Cartan Involutions and Cartan Decompositions of a Semi-Simple Lie Algebra
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1 Utah State University Tutorials on... in hour or less Differential Geometry Software Project Cartan Involutions and Cartan Decompositions of a Semi-Simple Lie Algebra Ian M. Anderson Utah State University Follow this and additional works at: Part of the Algebra Commons and the Geometry and Topology Commons Recommended Citation Anderson Ian M. "Cartan Involutions and Cartan Decompositions of a Semi-Simple Lie Algebra" (205). Tutorials on... in hour or less. Paper 5. This Article is brought to you for free and open access by the Differential Geometry Software Project at DigitalCommons@USU. It has been accepted for inclusion in Tutorials on... in hour or less by an authorized administrator of DigitalCommons@USU. For more information please contact rebecca.nelson@usu.edu.
2 Tutorial Series digitalcommons.usu. edu/dg Cartan Involutions and Cartan Decompositions of a Semi-Simple Lie Algebra Synopsis Let g be a semi-simple Lie algebra with Killing form $ $. The related concepts of Cartan involution and Cartan decomposition play an important role in the structure theory for real semi-simple Lie algebras. In this worksheet we shall review the basic definitions and properties of Cartan involutions and Cartan decompositions and illustrate these using the DifferentialGeometry software package for Lie algebras. We discuss in some detail the method given in [] page 203 for the construction of a Cartan involution for any semisimple Lie algebra. Definitions and Properties Definition. A Cartan involution of g is a Lie algebra automorphism Q : g / g with Q 2 bilinear form B Q x y = K x Q y is positive-definite. = Id and such that the symmetric Definition 2. A Cartan decomposition is a vector space decomposition g = t p where t is a subalgebra p a subspace [t p] p [p p] t and the Killing form is negative-definite on t and positive-definite on p. We remark that these properties imply that the decomposition g = t p defines a symmetric pair. The subalgebra t is called the compact part of the Cartan decomposition and the subspace p is called the positive part. Theorem. Given a Cartan decomposition the linear transformation which is the identity Id on t and KId on p is a Cartan
3 involution. Conversely given a Cartan involution Q the + - eigenspaces E = p and E K = t define a Cartan decomposition. Theorem 2. Any two Cartan involutions Q and Q 2 on g are related by an inner automorphism f : g / g that is Q 2 = f Q f K. Theorem 3. If a matrix representation of the algebra is available and the set of matrices defining this representation are closed under transposition (or under conjugate-transposition) then the decomposition into skew-hermitian and hermitian matrices will give a Cartan decomposition.. Cartan Involutions for Matrix Algebras In this section we illustrate Theorem 3 in the Synopsis using the standard representations of sl 3 and su 2 2. > with(differentialgeometry): with(liealgebras): Example : sl 3 The simplest examples of Cartan decompositions are provided by the algebras sl n. The compact part of the Cartan decomposition is given by the the skew-symmetric matrices and the positive part is given by the trace-free symmetric matrices. Retrieve the structure equations for sl 3. We use the keyword arguments labelformat and labels to generate labels for vectors which match their matrix definitions as elementary matrices. > > LD := SimpleLieAlgebraData("sl(3)" sl3 labelformat = "gl" labels = ['E' 'theta']): DGsetup(LD); Lie algebra: sl3 (2..) MultiplicationTable("LieTable");
4 sl3 E E22 E2 E3 E2 E23 E3 E E 0 0 E2 2 E3 K E2 E23 K 2 E3 K E E K E2 E3 E2 2 E23 K E3 K 2 E E2 K E2 E2 0 0 E K E22 E3 K E 0 E3 K 2 E3 K E3 0 0 K E23 0 E E2 E2 E2 K E2 K E C E22 E23 K E3 (2..2) E23 K E23 K 2 E23 K E3 E2 E22 E3 2 E3 E3 E K E 0 K E2 0 0 E E 2 E 0 K E2 E3 K E The matrices which correspond to the vectors E E22 E2 E3... are given by the standard representation. M := StandardRepresentation(sl3); M := (2..3) 0 0 K 0 0 K In this simple example we can calculate the Cartan decomposition by hand. The compact part is given by the skewsymmetric matrices in the span of M. T := evaldg([e2 - E2 E3 - E3 E23 - E]); T := E2 K E2 E3 K E3 E23 K E (2..4) The positive part is given by the symmetric matrices. P := evaldg([e E22 E2 + E2 E3 + E3 E23 + E]); P := E E22 E2 C E2 E3 C E3 E23 C E (2..5) Let's check that this is correct. We see that the Killing form is negative-definite on T and positive-definite on P.
