUP THEORY: SOLUTIONS TONY FENG These are hints/solutions/commentary on the problems. They are not a model for what to actually write on the quals. 1. 2010 FALL MORNING 5 (i) Note that G acts

Notes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop. Eric Sommers

Notes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop Eric Sommers 17 July 2009 2 Contents 1 Background 5 1.1 Linear algebra......................................... 5 1.1.1

LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori

LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI 1. Maximal Tori By a torus we mean a compact connected abelian Lie group, so a torus is a Lie group that is isomorphic to T n = R n /Z n. Definition 1.1.

ON NEARLY SEMIFREE CIRCLE ACTIONS

ON NEARLY SEMIFREE CIRCLE ACTIONS DUSA MCDUFF AND SUSAN TOLMAN Abstract. Recall that an effective circle action is semifree if the stabilizer subgroup of each point is connected. We show that if (M, ω)

Topological properties

CHAPTER 4 Topological properties 1. Connectedness Definitions and examples Basic properties Connected components Connected versus path connected, again 2. Compactness Definition and first examples Topological

Math 249B. Tits systems

Math 249B. Tits systems 1. Introduction Let (G, T ) be a split connected reductive group over a field k, and Φ = Φ(G, T ). Fix a positive system of roots Φ + Φ, and let B be the unique Borel k-subgroup

V (v i + W i ) (v i + W i ) is path-connected and hence is connected.

Math 396. Connectedness of hyperplane complements Note that the complement of a point in R is disconnected and the complement of a (translated) line in R 2 is disconnected. Quite generally, we claim that

10. Smooth Varieties. 82 Andreas Gathmann

82 Andreas Gathmann 10. Smooth Varieties Let a be a point on a variety X. In the last chapter we have introduced the tangent cone C a X as a way to study X locally around a (see Construction 9.20). It

arxiv:math/ v1 [math.rt] 1 Jul 1996

SUPERRIGID SUBGROUPS OF SOLVABLE LIE GROUPS DAVE WITTE arxiv:math/9607221v1 [math.rt] 1 Jul 1996 Abstract. Let Γ be a discrete subgroup of a simply connected, solvable Lie group G, such that Ad G Γ has

descends to an F -torus S T, and S M since S F ) 0 red T F

Math 249B. Basics of reductivity and semisimplicity In the previous course, we have proved the important fact that over any field k, a non-solvable connected reductive group containing a 1-dimensional

Math 396. Bijectivity vs. isomorphism

Math 396. Bijectivity vs. isomorphism 1. Motivation Let f : X Y be a C p map between two C p -premanifolds with corners, with 1 p. Assuming f is bijective, we would like a criterion to tell us that f 1

CONSEQUENCES OF THE SYLOW THEOREMS

CONSEQUENCES OF THE SYLOW THEOREMS KEITH CONRAD For a group theorist, Sylow s Theorem is such a basic tool, and so fundamental, that it is used almost without thinking, like breathing. Geoff Robinson 1.

Math 396. Subbundles and quotient bundles

Math 396. Subbundles and quotient bundles 1. Motivation We want to study the bundle analogues of subspaces and quotients of finite-dimensional vector spaces. Let us begin with some motivating examples.

is maximal in G k /R u (G k

Algebraic Groups I. Grothendieck s theorem on tori (based on notes by S. Lichtenstein) 1. Introduction In the early days of the theory of linear algebraic groups, the ground field was assumed to be algebraically

B 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X.

Math 6342/7350: Topology and Geometry Sample Preliminary Exam Questions 1. For each of the following topological spaces X i, determine whether X i and X i X i are homeomorphic. (a) X 1 = [0, 1] (b) X 2

Commutative Banach algebras 79

8. Commutative Banach algebras In this chapter, we analyze commutative Banach algebras in greater detail. So we always assume that xy = yx for all x, y A here. Definition 8.1. Let A be a (commutative)

Definitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations

Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of

BASIC GROUP THEORY : G G G,

BASIC GROUP THEORY 18.904 1. Definitions Definition 1.1. A group (G, ) is a set G with a binary operation : G G G, and a unit e G, possessing the following properties. (1) Unital: for g G, we have g e

Math 210B. Profinite group cohomology

Math 210B. Profinite group cohomology 1. Motivation Let {Γ i } be an inverse system of finite groups with surjective transition maps, and define Γ = Γ i equipped with its inverse it topology (i.e., the

REPRESENTATION THEORY WEEK 7

REPRESENTATION THEORY WEEK 7 1. Characters of L k and S n A character of an irreducible representation of L k is a polynomial function constant on every conjugacy class. Since the set of diagonalizable

Math 248B. Applications of base change for coherent cohomology

Math 248B. Applications of base change for coherent cohomology 1. Motivation Recall the following fundamental general theorem, the so-called cohomology and base change theorem: Theorem 1.1 (Grothendieck).

