DOCSLIB.ORG

The Modular Group and the Fundamental Domain Seminar on Modular Forms Spring 2019

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Modular Group and the Fundamental Domain Seminar on Modular Forms Spring 2019 The modular group and the fundamental domain Seminar on Modular Forms Spring 2019 Johannes Hruza and Manuel Trachsler March 13, 2019 1 The Group SL2(Z) and the fundamental do- main Definition 1. For a commutative Ring R we define GL2(R) as the following set: a b GL (R) := A = for which det(A) = ad − bc 2 R∗ : (1) 2 c d We define SL2(R) to be the set of all B 2 GL2(R) for which det(B) = 1. Lemma 1. SL2(R) is a subgroup of GL2(R). Proof. Recall that the kernel of a group homomorphism is a subgroup. Observe ∗ that det is a group homomorphism det : GL2(R) ! R and thus SL2(R) is its kernel by definition. ¯ Let R = R. Then we can define an action of SL2(R) on C ( = C [ f1g ) by az + b a a b A:z := and A:1 := ;A = 2 SL (R); z 2 : (2) cz + d c c d 2 C Definition 2. The upper half-plane of C is given by H := fz 2 C j Im(z) > 0g. Restricting this action to H gives us another well defined action ":" : SL2(R)× H 7! H called the fractional linear transformation. Indeed, for any z 2 H the imaginary part of A:z is positive: az + b (az + b)(cz¯ + d) Im(z) Im(A:z) = Im = Im = > 0: (3) cz + d jcz + dj2 jcz + dj2 a b Lemma 2. For A = c d 2 SL2(R) the map µA : H ! H defined by z 7! A:z is the identity if and only if A = ±I. az+b 2 Proof. Assume z = cz+d for all z 2 H which reduces to cz + (d − a)z + b = 0. a b Having assumed A = c d is in SL2(R), it follows that the only solutions satisfying ad − bc = 1 are b = c = 0 and a = d so a = d = ±1 and hence ±I are the only elements of SL2(R) that act trivially on the upper half plane. 1 Definition 3. The full modular group is given by: Γ := SL2(Z) ≤ SL2(R): (4) In order to make the group act faithfully on H (recall that this means that there are no group elements A 2 ΓnfIdg such that A:z = z; 8z 2 H). We define the modular group Γ¯ := Γ={±Ig. The fact that Γ is indeed a subgroup of SL2(R) can be verified through check- a b −1 1 d −b ing that the inverse of an element in Γ lies in Γ too: c d = ad−bc −c a 1 and ad−bc = 1 2 Z by assumption. Also since Γ is a group it follows that Γ¯ = Γ={± Ig is a group, as {±Ig is normal in Γ. Definition 4. Let N > 0. Then we define the principle subgroup of level N by ( ) a b Γ(N) := 2 SL ( )ja ≡ d ≡ 1(mod N); b ≡ c ≡ 0(mod N) : (5) c d 2 Z First we show some basic properties of these subgroups: Lemma 3. Let N > 0. Then Γ(N) is a normal subgroup of Γ. Proof. We claim that Γ(N) is the kernel of a group homomorhism and hence normal. As a candidate we choose as follows: : SL2(Z) ! SL2(Z=NZ) (6) a b a (mod N) b (mod N) 7! : (7) c d c (mod N) d (mod N) This is indeed a group homomorphism, as it is one in every component. Its kernel is then exactly Γ(N). We are done. Remark: For all N > 2, the kernel Γ(¯ N) of the homomorphism : SL2(Z) ! SL2(Z=NZ) defined as in Lemma 3 is in fact equal to Γ(N). In- deed, for N > 2 we have −1 6≡ 1 mod(N) and thus −I 62 Γ(N). Definition 5. Let C subgroup of Γ. We call C a congruence subgroup of level N if C contains Γ(N). Similarly we may call a subgroup C0 of Γ¯ a congruence subgroup of level N if it contains Γ(¯ N). The most important examples of congruence subgroups, Γ0(N) and Γ1(N) will be defined here but are of little relevance for this lecture: ( ) a b Γ (N) := 2 Γjc ≡ 0 (mod N) (8) 0 c d ( ) a b Γ (N) := 2 Γ (N)ja ≡ 1 (mod N) : (9) 1 c d 0 The fact that these are indeed subgroups with Γ(N) ⊂ Γ1(N) ⊂ Γ0(N) ⊂ Γ will not be proven here. We now continue with the most important definition of the lecture, the fundamental domain. 