5 Killing(T); Killing(P); K K K (2..6) (2..7) We see that T T 3 T ; TT := BracketOfSubspaces(TT); TT := K E23 C E E3 K E3 K E2 C E2 GetComponents(TT T orfalse = "on"); (2..8) (2..9) that T P 3 P; TP := BracketOfSubspaces(T P); TP := K E2 K E2 2 E K 2 E22 K E23 K E E3 C E3 2 E GetComponents(TP P orfalse = "on"); (2..0) (2..) and that P P 3 T: PP := BracketOfSubspaces(P P); PP := E2 K E2 2 E3 K 2 E3 E23 K E (2..2)
6 GetComponents(PP T orfalse = "on"); (2..3) We conclude the decomposition defined by T P defines a symmetric pair. The corresponding Cartan involution is easily constructed. Theta := LinearTransformation([[T[] T[]] [T[2] T[2]] [T[3] T[3]] [P[] - P[]] [P[2] - P[2]] [P[3] - P[3]] [P[4] -P[4]] [P[5]- P[5]]]); Q := E / K E E22 / K E22 E2 / K E2 E3 / K E3 E2 / K E2 E23 / K E E3 / K E3 E / K E23 Query(Theta "Homomorphism"); ComposeTransformations(Theta Theta); E / E E22 / E22 E2 / E2 E3 / E3 E2 / E2 E23 / E23 E3 / E3 E / E (2..4) (2..5) (2..6) These steps for finding the Cartan decomposition and Cartan involution can be reproduced with the commands CartanDecomposition and CartanInvolution > T P := CartanDecomposition(M sl3); T P := E2 K E2 E3 K E3 E23 K E E E22 E2 C E2 E3 C E3 E23 C E > CartanInvolution(T P); E / K E E22 / K E22 E2 / K E2 E3 / K E3 E2 / K E2 E23 / K E E3 / K E3 E / K E23 (2..7) (2..8) Example 2: su(22) For our second example we consider the 0-dimensional real Lie algebra su 2 2. Two different (but isomorphic) versions of this algebra are available with the command SimpleLieAlgebraData. In version su 2 2 is defined as the Lie algebra of complex matrices which are skew-symmetric with respect to the matrix
7 Version 2 is the set of matrices which are skew-symmetric with respect to the matrix K 0. K Initialize the second version of su 2 2. LD2 := SimpleLieAlgebraData("su(22)" su22 version = 2): DGsetup(LD2); Lie algebra: su22 (2.2.) Here is the defining or standard representation. su22 > M := StandardRepresentation(su22); 0 K I I 0 M := 0 I KI (2.2.2) I I I 0 0 KI 0 0 I 0 0 K 0 I 0 KI I KI 0 KI 0 0 KI I 0 0 0
8 I 0 KI Note that this representation is closed under complex conjugation but not under Hermitian conjugation. su22 > su22 > Query(M "ClosedUnderConjugation"); Query(M "ClosedUnderHermitianTransposition"); false (2.2.3) (2.2.4) Here is the Cartan decomposition computed from the standard representation. su22 > T P := CartanDecomposition(M su22); T P := e e2 e e2 e3 e4 e5 e3 e4 e5 e6 e7 e8 e9 e0 (2.2.5) Let's check that the matrices corresponding to T are skew-hermitian su22 > Tmatrices := [M[] M[2] M[] M[2] M[3] M[4] M[5]]; 0 K I 0 I 0 Tmatrices := 0 I 0 0 K 0 I 0 KI I KI 0 (2.2.6) I su22 > I 0 KI seq(t + t^* t = Tmatrices);