Some algebraic properties of. compact topological groups

Some algebraic properties of compact topological groups 1 Compact topological groups: examples connected: S 1, circle group. SO(3, R), rotation group not connected: Every finite group, with the discrete

Math 121 Homework 4: Notes on Selected Problems

Math 121 Homework 4: Notes on Selected Problems 11.2.9. If W is a subspace of the vector space V stable under the linear transformation (i.e., (W ) W ), show that induces linear transformations W on W

Math 594. Solutions 5

Math 594. Solutions 5 Book problems 6.1: 7. Prove that subgroups and quotient groups of nilpotent groups are nilpotent (your proof should work for infinite groups). Give an example of a group G which possesses

20 Group Homomorphisms

20 Group Homomorphisms In Example 1810(d), we have observed that the groups S 4 /V 4 and S 3 have almost the same multiplication table They have the same structure In this paragraph, we study groups with

c ij x i x j c ij x i y j

Math 48A. Class groups for imaginary quadratic fields In general it is a very difficult problem to determine the class number of a number field, let alone the structure of its class group. However, in

Math 396. Quotient spaces

Math 396. Quotient spaces. Definition Let F be a field, V a vector space over F and W V a subspace of V. For v, v V, we say that v v mod W if and only if v v W. One can readily verify that with this definition

1 Smooth manifolds and Lie groups

An undergraduate approach to Lie theory Slide 1 Andrew Baker, Glasgow Glasgow, 12/11/1999 1 Smooth manifolds and Lie groups A continuous g : V 1 V 2 with V k R m k open is called smooth if it is infinitely

A MARSDEN WEINSTEIN REDUCTION THEOREM FOR PRESYMPLECTIC MANIFOLDS

A MARSDEN WEINSTEIN REDUCTION THEOREM FOR PRESYMPLECTIC MANIFOLDS FRANCESCO BOTTACIN Abstract. In this paper we prove an analogue of the Marsden Weinstein reduction theorem for presymplectic actions of

These notes are incomplete they will be updated regularly.

These notes are incomplete they will be updated regularly. LIE GROUPS, LIE ALGEBRAS, AND REPRESENTATIONS SPRING SEMESTER 2008 RICHARD A. WENTWORTH Contents 1. Lie groups and Lie algebras 2 1.1. Definition

TEST CODE: PMB SYLLABUS

TEST CODE: PMB SYLLABUS Convergence and divergence of sequence and series; Cauchy sequence and completeness; Bolzano-Weierstrass theorem; continuity, uniform continuity, differentiability; directional

Solutions to Problem Set 1

Solutions to Problem Set 1 18.904 Spring 2011 Problem 1 Statement. Let n 1 be an integer. Let CP n denote the set of all lines in C n+1 passing through the origin. There is a natural map π : C n+1 \ {0}

MAT 445/ INTRODUCTION TO REPRESENTATION THEORY

MAT 445/1196 - INTRODUCTION TO REPRESENTATION THEORY CHAPTER 1 Representation Theory of Groups - Algebraic Foundations 1.1 Basic definitions, Schur s Lemma 1.2 Tensor products 1.3 Unitary representations

LIE ALGEBRAS: LECTURE 3 6 April 2010

LIE ALGEBRAS: LECTURE 3 6 April 2010 CRYSTAL HOYT 1. Simple 3-dimensional Lie algebras Suppose L is a simple 3-dimensional Lie algebra over k, where k is algebraically closed. Then [L, L] = L, since otherwise

Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35

Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35 1. Let R be a commutative ring with 1 0. (a) Prove that the nilradical of R is equal to the intersection of the prime