2 Definition 6. Let G be a group acting on a topological space X. A fundamental domain F for the action of G is a (closed) subset, such that exactly one point of each orbit is contained in (the interior of) F . Applying this to our situation we call a closed subset F ⊂ H a fundamental domain for a subgroup G of Γ if every z 2 HnF is G-equivalent to one point in ◦ ◦ the interior of F (denoted F ) and no two points z1; z2 2 F are G-equivalent. We will prove and describe the fundamental domain of the whole group Γ itself before turning to subgroups. Theorem 1. A fundamental domain of Γ under fractional linear transforma- tions (2) is given by 1 1 F := fz 2 j − ≤ Re(z) ≤ and jzj ≥ 1g: H 2 2 Figure 1: Fundamental domain of Γ and approximating γ = T ◦ S ◦ T −1 such that γ:x 2 F . Proof. The idea is to fix any point z 2 H and then to find a combination of translations and inversions until z is mapped to F . This process is demonstrated for some x in Figure 1. Fix z 2 H. Define Γ0 to be the subgroup generated by T and S, where T and S are: 1 1 T := so that T:z = z + 1; (10) 0 1 3 0 −1 1 S := so that S:z = − : (11) 1 0 z a b 0 −2 Now let γ = c d 2 Γ . Then by (3) we have Im(γ:z) = Im(z)jcz+dj and thus Im(γ:z) > 0. By choosing the factors c and d so that jcz + dj is as small as possible but not zero. This is possible as cz +d must be on the lattice generated by 1 and z. Hence jcz + dj must be bigger or equal than one of the following: fj ± z + 0j; j ± z ± 1j; j0 + ±1jg by construction. But this set is finite hence has a minimum. So we are able to find a γ such that Im(γ:z) is maximal. In a next j 0 1 1 step find a suitable j 2 N [ f0g so that (T γ):z 2 fz 2 Hj − 2 ≤ Re(z) ≤ 2 g. Define T jγ =: γ0 and claim that γ0:z lies in F . If this was not the case, so jγ0:zj < 1, then again by (3) we would have Im((Sγ0):z) = Im(γ0:z)jγ0:zj−2 > Im(γ0:z) contradicting the maximality re- quirement we set above. Hence 9γ 2 Γ0 such that γ:z 2 F; 8z 2 H. We now have to prove that no two points in the interior of F share the same Orbit. For this assume z1; z2 2 F and wlog that Im(z2) ≥ Im(z1) and lastly that there exists an A 2 Γ so that z2 = A:z1. Having assumed that it follows that −2 Im(z2) = Im(A:z1) = Im(z1)jcz1 + dj ≥ Im(z1): by (3) and hence jcz + dj2 ≤ 1. As all entries in A must lie in and p 1 Z 3 Im(z1) ≥ 2 , it follows that the absolute value of c must be less than 2 oth- 1 erwise 1 ≤ Im(cz1) = Im(cz1 + d) ≤ jcz1 + dj. Moreover, we have jdj − 2 ≤ jd + Re(z)cj ≤ jcz + dj ≤ 1. This slims down the possibilities for A and we can deal with them case by case. Case 1: c = 0, d = ±1. By enforcing the ad − bc = 1 constraint we see that j 1 j then A or −A must be equal to T = 0 1 . But for A:z1 to lie in F again only j 2 {−1; 0; 1g come in question. For j = 0, we have z1 = z2. For j = ±1 recall the definition of T and we see that z2 and z1 must lie on the boundary lines 1 Re(z) = ± 2 of F and hence not in the interior of F . Case 2: c = ±1, d = 0. Then z1 lies on the unit circle, as we have jcz1j ≤ 1 a −1 a and jz1j ≥ 1. To satisfy ad−bc = 1, A has to be of the form A = ± 1 0 = T S a for some a 2 Z. For jaj > 2 we see that T S:z1 2= F , thus a 2 {−1; 0; 1g. 1 For a = 0 we get A = ±S and hence that z1; z2 lie in S \ F and are symmetric with respect to i as jz j = 1. R 1 p 1 −3 For a = 1 we have A = ±TS, but then z1 = + . The case a = −1 is p 2 2 1 −3 similar: We get A = ±ST and z1 = − 2 + 2 . p 1 −3 Case 3: Suppose c = d = ±1. Since jcz+dj ≤ 1, we see that z1 = − 2 + 2 , as for any other z1 2 F we find that j ± z1 ± 1j > 1.