9 (2.2.7) Check that the matrices corresponding to P are hermitian. su22 > Pmatrices := DGzip(convert(P DGlist) M "plus"); I 0 I 0 Pmatrices := KI I 0 0 KI 0 0 (2.2.8) KI 0 0 I 0 0 KI 0 0 su22 > seq(t - t^* t = Pmatrices); (2.2.9) Conjugacy of Cartan Involutions We use the symplectic Lie algebra sp 4 R to illustrate the fact that if Q : g / g is a Cartan involution and f : g / g is an
10 automorphism then f Q f K is also a Cartan involution. This section is independent of other sections. > with(differentialgeometry): with(liealgebras): Retrieve the structure equations for sp 4 R and initialize. > LD2 := SimpleLieAlgebraData("sp(4R)" sp4r); LD2 := e e2 = e2 e e3 = K e3 e e4 = 0 e e5 = 2 e5 e e6 = e6 e e7 = 0 e e8 = K 2 e8 e e9 = K e9 e e0 = 0 e2 e3 = e K e4 e2 e4 = e2 e2 e5 = 0 e2 e6 = 2 e5 e2 e7 = e6 e2 e8 = K e9 e2 e9 = K 2 e0 e2 e0 = 0 e3 e4 = K e3 e3 e5 = e6 e3 e6 = 2 e7 e3 e7 = 0 e3 e8 = 0 e3 e9 = K 2 e8 e3 e0 = K e9 e4 e5 = 0 e4 e6 = e6 e4 e7 = 2 e7 e4 e8 = 0 e4 e9 = K e9 e4 e0 = K 2 e0 e5 e6 = 0 e5 e7 = 0 e5 e8 = e e5 e9 = e2 e5 e0 = 0 e6 e7 = 0 e6 e8 = e3 e6 e9 = e C e4 e6 e0 = e2 e7 e8 = 0 e7 e9 = e3 e7 e0 = e4 e8 e9 = 0 e8 e0 = 0 e9 e0 = 0 DGsetup(LD2); Lie algebra: sp4r (3.) (3.2) One has immediate access to many properties of a simple Lie algebra that is created by the SimpleLieAlgebraData command. The command SimpleLieAlgebraProperties returns a record whose exports give the Lie algebra properties. sp4r > P := SimpleLieAlgebraProperties(sp4R); P := Record CartanSubalgebra = e e4 CartanMatrix KillingForm SimpleRoots = K 0 2 PositiveRoots = K (3.3) SimpleRootSpaces = table K = e2 0 2 = e7 PositiveRootSpaces = table K = e2 0 2 = e7 2 0 = e5 = e6 NegativeRoots = K 0 K2 K K K2 0 NegativeRootSpaces = table 0 K2 = e0 K2 0 = e8 K K = e9 K = e3 RootSpaceDecomposition = table K = e2 0 K2 = e0 0 2 = e7 K2 0 = e8 K K = e9 2 0 = e5 K = e3 = e6 BorelSubalgebra = e e4 e2 e7 e6 e5
