Annihilators of Simple Tensor Modules
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DIFFERENTIAL GEOMETRY FINAL PROJECT 1. Introduction for This
DIFFERENTIAL GEOMETRY FINAL PROJECT KOUNDINYA VAJJHA 1. Introduction For this project, we outline the basic structure theory of Lie groups relating them to the concept of Lie algebras. Roughly, a Lie algebra encodes the \infinitesimal” structure of a Lie group, but is simpler, being a vector space rather than a nonlinear manifold. At the local level at least, the Fundamental Theorems of Lie allow one to reconstruct the group from the algebra. 2. The Category of Local (Lie) Groups The correspondence between Lie groups and Lie algebras will be local in nature, the only portion of the Lie group that will be of importance is that portion of the group close to the group identity 1. To formalize this locality, we introduce local groups: Definition 2.1 (Local group). A local topological group is a topological space G, with an identity element 1 2 G, a partially defined but continuous multiplication operation · :Ω ! G for some domain Ω ⊂ G × G, a partially defined but continuous inversion operation ()−1 :Λ ! G with Λ ⊂ G, obeying the following axioms: • Ω is an open neighbourhood of G × f1g Sf1g × G and Λ is an open neighbourhood of 1. • (Local associativity) If it happens that for elements g; h; k 2 G g · (h · k) and (g · h) · k are both well-defined, then they are equal. • (Identity) For all g 2 G, g · 1 = 1 · g = g. • (Local inverse) If g 2 G and g−1 is well-defined in G, then g · g−1 = g−1 · g = 1. A local group is said to be symmetric if Λ = G, that is, every element g 2 G has an inverse in G.A local Lie group is a local group in which the underlying topological space is a smooth manifold and where the associated maps are smooth maps on their domain of definition. -
The Algebraic Method in Representation Theory
THE ALGEBRAIC METHOD IN REPRESENTATION THEORY (ENVELOPING ALGEBRAS) Anthony JOSEPH ~) Institut des Hautes Etudes Scientifiques 91440 - BURES-sur-YVETTE (France) ~) Permanent address : Department of Physics and Astronomy, Tel-Aviv University, Ramit Avic, Israel 98 THE ALGEBRAIC METHOD IN REPRESENTATION THEORY ( ENVELOPING ALGEBRAS) I Introduction In Lie algebra theory a basic open problem is to classify all irreducible representations (up to equivalence). For the category of finite dimensional representations the answer is known and is classical. By contrast a full classification for infinite dimensional representations appears to be impossible. For example, this is evidenced by the work of Arnal and Pinczon [2] on s~(2) and by the work of MeConnell and Robson [34] on A 1 which can be used to show [5] that the Heisenberg Lie algebra [18], 4.6.1, admits infinitely many inequivalent irreducible representations all very different from the standard one. One way out of this difficulty is to consider only representations which integrate to unitary (or just bounded) representations of the corresponding Lie group(s). This has physical justification through Wigner's theorem. We call it the analytic approach. In it the Lie algebra plays only a subservient role. About ten years ago, Di~nier proposed a purely algebraic way out which has since then generated a new area of mathematics called enveloping algebras. The philosophy is to find a less refined classification than equivalence classes rather than to exclude representations. We call it the algebraic approach. Except for finite dimensional representations, or for nilpotent Lie algebras, the exact relationship between these two approaches is not yet known, though one can find many analogies. -
10 Group Theory and Standard Model