Load more

Recommended publications
  • Math 5111 (Algebra 1) Lecture #14 of 24 ∼ October 19Th, 2020
    Math 5111 (Algebra 1) Lecture #14 of 24 ∼ October 19th, 2020 Group Isomorphism Theorems + Group Actions The Isomorphism Theorems for Groups Group Actions Polynomial Invariants and An This material represents x3.2.3-3.3.2 from the course notes. Quotients and Homomorphisms, I Like with rings, we also have various natural connections between normal subgroups and group homomorphisms. To begin, observe that if ' : G ! H is a group homomorphism, then ker ' is a normal subgroup of G. In fact, I proved this fact earlier when I introduced the kernel, but let me remark again: if g 2 ker ', then for any a 2 G, then '(aga−1) = '(a)'(g)'(a−1) = '(a)'(a−1) = e. Thus, aga−1 2 ker ' as well, and so by our equivalent properties of normality, this means ker ' is a normal subgroup. Thus, we can use homomorphisms to construct new normal subgroups. Quotients and Homomorphisms, II Equally importantly, we can also do the reverse: we can use normal subgroups to construct homomorphisms. The key observation in this direction is that the map ' : G ! G=N associating a group element to its residue class / left coset (i.e., with '(a) = a) is a ring homomorphism. Indeed, the homomorphism property is precisely what we arranged for the left cosets of N to satisfy: '(a · b) = a · b = a · b = '(a) · '(b). Furthermore, the kernel of this map ' is, by definition, the set of elements in G with '(g) = e, which is to say, the set of elements g 2 N. Thus, kernels of homomorphisms and normal subgroups are precisely the same things.
    [Show full text]
  • The Fundamental Groupoid of the Quotient of a Hausdorff
    The fundamental groupoid of the quotient of a Hausdorff space by a discontinuous action of a discrete group is the orbit groupoid of the induced action Ronald Brown,∗ Philip J. Higgins,† Mathematics Division, Department of Mathematical Sciences, School of Informatics, Science Laboratories, University of Wales, Bangor South Rd., Gwynedd LL57 1UT, U.K. Durham, DH1 3LE, U.K. October 23, 2018 University of Wales, Bangor, Maths Preprint 02.25 Abstract The main result is that the fundamental groupoidof the orbit space of a discontinuousaction of a discrete groupon a Hausdorffspace which admits a universal coveris the orbit groupoid of the fundamental groupoid of the space. We also describe work of Higgins and of Taylor which makes this result usable for calculations. As an example, we compute the fundamental group of the symmetric square of a space. The main result, which is related to work of Armstrong, is due to Brown and Higgins in 1985 and was published in sections 9 and 10 of Chapter 9 of the first author’s book on Topology [3]. This is a somewhat edited, and in one point (on normal closures) corrected, version of those sections. Since the book is out of print, and the result seems not well known, we now advertise it here. It is hoped that this account will also allow wider views of these results, for example in topos theory and descent theory. Because of its provenance, this should be read as a graduate text rather than an article. The Exercises should be regarded as further propositions for which we leave the proofs to the reader.