11 CartanDecomposition = e2 K e3 e7 K e0 e6 K e9 e5 K e8 e4 e2 C e3 e e7 C e0 e6 C e9 e5 C e8 CartanInvolution = e / K e e2 / K e3 e3 / K e2 e4 / K e4 e5 / K e8 e6 / K e9 e7 / K e0 e8 / K e5 e9 / K e6 e0 / K e7 Here is a Cartan involution. sp4r > Theta := P:-CartanInvolution; Q := e / K e e2 / K e3 e3 / K e2 e4 / K e4 e5 / K e8 e6 / K e9 e7 / K e0 e8 / K e5 e9 / K e6 e0 / K e7 (3.4) Here is the corresponding Cartan decomposition. sp4r > T P := P:-CartanDecomposition; T P := e2 K e3 e7 K e0 e6 K e9 e5 K e8 e4 e2 C e3 e e7 C e0 e6 C e9 e5 C e8 (3.5) For example we check that e2 Ke3 2 T is in the + eigenspace of Q and e6 C e9 2 P is in the - eigenspace of Q sp4r > sp4r > ApplyLinearTransformation(Theta e2 - e3); e2 K e3 ApplyLinearTransformation(Theta e6 + e9); K e6 K e9 (3.6) (3.7) We create an automorphism of sp 4 R using the matrix exponential of an adjoint matrix. sp4r > phi := LinearTransformation(sp4R sp4r AdjointExp(e3 + e5)); f := e / e C e3 K 2 e5 K 3 2 e6 K e7 e2 / K e C e2 K e3 C e4 C e5 C 2 3 e6 C 3 e7 e3 / e3 K e6 K e7 e4 / (3.8) K e3 C e4 C 2 e6 C 3 e7 e5 / e5 C e6 C e7 e6 / e6 C 2 e7 e7 / e7 e8 / e C 2 e3 K e5 K 2 e6 K 4 e7 C e8 e9 / K 3 2 e C e2 K 2 3 e3 C 2 e4 C e5 C 5 2 e6 C 6 e7 K 2 e8 C e9 e0 / 2 e K 2 e2 C 6 e3
12 K 6 e4 K 4 e5 K 2 e6 K 36 e7 C e8 K e9 C e0 Conjugate the Cartan involution Q by f. sp4r > Theta2 := ComposeTransformations(InverseTransformation(phi) Theta phi); Q2 := e / K 27 4 e K 3 e2 C e3 C 6 2 e4 K 27 4 e5 C 3 8 e6 K 9 9 e7 C 6 e8 C 5 2 e9 C e0 e2 / 25 6 e C 6 e2 K 5 e3 K 43 8 e4 C 9 4 C 5 3 e4 K 9 4 e5 C 7 6 K 0 72 e6 K 9 36 e5 K 6 36 e6 C e7 K 8 3 e8 K e9 K 3 e0 e3 / K 3 e K 5 2 e2 C 6 e3 e7 C 3 e8 C 2 e9 C e0 e4 / 23 2 e C 5 3 e2 K 43 8 e3 K e4 C 9 2 e5 e6 C 4 54 e7 K 4 3 e8 K 5 6 e9 K 3 e0 e5 / 3 e C 3 2 e2 K 4 3 e3 K 2 3 e4 C 9 4 e5 K e6 C 4 9 e7 K 4 e8 (3.9) K 2 e9 K e0 e6 / 5 2 e C 2 e2 K e3 K 5 6 e4 C 3 2 e5 K 7 2 e6 C 2 9 e7 K 4 e8 K 3 e9 K 2 e0 e7 / 2 e C 2 e2 K 6 e3 K 6 e4 C 4 e5 K 2 e6 C 36 K 8 6 K C e5 C 57 6 e7 K e8 K e9 K e0 e8 / K 27 8 e K 9 8 e2 C 9 8 e6 K e7 C 9 4 e8 C 3 4 e9 C 4 e0 e9 / 3 8 e C 7 6 e6 C e7 K 2 e8 K 7 2 e9 K 6 e6 K 68 4 e7 C 4 9 e8 C 9 e9 C 36 e0 e0 e0 / K 9 8 e K 9 72 e2 K 6 36 e2 C 4 27 e3 C 9 24 e4 e3 K 0 72 e4 C 57 8 e5 e3 C 4 08 e4 K e5 Check that Q2 is a Cartan involution. sp4r > Query(Theta2 "CartanInvolution"); (3.0) We can calculate the corresponding Cartan decomposition using either the DGEigentensor command in the Tensor
13 package or the command CartanDecomposition. sp4r > EigenSpaces := C sp4r > EigenSpaces := Tensor:-DGEigenTensors(Theta2 [e e2 e3 e4 e5 e6 e7 e8 e9 e0]); K K K K K K e3 C e C e3 C 3 2 e4 C e5 2 8 e C 3 2 e4 C e9 K 7 e C 5 6 e2 K 6 K 3 4 e K 3 2 e2 C 9 2 e3 C 27 4 e4 C e e2 K 85 2 e3 K e4 C e0 K 2 6 e K 45 8 e2 e3 K 63 e4 C e8 K 9 8 e K 9 4 e2 C 5 4 e3 C 57 8 e C 43 2 e2 K 37 6 K 7 4 e2 C 37 6 