Physics 129b Lecture 18 Caltech, 03/05/20 10 Group Theory and Standard Model Group theory played a big role in the development of the Standard model, which explains the origin of all fundamental particles we see in nature. In order to understand how that works, we need to learn about a new Lie group: SU(3). 10.1 SU(3) and more about Lie groups SU(3) is the group of special (det U = 1) unitary (UU y = I) matrices of dimension three. What are the generators of SU(3)? If we want three dimensional matrices X such that U = eiθX is unitary (eigenvalues of absolute value 1), then X need to be Hermitian (real eigenvalue). Moreover, if U has determinant 1, X has to be traceless. Therefore, the generators of SU(3) are the set of traceless Hermitian matrices of dimension 3. Let's count how many independent parameters we need to characterize this set of matrices (what is the dimension of the Lie algebra). 3 × 3 complex matrices contains 18 real parameters. If it is to be Hermitian, then the number of parameters reduces by a half to 9. If we further impose traceless-ness, then the number of parameter reduces to 8. Therefore, the generator of SU(3) forms an 8 dimensional vector space. We can choose a basis for this eight dimensional vector space as 00 1 01 00 −i 01 01 0 01 00 0 11 λ1 = @1 0 0A ; λ2 = @i 0 0A ; λ3 = @0 −1 0A ; λ4 = @0 0 0A (1) 0 0 0 0 0 0 0 0 0 1 0 0 00 0 −i1 00 0 01 00 0 0 1 01 0 0 1 1 λ5 = @0 0 0 A ; λ6 = @0 0 1A ; λ7 = @0 0 −iA ; λ8 = p @0 1 0 A (2) i 0 0 0 1 0 0 i 0 3 0 0 −2 They are called the Gell-Mann matrices. -
Representation Theory
M392C NOTES: REPRESENTATION THEORY ARUN DEBRAY MAY 14, 2017 These notes were taken in UT Austin's M392C (Representation Theory) class in Spring 2017, taught by Sam Gunningham. I live-TEXed them using vim, so there may be typos; please send questions, comments, complaints, and corrections to [email protected]. Thanks to Kartik Chitturi, Adrian Clough, Tom Gannon, Nathan Guermond, Sam Gunningham, Jay Hathaway, and Surya Raghavendran for correcting a few errors. Contents 1. Lie groups and smooth actions: 1/18/172 2. Representation theory of compact groups: 1/20/174 3. Operations on representations: 1/23/176 4. Complete reducibility: 1/25/178 5. Some examples: 1/27/17 10 6. Matrix coefficients and characters: 1/30/17 12 7. The Peter-Weyl theorem: 2/1/17 13 8. Character tables: 2/3/17 15 9. The character theory of SU(2): 2/6/17 17 10. Representation theory of Lie groups: 2/8/17 19 11. Lie algebras: 2/10/17 20 12. The adjoint representations: 2/13/17 22 13. Representations of Lie algebras: 2/15/17 24 14. The representation theory of sl2(C): 2/17/17 25 15. Solvable and nilpotent Lie algebras: 2/20/17 27 16. Semisimple Lie algebras: 2/22/17 29 17. Invariant bilinear forms on Lie algebras: 2/24/17 31 18. Classical Lie groups and Lie algebras: 2/27/17 32 19. Roots and root spaces: 3/1/17 34 20. Properties of roots: 3/3/17 36 21. Root systems: 3/6/17 37 22. Dynkin diagrams: 3/8/17 39 23. -
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). -
Over Noncommutative Noetherian Rings
transactions of the american mathematical society Volume 323, Number 2, February 1991 GENERATING MODULES EFFICIENTLY OVER NONCOMMUTATIVE NOETHERIAN RINGS S. C. COUTINHO Abstract. The Forster-Swan Theorem gives an upper bound on the number of generators of a module over a commutative ring in terms of local data. Stafford showed that this theorem could be generalized to arbitrary right and left noethe- rian rings. In this paper a similar result is proved for right noetherian rings with finite Krull dimension. A new dimension function—the basic dimension—is the main tool used in the proof of this result. Introduction The proof of many interesting results in commutative algebra and algebraic 7C-theory are applications of local-global principles. These are results of the following type: if a property is true for all the localisations of an 7?-module M at each prime ideal of 7?, then it is true for M itself. For example, a principle of this kind lies at the heart of Quillen's solution of Serre's conjecture [8]. A further example of a local-global principle is the Forster-Swan Theorem. Suppose that M is a finitely generated right module over a right noetherian ring R. Let us denote the minimal number of generators of M by gR(M). For a commutative ring R, the Forster-Swan Theorem gives an upper bound for gR(M) in terms of local data; more precisely gR(M) < sup{gR (Mp) + Kdim(7?/P) : P a J -prime ideal of 7? } . For a proof of this theorem and some of its applications to commutative algebra and algebraic geometry, see [8]. -
Lectures on Non-Commutative Rings