    [Show full text]
  • GROUP ACTIONS 1. Introduction the Groups Sn, An, and (For N ≥ 3)
    GROUP ACTIONS KEITH CONRAD 1. Introduction The groups Sn, An, and (for n ≥ 3) Dn behave, by their definitions, as permutations on certain sets. The groups Sn and An both permute the set f1; 2; : : : ; ng and Dn can be considered as a group of permutations of a regular n-gon, or even just of its n vertices, since rigid motions of the vertices determine where the rest of the n-gon goes. If we label the vertices of the n-gon in a definite manner by the numbers from 1 to n then we can view Dn as a subgroup of Sn. For instance, the labeling of the square below lets us regard the 90 degree counterclockwise rotation r in D4 as (1234) and the reflection s across the horizontal line bisecting the square as (24). The rest of the elements of D4, as permutations of the vertices, are in the table below the square. 2 3 1 4 1 r r2 r3 s rs r2s r3s (1) (1234) (13)(24) (1432) (24) (12)(34) (13) (14)(23) If we label the vertices in a different way (e.g., swap the labels 1 and 2), we turn the elements of D4 into a different subgroup of S4. More abstractly, if we are given a set X (not necessarily the set of vertices of a square), then the set Sym(X) of all permutations of X is a group under composition, and the subgroup Alt(X) of even permutations of X is a group under composition. If we list the elements of X in a definite order, say as X = fx1; : : : ; xng, then we can think about Sym(X) as Sn and Alt(X) as An, but a listing in a different order leads to different identifications 1 of Sym(X) with Sn and Alt(X) with An.
    [Show full text]
  • An Introduction to Orbifolds
    An introduction to orbifolds Joan Porti UAB Subdivide and Tile: Triangulating spaces for understanding the world Lorentz Center November 2009 An introduction to orbifolds – p.1/20 Motivation • Γ < Isom(Rn) or Hn discrete and acts properly discontinuously (e.g. a group of symmetries of a tessellation). • If Γ has no fixed points ⇒ Γ\Rn is a manifold. • If Γ has fixed points ⇒ Γ\Rn is an orbifold. An introduction to orbifolds – p.2/20 Motivation • Γ < Isom(Rn) or Hn discrete and acts properly discontinuously (e.g. a group of symmetries of a tessellation). • If Γ has no fixed points ⇒ Γ\Rn is a manifold. • If Γ has fixed points ⇒ Γ\Rn is an orbifold. ··· (there are other notions of orbifold in algebraic geometry, string theory or using grupoids) An introduction to orbifolds – p.2/20 Examples: tessellations of Euclidean plane Γ= h(x, y) → (x + 1, y), (x, y) → (x, y + 1)i =∼ Z2 Γ\R2 =∼ T 2 = S1 × S1 An introduction to orbifolds – p.3/20 Examples: tessellations of Euclidean plane Rotations of angle π around red points (order 2) An introduction to orbifolds – p.3/20 Examples: tessellations of Euclidean plane Rotations of angle π around red points (order 2) 2 2 An introduction to orbifolds – p.3/20 Examples: tessellations of Euclidean plane Rotations of angle π around red points (order 2) 2 2 2 2 2 2 2 2 2 2 2 2 An introduction to orbifolds – p.3/20 Example: tessellations of hyperbolic plane Rotations of angle π, π/2 and π/3 around vertices (order 2, 4, and 6) An introduction to orbifolds – p.4/20 Example: tessellations of hyperbolic plane Rotations of angle π, π/2 and π/3 around vertices (order 2, 4, and 6) 2 4 2 6 An introduction to orbifolds – p.4/20 Definition Informal Definition • An orbifold O is a metrizable topological space equipped with an atlas modelled on Rn/Γ, Γ < O(n) finite, with some compatibility condition.