e3 C 5 4 e4 K 4 e5 C 6 e6 C e9 K 6 C e8 T P := e C e C 3 4 e2 K 3 2 e3 K 3 4 e4 C 3 4 e5 K 5 4 e6 C e7 T P := CartanDecomposition(Theta2); 2 e5 C 6 e6 K 2 4 e7 K e8 C 3 4 e9 C 9 4 e0 e2 C 8 e3 K 5 8 e5 C 45 8 e6 K 3 e7 C 3 2 e9 C 9 2 e0 e4 C 63 6 e3 K 4 2 e4 C 6 e C 5 6 e2 K 8 e5 K 89 6 e4 C e7 e5 K e6 C e0 K 4 e e3 K 5 6 e4 K 7 6 e5 K 6 e6 e5 K e6 C 85 6 e7 K 2 e8 K 9 2 e9 K 5 2 e0 e6 C e7 K 3 e8 K 27 4 e9 K 57 4 e0 e (3.) (3.2)
14 C 9 6 C e7 K e8 C e9 C e0 e4 K 9 6 K e0 e6 K 0 3 e7 C e9 C e0 e2 K 4 9 e7 K e8 K e8 K e9 K e0 e7 K e8 K e9 K e0 e5 K 4 3 e9 K e0 e3 C 9 2 e7 C e8 C e9 e7 C e8 The individual eigenvectors produced by these 2 commands are not the same but the eigenspaces spanned by these eigenvectors can be shown to be the same using DGequal. 3. A General Algorithm for Constructing a Cartan Involution This section provides a detailed illustration of the method given in [] page 203 for the construction of a Cartan involution for any semi-simple Lie algebra. The steps are as follows: Step. Construct the complexification of the g C of the Lie algebra g. This is a real Lie algebra whose elements are x C i y where x y 2 g. Let s : g C / g C denote the standard conjugation map s (x C i y = x Ki y. A Lie algebra homomorphism y : g C / g C which commutes with s will preserve the eigenspaces of s and therefore restricts to a homomorphism of the underlying original Lie algebra g. Step 2. Construct a compact form u of g and identify u with a subalgebra of g C. Step 3. Construct the conjugation map t : g C / g C defined with respect to u. A key observation is that t is a Cartan involution of g C. The problem is that t does not commute with s and therefore the Cartan involution t of g C which we have just constructed does not restrict to a mapping of g. Step 4. Construct the map y = s t s t and check that the eigenvalues of y are all positive. Step 5. Then Q = y / 4 t y K / 4 is a Cartan involution of g C which commutes with s. This automorphism restricts to a
15 Cartan involution of g. We illustrate these steps using the Lie algebra su 2. Of course this is a matrix algebra and a Cartan involution can be found using Theorem 3 as was illustrated in Section. But the objective here is to calculate a Cartan involution for su 2 as an abstract Lie algebra. This section is independent of the other sections of this worksheet. Step. The Lie algebra su(2) and its complexification su 2 C > with(differentialgeometry): with(liealgebras): The Lie algebra