Lectures on Non-Commutative Rings by Frank W. Anderson Mathematics 681 University of Oregon Fall, 2002 This material is free. However, we retain the copyright. You may not charge to redistribute this material, in whole or part, without written permission from the author. Preface. This document is a somewhat extended record of the material covered in the Fall 2002 seminar Math 681 on non-commutative ring theory. This does not include material from the informal discussion of the representation theory of algebras that we had during the last couple of lectures. On the other hand this does include expanded versions of some items that were not covered explicitly in the lectures. The latter mostly deals with material that is prerequisite for the later topics and may very well have been covered in earlier courses. For the most part this is simply a cleaned up version of the notes that were prepared for the class during the term. In this we have attempted to correct all of the many mathematical errors, typos, and sloppy writing that we could nd or that have been pointed out to us. Experience has convinced us, though, that we have almost certainly not come close to catching all of the goofs. So we welcome any feedback from the readers on how this can be cleaned up even more. One aspect of these notes that you should understand is that a lot of the substantive material, particularly some of the technical stu, will be presented as exercises. Thus, to get the most from this you should probably read the statements of the exercises and at least think through what they are trying to address. -
Group Theory - QMII 2017
Group Theory - QMII 2017 Reminder Last time we said that a group element of a matrix lie group can be written as an exponent: a U = eiαaX ; a = 1; :::; N: We called Xa the generators, we have N of them, they span a basis for the Lie algebra, and they can be found by taking the derivative with respect to αa at α = 0. The generators are closed under the Lie product (∼ [·; ·]), and related by the structure constants [Xa;Xb] = ifabcXc: (1) We end with the Jacobi identity [Xa; [Xb;Xc]] + [Xb; [Xc;Xa]] + [Xc; [Xa;Xb]] = 0: (2) 1 Representations of Lie Algebra 1.1 The adjoint rep. part- I One of the most important representations is the adjoint. There are two equivalent ways to defined it. Here we follow the definition by Georgi. By plugging Eq. (1) into the Jacobi identity Eq. (2) we get the following: fbcdfade + fabdfcde + fcadfbde = 0 (3) Proof: 0 = [Xa; [Xb;Xc]] + [Xb; [Xc;Xa]] + [Xc; [Xa;Xb]] = ifbcd [Xa;Xd] + ifcad [Xb;Xd] + ifabd [Xc;Xd] = − (fbcdfade + fcadfbde + fabdfcde) Xe (4) 1 Now we can define a set of matrices Ta s.t. [Ta]bc = −ifabc: (5) Then by using the relation of the structure constants Eq. (3) we get [Ta;Tb] = ifabcTc: (6) Proof: [Ta;Tb]ce = [Ta]cd[Tb]de − [Tb]cd[Ta]de = −facdfbde + fbcdfade = fcadfbde + fbcdfade = −fabdfcde = fabdfdce = ifabd[Td]ce (7) which means that: [Ta;Tb] = ifabdTd: (8) Therefore the structure constants themselves generate a representation. This representation is called the adjoint representation. The adjoint of su(2) We already found that the structure constants of su(2) are given by the Levi-Civita tensor. -
Groups and Representations the Material Here Is Partly in Appendix a and B of the Book
Groups and representations The material here is partly in Appendix A and B of the book. 1 Introduction The concept of symmetry, and especially gauge symmetry, is central to this course. Now what is a symmetry: you have something, e.g. a vase, and you do something to it, e.g. turn it by 29 degrees, and it still looks the same then we call the operation performed, i.e. the rotation, a symmetryoperation on the object. But if we first rotate it by 29 degrees and then by 13 degrees and it still looks the same then also the combined operation of a rotation by 42 degrees is a symmetry operation as well. The mathematics that is involved here is that of groups and their representations. The symmetry operations form the group and the objects the operations work on are in representations of the group. 2 Groups A group is a set of elements g where there exist an operation ∗ that combines two group elements and the results is a third: ∃∗ : 8g1; g2 2 G : g1 ∗ g2 = g3 2 G (1) This operation must be associative: 8g1; g2; g3 2 G :(g1 ∗ g2) ∗ g3 = g1 ∗ (g2 ∗ g3) (2) and there exists an element unity: 91 2 G : 8g 2 G : g ∗ 1 = 1 ∗ g = g (3) and for every element in G there exists an inverse: 8g 2 G : 9g−1 2 G : g ∗ g−1 = g−1 ∗ g = 1 (4) This is the general definition of a group. Now if all elements of a group commute, i.e. -