8. COMPACT LIE GROUPS AND REPRESENTATIONS

8. COMPACT LIE GROUPS AND REPRESENTATIONS. Abelian Lie groups.. Theorem. Assume G is a Lie group and g its Lie algebra. Then G 0 is abelian iff g is abelian... Proof.. Let 0 U g and e V G small (symmetric)

Lecture 11 The Radical and Semisimple Lie Algebras

18.745 Introduction to Lie Algebras October 14, 2010 Lecture 11 The Radical and Semisimple Lie Algebras Prof. Victor Kac Scribe: Scott Kovach and Qinxuan Pan Exercise 11.1. Let g be a Lie algebra. Then

Homework 3 MTH 869 Algebraic Topology

Homework 3 MTH 869 Algebraic Topology Joshua Ruiter February 12, 2018 Proposition 0.1 (Exercise 1.1.10). Let (X, x 0 ) and (Y, y 0 ) be pointed, path-connected spaces. Let f : I X y 0 } and g : I x 0 }

Notes 10: Consequences of Eli Cartan s theorem.

Notes 10: Consequences of Eli Cartan s theorem. Version 0.00 with misprints, The are a few obvious, but important consequences of the theorem of Eli Cartan on the maximal tori. The first one is the observation

p,q H (X), H (Y ) ), where the index p has the same meaning as the

There are two Eilenberg-Moore spectral sequences that we shall consider, one for homology and the other for cohomology. In contrast with the situation for the Serre spectral sequence, for the Eilenberg-Moore

Lie Algebras. Shlomo Sternberg

Lie Algebras Shlomo Sternberg March 8, 2004 2 Chapter 5 Conjugacy of Cartan subalgebras It is a standard theorem in linear algebra that any unitary matrix can be diagonalized (by conjugation by unitary

(2) For x π(p ) the fiber π 1 (x) is a closed C p subpremanifold in P, and for all ξ π 1 (x) the kernel of the surjection dπ(ξ) : T ξ (P ) T x (X) is

Math 396. Submersions and transverse intersections Fix 1 p, and let f : X X be a C p mapping between C p premanifolds (no corners or boundary!). In a separate handout there is proved an important local

COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY

COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY BRIAN OSSERMAN Classical algebraic geometers studied algebraic varieties over the complex numbers. In this setting, they didn t have to worry about the Zariski

Since G is a compact Lie group, we can apply Schur orthogonality to see that G χ π (g) 2 dg =

Problem 1 Show that if π is an irreducible representation of a compact lie group G then π is also irreducible. Give an example of a G and π such that π = π, and another for which π π. Is this true for

ON RIGIDITY OF ALGEBRAIC DYNAMICAL SYSTEMS

ON RIGIDITY OF ALGEBRAIC DYNAMICAL SYSTEMS ALEX CLARK AND ROBBERT FOKKINK Abstract. We study topological rigidity of algebraic dynamical systems. In the first part of this paper we give an algebraic condition

Math 215B: Solutions 1

Math 15B: Solutions 1 Due Thursday, January 18, 018 (1) Let π : X X be a covering space. Let Φ be a smooth structure on X. Prove that there is a smooth structure Φ on X so that π : ( X, Φ) (X, Φ) is an

LECTURE 4: REPRESENTATION THEORY OF SL 2 (F) AND sl 2 (F)

LECTURE 4: REPRESENTATION THEORY OF SL 2 (F) AND sl 2 (F) IVAN LOSEV In this lecture we will discuss the representation theory of the algebraic group SL 2 (F) and of the Lie algebra sl 2 (F), where F is

REPRESENTATIONS OF U(N) CLASSIFICATION BY HIGHEST WEIGHTS NOTES FOR MATH 261, FALL 2001

9 REPRESENTATIONS OF U(N) CLASSIFICATION BY HIGHEST WEIGHTS NOTES FOR MATH 261, FALL 21 ALLEN KNUTSON 1 WEIGHT DIAGRAMS OF -REPRESENTATIONS Let be an -dimensional torus, ie a group isomorphic to The we

fy (X(g)) Y (f)x(g) gy (X(f)) Y (g)x(f)) = fx(y (g)) + gx(y (f)) fy (X(g)) gy (X(f))

1. Basic algebra of vector fields Let V be a finite dimensional vector space over R. Recall that V = {L : V R} is defined to be the set of all linear maps to R. V is isomorphic to V, but there is no canonical