su 2 is the real 8-dimensional Lie algebra of trace-free 3 # 3 complex matrices which preserve a Minkowski inner product. We initialize the Lie algebra su 2 with the commands SimpleLieAlgebraData and DGsetup > LD := SimpleLieAlgebraData("su(2)" su2); LD := e e2 = 0 e e3 = 3 e7 e e4 = 3 e8 e e5 = 0 e e6 = 0 e e7 = K 3 e3 e e8 = K 3 e4 e2 e3 = K e3 e2 e4 = e4 e2 e5 = K 2 e5 e2 e6 = 2 e6 e2 e7 = K e7 e2 e8 = e8 e3 e4 = e2 e3 e5 = 0 e3 e6 = K e8 e3 e7 = 2 e5 e3 e8 = e e4 e5 = K e7 e4 e6 = 0 e4 e7 = e e4 e8 = 2 e6 e5 e6 = e2 e5 e7 = 0 e5 e8 = K e3 e6 e7 = K e4 e6 e8 = 0 e7 e8 = e2 > DGsetup(LD); Lie algebra: su2 (4..) (4..2) The complexification of su 2 is a 6-dimensional real Lie algebra whose underlying vector space is the direct sum su 2 4 i su 2. The command complexify creates a Lie algebra using a basis adapted to this direct sum. su2 > LD2 := Complexify(su2 su2c); LD2 := e e2 = 0 e e3 = 3 e7 e e4 = 3 e8 e e5 = 0 e e6 = 0 e e7 = K 3 e3 e e8 = K 3 e4 e e9 = 0 e e0 = 0 e e = 3 e5 e e2 = 3 e6 e e3 = 0 e e4 = 0 e e5 = K 3 e e e6 = K 3 e2 e2 e3 = K e3 e2 e4 = e4 e2 e5 = K 2 e5 e2 e6 = 2 e6 e2 e7 = K e7 e2 e8 = e8 e2 e9 = 0 e2 e0 = 0 e2 e = K e e2 e2 = e2 e2 e3 = K 2 e3 e2 e4 = 2 e4 e2 e5 = K e5 e2 e6 (4..3)
16 = e6 e3 e4 = e2 e3 e5 = 0 e3 e6 = K e8 e3 e7 = 2 e5 e3 e8 = e e3 e9 = K 3 e5 e3 e0 = e e3 e = 0 e3 e2 = e0 e3 e3 = 0 e3 e4 = K e6 e3 e5 = 2 e3 e3 e6 = e9 e4 e5 = K e7 e4 e6 = 0 e4 e7 = e e4 e8 = 2 e6 e4 e9 = K 3 e6 e4 e0 = K e2 e4 e = K e0 e4 e2 = 0 e4 e3 = K e5 e4 e4 = 0 e4 e5 = e9 e4 e6 = 2 e4 e5 e6 = e2 e5 e7 = 0 e5 e8 = K e3 e5 e9 = 0 e5 e0 = 2 e3 e5 e = 0 e5 e2 = e5 e5 e3 = 0 e5 e4 = e0 e5 e5 = 0 e5 e6 = K e e6 e7 = K e4 e6 e8 = 0 e6 e9 = 0 e6 e0 = K 2 e4 e6 e = e6 e6 e2 = 0 e6 e3 = K e0 e6 e4 = 0 e6 e5 = K e2 e6 e6 = 0 e7 e8 = e2 e7 e9 = 3 e e7 e0 = e5 e7 e = K 2 e3 e7 e2 = K e9 e7 e3 = 0 e7 e4 = e2 e7 e5 = 0 e7 e6 = e0 e8 e9 = 3 e2 e8 e0 = K e6 e8 e = K e9 e8 e2 = K 2 e4 e8 e3 = e e8 e4 = 0 e8 e5 = K e0 e8 e6 = 0 e9 e0 = 0 e9 e = K 3 e7 e9 e2 = K 3 e8 e9 e3 = 0 e9 e4 = 0 e9 e5 = 3 e3 e9 e6 = 3 e4 e0 e = e3 e0 e2 = K e4 e0 e3 = 2 e5 e0 e4 = K 2 e6 e0 e5 = e7 e0 e6 = K e8 e e2 = K e2 e e3 = 0 e e4 = e8 e e5 = K 2 e5 e e6 = K e e2 e3 = e7 e2 e4 = 0 e2 e5 = K e e2 e6 = K 2 e6 e3 e4 = K e2 e3 e5 = 0 e3 e6 = e3 e4 e5 = e4 e4 e6 = 0 e5 e6 = K e2 To help remind us of the construction of the complexification we will use the following labels for the basis vectors: su2 > su(2) > su2cvectorlabels := [E E2 E3 E4 E5 E6 E7 E8 ie ie2 ie3 ie4 ie5 