Characterization of SU(N)
University of Rochester Group Theory for Physicists Professor Sarada Rajeev Characterization of SU(N) David Mayrhofer PHY 391 Independent Study Paper December 13th, 2019 1 Introduction At this point in the course, we have discussed SO(N) in detail. We have de- termined the Lie algebra associated with this group, various properties of the various reducible and irreducible representations, and dealt with the specific cases of SO(2) and SO(3). Now, we work to do the same for SU(N). We de- termine how to use tensors to create different representations for SU(N), what difficulties arise when moving from SO(N) to SU(N), and then delve into a few specific examples of useful representations. 2 Review of Orthogonal and Unitary Matrices 2.1 Orthogonal Matrices When initially working with orthogonal matrices, we defined a matrix O as orthogonal by the following relation OT O = 1 (1) This was done to ensure that the length of vectors would be preserved after a transformation. This can be seen by v ! v0 = Ov =) (v0)2 = (v0)T v0 = vT OT Ov = v2 (2) In this scenario, matrices then must transform as A ! A0 = OAOT , as then we will have (Av)2 ! (A0v0)2 = (OAOT Ov)2 = (OAOT Ov)T (OAOT Ov) (3) = vT OT OAT OT OAOT Ov = vT AT Av = (Av)2 Therefore, when moving to unitary matrices, we want to ensure similar condi- tions are met. 2.2 Unitary Matrices When working with quantum systems, we not longer can restrict ourselves to purely real numbers. Quite frequently, it is necessarily to extend the field we are with with to the complex numbers. -
Covariant Dirac Operators on Quantum Groups
Covariant Dirac Operators on Quantum Groups Antti J. Harju∗ Abstract We give a construction of a Dirac operator on a quantum group based on any simple Lie alge- bra of classical type. The Dirac operator is an element in the vector space Uq (g) ⊗ clq(g) where the second tensor factor is a q-deformation of the classical Clifford algebra. The tensor space Uq (g) ⊗ clq(g) is given a structure of the adjoint module of the quantum group and the Dirac operator is invariant under this action. The purpose of this approach is to construct equivariant Fredholm modules and K-homology cycles. This work generalizes the operator introduced by Bibikov and Kulish in [1]. MSC: 17B37, 17B10 Keywords: Quantum group; Dirac operator; Clifford algebra 1. The Dirac operator on a simple Lie group G is an element in the noncommutative Weyl algebra U(g) cl(g) where U(g) is the enveloping algebra for the Lie algebra of G. The vector space g generates a Clifford⊗ algebra cl(g) whose structure is determined by the Killing form of g. Since g acts on itself by the adjoint action, U(g) cl(g) is a g-module. The Dirac operator on G spans a one dimensional invariant submodule of U(g)⊗ cl(g) which is of the first order in cl(g) and in U(g). Kostant’s Dirac operator [10] has an additional⊗ cubical term in cl(g) which is constant in U(g). The noncommutative Weyl algebra is given an action on a Hilbert space which is also a g-module. -
Chapter 3 Adjoint Representations and the Derivative Of
Chapter 3 Adjoint Representations and the Derivative of exp 3.1 The Adjoint Representations Ad and ad Given any two vector spaces E and F ,recallthatthe vector space of all linear maps from E to F is denoted by Hom(E,F). The vector space of all invertible linear maps from E to itself is a group denoted GL(E). When E = Rn,weoftendenoteGL(Rn)byGL(n, R) (and if E = Cn,weoftendenoteGL(Cn)byGL(n, C)). The vector space Mn(R)ofalln n matrices is also ⇥ denoted by gl(n, R)(andMn(C)bygl(n, C)). 121 122 CHAPTER 3. ADJOINT REPRESENTATIONS AND THE DERIVATIVE OF exp Then, GL(gl(n, R)) is the vector space of all invertible linear maps from gl(n, R)=Mn(R)toitself. For any matrix A MA(R)(orA MA(C)), define the 2 2 maps LA :Mn(R) Mn(R)andRA :Mn(R) Mn(R) by ! ! LA(B)=AB, RA(B)=BA, for all B Mn(R). 2 Observe that LA RB = RB LA for all A, B Mn(R). ◦ ◦ 2 For any matrix A GL(n, R), let 2 AdA :Mn(R) Mn(R)(conjugationbyA) ! be given by 1 AdA(B)=ABA− for all B Mn(R). 2 Observe that AdA = LA R 1 and that AdA is an ◦ A− invertible linear map with inverse Ad 1. A− 3.1. ADJOINT REPRESENTATIONS Ad AND ad 123 The restriction of AdA to invertible matrices B GL(n, R) yields the map 2 AdA : GL(n, R) GL(n, R) ! also given by 1 AdA(B)=ABA− for all B GL(n, R).