    [Show full text]
  • Lie Group and Geometry on the Lie Group SL2(R)
    INDIAN INSTITUTE OF TECHNOLOGY KHARAGPUR Lie group and Geometry on the Lie Group SL2(R) PROJECT REPORT – SEMESTER IV MOUSUMI MALICK 2-YEARS MSc(2011-2012) Guided by –Prof.DEBAPRIYA BISWAS Lie group and Geometry on the Lie Group SL2(R) CERTIFICATE This is to certify that the project entitled “Lie group and Geometry on the Lie group SL2(R)” being submitted by Mousumi Malick Roll no.-10MA40017, Department of Mathematics is a survey of some beautiful results in Lie groups and its geometry and this has been carried out under my supervision. Dr. Debapriya Biswas Department of Mathematics Date- Indian Institute of Technology Khargpur 1 Lie group and Geometry on the Lie Group SL2(R) ACKNOWLEDGEMENT I wish to express my gratitude to Dr. Debapriya Biswas for her help and guidance in preparing this project. Thanks are also due to the other professor of this department for their constant encouragement. Date- place-IIT Kharagpur Mousumi Malick 2 Lie group and Geometry on the Lie Group SL2(R) CONTENTS 1.Introduction ................................................................................................... 4 2.Definition of general linear group: ............................................................... 5 3.Definition of a general Lie group:................................................................... 5 4.Definition of group action: ............................................................................. 5 5. Definition of orbit under a group action: ...................................................... 5 6.1.The general linear
    [Show full text]
  • Definition 1.1. a Group Is a Quadruple (G, E, ⋆, Ι)
    GROUPS AND GROUP ACTIONS. 1. GROUPS We begin by giving a definition of a group: Definition 1.1. A group is a quadruple (G, e, ?, ι) consisting of a set G, an element e G, a binary operation ?: G G G and a map ι: G G such that ∈ × → → (1) The operation ? is associative: (g ? h) ? k = g ? (h ? k), (2) e ? g = g ? e = g, for all g G. ∈ (3) For every g G we have g ? ι(g) = ι(g) ? g = e. ∈ It is standard to suppress the operation ? and write gh or at most g.h for g ? h. The element e is known as the identity element. For clarity, we may also write eG instead of e to emphasize which group we are considering, but may also write 1 for e where this is more conventional (for the group such as C∗ for example). Finally, 1 ι(g) is usually written as g− . Remark 1.2. Let us note a couple of things about the above definition. Firstly clo- sure is not an axiom for a group whatever anyone has ever told you1. (The reason people get confused about this is related to the notion of subgroups – see Example 1.8 later in this section.) The axioms used here are not “minimal”: the exercises give a different set of axioms which assume only the existence of a map ι without specifying it. We leave it to those who like that kind of thing to check that associa- tivity of triple multiplications implies that for any k N, bracketing a k-tuple of ∈ group elements (g1, g2, .
    [Show full text]
  • Presentations of Groups Acting Discontinuously on Direct Products of Hyperbolic Spaces
    Presentations of Groups Acting Discontinuously on Direct Products of Hyperbolic Spaces∗ E. Jespers A. Kiefer Á. del Río November 6, 2018 Abstract The problem of describing the group of units U(ZG) of the integral group ring ZG of a finite group G has attracted a lot of attention and providing presentations for such groups is a fundamental problem. Within the context of orders, a central problem is to describe a presentation of the unit group of an order O in the simple epimorphic images A of the rational group algebra QG. Making use of the presentation part of Poincaré’s Polyhedron Theorem, Pita, del Río and Ruiz proposed such a method for a large family of finite groups G and consequently Jespers, Pita, del Río, Ruiz and Zalesskii described the structure of U(ZG) for a large family of finite groups G. In order to handle many more groups, one would like to extend Poincaré’s Method to discontinuous subgroups of the group of isometries of a direct product of hyperbolic spaces. If the algebra A has degree 2 then via the Galois embeddings of the centre of the algebra A one considers the group of reduced norm one elements of the order O as such a group and thus one would obtain a solution to the mentioned problem. This would provide presentations of the unit group of orders in the simple components of degree 2 of QG and in particular describe the unit group of ZG for every group G with irreducible character degrees less than or equal to 2.