ie6 ie7 ie8]; su2cvectorlabels := E E2 E3 E4 E5 E6 E7 E8 ie ie2 ie3 ie4 ie5 ie6 ie7 ie8 DGsetup(LD2 su2cvectorlabels [xi]); Lie algebra: su2c (4..4) (4..5) We shall need the tautological mapping i : su 2 5 C /su 2 C which maps a vector in su 2 with complex coefficients to its counterpart in su 2 C. su2 > su2 > su2 > su2 > su2 > iota := proc(x) local A B; A:= convert(dgre(x) DGlist); B := convert(dgim(x) DGlist); DGzip([op(A) op(b)] su2cvectorlabels "plus"); end:
17 Let's check that this procedure works iota(e) iota(i*e) iota( 2*e + 3*I*e2); E ie 2 E C 3 ie2 (4..6) We need the map which defines complex conjugation on su 2 C - it is the identity on the real part of su 2 C and minus the identity on the imaginary part of su 2 C. This map is not a Cartan involution. sigma := LinearTransformation(su2C su2c LinearAlgebra:-DiagonalMatrix( [$8 -$8])); s := E / E E2 / E2 E3 / E3 E4 / E4 E5 / E5 E6 / E6 E7 / E7 E8 / E8 ie / K ie ie2 / su2 > K ie2 ie3 / K ie3 ie4 / K ie4 ie5 / K ie5 ie6 / K ie6 ie7 / K ie7 ie8 / K ie8 Query(sigma "Homomorphism"); Query(sigma "CartanInvolution"); false (4..7) (4..8) (4..9) Step 2. A compact form u of su(2) and the Cartan involution t of su 2 C To calculate a compact form u of su 2 we need a Cartan subalgebra the associated root space decomposition and a choice of positive roots. RSD := RootSpaceDecomposition(CSA); RSD := table K3 I K = e3 C I e7 0 2 = e6 3 I = e4 K I e8 0 K2 = e5 3 I K = e3 K I e7 K3 I = e4 C I e8 su2 > CSA := CartanSubalgebra(su2); CSA := PosRts := PositiveRoots(RSD); PosRts := e e I 3 I K (4.2.) (4.2.2) (4.2.3)
18 Here are complex bases defining the real split form and the real compact form of su 2. SplitBasis CompactBasis := SplitAndCompactForms(CSA RSD PosRts); SplitBasis CompactBasis := e2 K I 2 e K 2 e2 K e6 K 4 e3 C I 4 e7 K I 4 e4 K 4 e8 K e5 K e4 K I e8 I e3 (4.2.4) K e7 I e2 2 e K I 2 e2 K e5 K e6 K 4 e3 K e4 C I 4 e7 K I e8 I e3 K I 4 e4 K e7 K 4 e8 I e5 K I e6 K I 4 e3 C I e4 K 4 e7 K e8 e3 C 4 e4 C I e7 K I 4 e8 su2 > LieAlgebraData(CompactBasis Compactsu2); e e2 = 0 e e3 = 2 e6 e e4 = K e7 e e5 = e8 e e6 = K 2 e3 e e7 = e4 e e8 = K e5 e2 e3 = K e6 e2 e4 = 2 e7 e2 e5 = e8 e2 e6 = e3 e2 e7 = K 2 e4 e2 e8 = K e5 e3 e4 = K e5 e3 e5 = e4 e3 e6 = 2 e e3 e7 = K e8 e3 e8 = e7 e4 e5 = K e3 e4 e6 = e8 e4 e7 = 2 e2 e4 e8 = K e6 e5 e6 = e7 e5 e7 = K e6 e5 e8 = 2 e C 2 e2 e6 e7 = e5 e6 e8 = e4 e7 e8 = K e3 (4.2.5) Step 3. The Cartan involution t of su 2 C associated to compact form u We take the complex