THE EULER CHARACTERISTIC OF A LIE GROUP

THE EULER CHARACTERISTIC OF A LIE GROUP JAY TAYLOR 1 Examples of Lie Groups The following is adapted from [2] We begin with the basic definition and some core examples Definition A Lie group is a smooth

1 Classifying Unitary Representations: A 1

Lie Theory Through Examples John Baez Lecture 4 1 Classifying Unitary Representations: A 1 Last time we saw how to classify unitary representations of a torus T using its weight lattice L : the dual of

L(C G (x) 0 ) c g (x). Proof. Recall C G (x) = {g G xgx 1 = g} and c g (x) = {X g Ad xx = X}. In general, it is obvious that

ALGEBRAIC GROUPS 61 5. Root systems and semisimple Lie algebras 5.1. Characteristic 0 theory. Assume in this subsection that chark = 0. Let me recall a couple of definitions made earlier: G is called reductive

Master Algèbre géométrie et théorie des nombres Final exam of differential geometry Lecture notes allowed

Université de Bordeaux U.F. Mathématiques et Interactions Master Algèbre géométrie et théorie des nombres Final exam of differential geometry 2018-2019 Lecture notes allowed Exercise 1 We call H (like

Margulis Superrigidity I & II

Margulis Superrigidity I & II Alastair Litterick 1,2 and Yuri Santos Rego 1 Universität Bielefeld 1 and Ruhr-Universität Bochum 2 Block seminar on arithmetic groups and rigidity Universität Bielefeld 22nd

9. The Lie group Lie algebra correspondence

9. The Lie group Lie algebra correspondence 9.1. The functor Lie. The fundamental theorems of Lie concern the correspondence G Lie(G). The work of Lie was essentially local and led to the following fundamental

SYMPLECTIC GEOMETRY: LECTURE 5

SYMPLECTIC GEOMETRY: LECTURE 5 LIAT KESSLER Let (M, ω) be a connected compact symplectic manifold, T a torus, T M M a Hamiltonian action of T on M, and Φ: M t the assoaciated moment map. Theorem 0.1 (The

ALGEBRAICALLY TRIVIAL, BUT TOPOLOGICALLY NON-TRIVIAL MAP. Contents 1. Introduction 1

ALGEBRAICALLY TRIVIAL, BUT TOPOLOGICALLY NON-TRIVIAL MAP HONG GYUN KIM Abstract. I studied the construction of an algebraically trivial, but topologically non-trivial map by Hopf map p : S 3 S 2 and a

LECTURE 15-16: PROPER ACTIONS AND ORBIT SPACES

LECTURE 15-16: PROPER ACTIONS AND ORBIT SPACES 1. Proper actions Suppose G acts on M smoothly, and m M. Then the orbit of G through m is G m = {g m g G}. If m, m lies in the same orbit, i.e. m = g m for

Math 121 Homework 5: Notes on Selected Problems

Math 121 Homework 5: Notes on Selected Problems 12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x 1,..., x n is any maximal set of linearly independent elements

Thus, the integral closure A i of A in F i is a finitely generated (and torsion-free) A-module. It is not a priori clear if the A i s are locally

Math 248A. Discriminants and étale algebras Let A be a noetherian domain with fraction field F. Let B be an A-algebra that is finitely generated and torsion-free as an A-module with B also locally free

QUATERNIONS AND ROTATIONS

QUATERNIONS AND ROTATIONS SVANTE JANSON 1. Introduction The purpose of this note is to show some well-known relations between quaternions and the Lie groups SO(3) and SO(4) (rotations in R 3 and R 4 )

3. The Sheaf of Regular Functions

24 Andreas Gathmann 3. The Sheaf of Regular Functions After having defined affine varieties, our next goal must be to say what kind of maps between them we want to consider as morphisms, i. e. as nice

SPLITTING FIELDS OF CHARACTERISTIC POLYNOMIALS IN ALGEBRAIC GROUPS

SPLITTING FIELDS OF CHARACTERISTIC POLYNOMIALS IN ALGEBRAIC GROUPS E. KOWALSKI In [K1] and earlier in [K2], questions of the following type are considered: suppose a family (g i ) i of matrices in some