    [Show full text]
  • Fundamental Domains for Genus-Zero and Genus-One Congruence Subgroups
    LMS J. Comput. Math. 13 (2010) 222{245 C 2010 Author doi:10.1112/S1461157008000041e Fundamental domains for genus-zero and genus-one congruence subgroups C. J. Cummins Abstract In this paper, we compute Ford fundamental domains for all genus-zero and genus-one congruence subgroups. This is a continuation of previous work, which found all such groups, including ones that are not subgroups of PSL(2; Z). To compute these fundamental domains, an algorithm is given that takes the following as its input: a positive square-free integer f, which determines + a maximal discrete subgroup Γ0(f) of SL(2; R); a decision procedure to determine whether a + + given element of Γ0(f) is in a subgroup G; and the index of G in Γ0(f) . The output consists of: a fundamental domain for G, a finite set of bounding isometric circles; the cycles of the vertices of this fundamental domain; and a set of generators of G. The algorithm avoids the use of floating-point approximations. It applies, in principle, to any group commensurable with the modular group. Included as appendices are: MAGMA source code implementing the algorithm; data files, computed in a previous paper, which are used as input to compute the fundamental domains; the data computed by the algorithm for each of the congruence subgroups of genus zero and genus one; and an example, which computes the fundamental domain of a non-congruence subgroup. 1. Introduction The modular group PSL(2; Z) := SL(2; Z)={±12g is a discrete subgroup of PSL(2; R) := SL(2; R)={±12g.
    [Show full text]
  • REPLICATING TESSELLATIONS* ANDREW Vincet Abstract
    SIAM J. DISC. MATH. () 1993 Society for Industrial and Applied Mathematics Vol. 6, No. 3, pp. 501-521, August 1993 014 REPLICATING TESSELLATIONS* ANDREW VINCEt Abstract. A theory of replicating tessellation of R is developed that simultaneously generalizes radix representation of integers and hexagonal addressing in computer science. The tiling aggregates tesselate Eu- clidean space so that the (m + 1)st aggregate is, in turn, tiled by translates of the ruth aggregate, for each m in exactly the same way. This induces a discrete hierarchical addressing systsem on R'. Necessary and sufficient conditions for the existence of replicating tessellations are given, and an efficient algorithm is provided to de- termine whether or not a replicating tessellation is induced. It is shown that the generalized balanced ternary is replicating in all dimensions. Each replicating tessellation yields an associated self-replicating tiling with the following properties: (1) a single tile T tesselates R periodically and (2) there is a linear map A, such that A(T) is tiled by translates of T. The boundary of T is often a fractal curve. Key words, tiling, self-replicating, radix representation AMS(MOS) subject classifications. 52C22, 52C07, 05B45, 11A63 1. Introduction. The standard set notation X + Y {z + y z E X, y E Y} will be used. For a set T c Rn denote by Tx z + T the translate of T to point z. Throughout this paper, A denotes an n-dimensional lattice in l'. A set T tiles a set R by translation by lattice A if R [-JxsA T and the intersection of the interiors of distinct tiles T and Tu is empty.
    [Show full text]
  • Construction of Free Subgroups in the Group of Units of Modular Group Algebras
    CONSTRUCTION OF FREE SUBGROUPS IN THE GROUP OF UNITS OF MODULAR GROUP ALGEBRAS Jairo Z. Gon¸calves1 Donald S. Passman2 Department of Mathematics Department of Mathematics University of S~ao Paulo University of Wisconsin-Madison 66.281-Ag Cidade de S. Paulo Van Vleck Hall 05389-970 S. Paulo 480 Lincoln Drive S~ao Paulo, Brazil Madison, WI 53706, U.S.A [email protected] [email protected] Abstract. Let KG be the group algebra of a p0-group G over a field K of characteristic p > 0; and let U(KG) be its group of units. If KG contains a nontrivial bicyclic unit and if K is not algebraic over its prime field, then we prove that the free product Zp ∗ Zp ∗ Zp can be embedded in U(KG): 1. Introduction Let KG be the group algebra of the group G over the field K; and let U(KG) be its group of units. Motivated by the work of Pickel and Hartley [4], and Sehgal ([7, pg. 200]) on the existence of free subgroups in the inte- gral group ring ZG; analogous conditions for U(KG) have been intensively investigated in [1], [2] and [3]. Recently Marciniak and Sehgal [5] gave a constructive method for produc- ing free subgroups in U(ZG); provided ZG contains a nontrivial bicyclic unit. In this paper we prove an analogous result for the modular group algebra KG; whenever K is not algebraic over its prime field GF (p): Specifically, if Zp denotes the cyclic group of order p, then we prove: 1- Research partially supported by CNPq - Brazil.
    [Show full text]
  • Special Unitary Group - Wikipedia
    Special unitary group - Wikipedia https://en.wikipedia.org/wiki/Special_unitary_group Special unitary group In mathematics, the special unitary group of degree n, denoted SU( n), is the Lie group of n×n unitary matrices with determinant 1. (More general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case.) The group operation is matrix multiplication. The special unitary group is a subgroup of the unitary group U( n), consisting of all n×n unitary matrices. As a compact classical group, U( n) is the group that preserves the standard inner product on Cn.[nb 1] It is itself a subgroup of the general linear group, SU( n) ⊂ U( n) ⊂ GL( n, C). The SU( n) groups find wide application in the Standard Model of particle physics, especially SU(2) in the electroweak interaction and SU(3) in quantum chromodynamics.[1] The simplest case, SU(1) , is the trivial group, having only a single element. The group SU(2) is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), there is a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is {+ I, − I}. [nb 2] SU(2) is also identical to one of the symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations. Contents Properties Lie algebra Fundamental representation Adjoint representation The group SU(2) Diffeomorphism with S 3 Isomorphism with unit quaternions Lie Algebra The group SU(3) Topology Representation theory Lie algebra Lie algebra structure Generalized special unitary group Example Important subgroups See also 1 of 10 2/22/2018, 8:54 PM Special unitary group - Wikipedia https://en.wikipedia.org/wiki/Special_unitary_group Remarks Notes References Properties The special unitary group SU( n) is a real Lie group (though not a complex Lie group).
    [Show full text]
  • Ideal Classes and SL 2
    IDEAL CLASSES AND SL2 KEITH CONRAD 1. Introduction A standard group action in complex analysis is the action of GL2(C) on the Riemann sphere C [ f1g by linear fractional transformations (M¨obiustransformations): a b az + b (1.1) z = : c d cz + d We need to allow the value 1 since cz + d might be 0. (If that happens, az + b 6= 0 since a b ( c d ) is invertible.) When z = 1, the value of (1.1) is a=c 2 C [ f1g. It is easy to see this action of GL2(C) on the Riemann sphere is transitive (that is, there is one orbit): for every a 2 C, a a − 1 (1.2) 1 = a; 1 1 a a−1 so the orbit of 1 passes through all points. In fact, since ( 1 1 ) has determinant 1, the action of SL2(C) on C [ f1g is transitive. However, the action of SL2(R) on the Riemann sphere is not transitive. The reason is the formula for imaginary parts under a real linear fractional transformation: az + b (ad − bc) Im(z) Im = cz + d jcz + dj2 a b a b when ( c d ) 2 GL2(R). Thus the imaginary parts of z and ( c d )z have the same sign when a b ( c d ) has determinant 1. The action of SL2(R) on the Riemann sphere has three orbits: R [ f1g, the upper half-plane h = fx + iy : y > 0g, and the lower half-plane. To see that the action of SL2(R) on h is transitive, pick x + iy with y > 0.
    [Show full text]