Bare-bones outline of eigenvalue theory and the Jordan canonical form

Bare-bones outline of eigenvalue theory and the Jordan canonical form April 3, 2007 N.B.: You should also consult the text/class notes for worked examples. Let F be a field, let V be a finite-dimensional

Math 210B. Artin Rees and completions

Math 210B. Artin Rees and completions 1. Definitions and an example Let A be a ring, I an ideal, and M an A-module. In class we defined the I-adic completion of M to be M = lim M/I n M. We will soon show

Two subgroups and semi-direct products

Two subgroups and semi-direct products 1 First remarks Throughout, we shall keep the following notation: G is a group, written multiplicatively, and H and K are two subgroups of G. We define the subset

TCC Homological Algebra: Assignment #3 (Solutions)

TCC Homological Algebra: Assignment #3 (Solutions) David Loeffler, d.a.loeffler@warwick.ac.uk 30th November 2016 This is the third of 4 problem sheets. Solutions should be submitted to me (via any appropriate

NONSINGULAR CURVES BRIAN OSSERMAN

NONSINGULAR CURVES BRIAN OSSERMAN The primary goal of this note is to prove that every abstract nonsingular curve can be realized as an open subset of a (unique) nonsingular projective curve. Note that

Math 396. From integral curves to integral manifolds 1. Integral manifolds for trivial line bundles Let M be a C manifold (without corners) and let E

Math 396. From integral curves to integral manifolds 1. Integral manifolds for trivial line bundles Let M be a C manifold (without corners) and let E T M be a subbundle of the tangent bundle. In class

Notes 2 for MAT4270 Connected components and universal covers π 0 and π 1.

Notes 2 for MAT4270 Connected components and universal covers π 0 and π 1. Version 0.00 with misprints, Connected components Recall thaty if X is a topological space X is said to be connected if is not

9. Finite fields. 1. Uniqueness

9. Finite fields 9.1 Uniqueness 9.2 Frobenius automorphisms 9.3 Counting irreducibles 1. Uniqueness Among other things, the following result justifies speaking of the field with p n elements (for prime

Boolean Algebras, Boolean Rings and Stone s Representation Theorem

Boolean Algebras, Boolean Rings and Stone s Representation Theorem Hongtaek Jung December 27, 2017 Abstract This is a part of a supplementary note for a Logic and Set Theory course. The main goal is to

INVARIANT PROBABILITIES ON PROJECTIVE SPACES. 1. Introduction

INVARIANT PROBABILITIES ON PROJECTIVE SPACES YVES DE CORNULIER Abstract. Let K be a local field. We classify the linear groups G GL(V ) that preserve an probability on the Borel subsets of the projective

Rings and groups. Ya. Sysak

Rings and groups. Ya. Sysak 1 Noetherian rings Let R be a ring. A (right) R -module M is called noetherian if it satisfies the maximum condition for its submodules. In other words, if M 1... M i M i+1...

Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra

Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................

MATH 221 NOTES BRENT HO. Date: January 3, 2009.

MATH 22 NOTES BRENT HO Date: January 3, 2009. 0 Table of Contents. Localizations......................................................................... 2 2. Zariski Topology......................................................................

A GLIMPSE OF ALGEBRAIC K-THEORY: Eric M. Friedlander

A GLIMPSE OF ALGEBRAIC K-THEORY: Eric M. Friedlander During the first three days of September, 1997, I had the privilege of giving a series of five lectures at the beginning of the School on Algebraic

algebraic groups. 1 See [1, 4.2] for the definition and properties of the invariant differential forms and the modulus character for

LECTURE 2: COMPACTNESS AND VOLUME FOR ADELIC COSET SPACES LECTURE BY BRIAN CONRAD STANFORD NUMBER THEORY LEARNING SEMINAR OCTOBER 11, 2017 NOTES BY DAN DORE Let k be a global field with adele ring A, G

Chapter 1. Smooth Manifolds

Chapter 1. Smooth Manifolds Theorem 1. [Exercise 1.18] Let M be a topological manifold. Then any two smooth atlases for M determine the same smooth structure if and only if their union is a smooth atlas.

Problems in Linear Algebra and Representation Theory

Problems in Linear Algebra and Representation Theory (Most of these were provided by Victor Ginzburg) The problems appearing below have varying level of difficulty. They are not listed in any specific