vectors defining the compact form of su 2 and map them into su 2 C. This gives us the compact part of a Cartan decomposition for su 2 C. T := ie2 T := map(iota CompactBasis); 2 E K 2 ie2 K E5 K E6 K 4 E3 K E4 C 4 ie7 K ie8 K E7 K 4 E8 C ie3 K 4 K 4 E7 K E8 K 4 ie3 C ie4 E3 C 4 E4 C ie7 K 4 ie8 ie4 ie5 K ie6 (4.3.) We take the complex vectors defining the compact of su 2 multiply by I and map into su 2 C. This gives us the
19 non-compact (or positive) part of a Cartan decomposition for su 2 C. su2 > P := map(iota [seq(evaldg(i*c) c = CompactBasis)]); P := K E2 2 E2 C 2 ie K ie5 K ie6 K 4 E7 C E8 K 4 ie3 K ie4 K E3 C 4 E4 K ie7 K 4 4 E3 K E4 K 4 ie7 K ie8 K E7 C 4 E8 C ie3 C 4 ie4 Now we create the transformation which is - on T and on P. ie8 K E5 C E6 (4.3.2) tau := LinearTransformation([seq([x x] x = T) seq([x -x] x = P)]); t := E / E E2 / K E2 E3 / 7 8 E4 C 5 8 ie8 E4 / 7 8 E3 C 5 8 ie7 E5 / E6 E6 / E5 E7 / 7 8 E8 (4.3.3) K 5 8 ie4 E8 / 7 8 E7 K 5 8 ie3 ie / K ie ie2 / ie2 ie3 / 5 8 E8 K 7 8 ie4 ie4 / 5 8 E7 K 7 8 ie3 ie5 / K ie6 ie6 / K ie5 ie7 / K 5 8 E4 K 7 8 ie8 ie8 / K 5 8 E3 K 7 8 ie7 Let's check that t a Cartan involution. Query(tau "CartanInvolution"); (4.3.4) Step 4. The automorphisms y and y / 4 Define the composition y = s + t + s + t. psi := ComposeTransformations(sigma tau sigma tau); y := E / E E2 / E2 E3 / E3 K ie7 E4 / E4 K ie8 E5 / E5 E6 / E6 E7 (4.4.)
20 / 257 / 255 E7 C 255 E8 C 257 ie3 E8 / 257 E8 C 255 ie4 ie5 / ie5 ie6 / ie6 ie7 / K 255 ie4 ie / ie ie2 / ie2 ie3 / 255 E3 C 257 E7 C 257 ie7 ie8 / K 255 ie3 ie4 E4 C 257 ie8 It is shown in reference [] that y has positive eigenvalues. We take the 4-th root of the Jacobian of y and use it to construct f = y / 4. A := Tools:-DGinfo(psi "JacobianMatrix"): phi := LinearTransformation(su2C su2c LinearAlgebra:-MatrixPower(A /4) ); f := E / E E2 / E2 E3 / 5 4 E3 K 3 4 ie7 E4 / 5 4 E4 K 3 4 ie8 E5 / E5 E6 / E6 E7 / 5 4 E7 (4.4.2) C 3 4 ie3 E8 / 5 4 E8 C 3 4 ie4 ie / ie ie2 / ie2 ie3 / 3 4 E7 C 5 4 ie3 ie4 / 3 4 E8 C 5 4 ie4 ie5 / ie5 ie6 / ie6 ie7 / K 3 4 E3 C 5 4 ie7 ie8 / K 3 4 E4 C 5 4 ie8 Check the result : DGequal(psi ComposeTransformations(phi phi phi phi)); (4.4.3) Step 5. A Cartan involution for su(2) We use the transformation y constructed in the previous step to define q = y+t +y K for su 2 C which commutes with s.. This is a Cartan involution theta := ComposeTransformations(phi tau InverseTransformation(phi)); q := E / E E2 / K E2 E3 / E4 E4 / E3 E5 / E6 E6 / E5 E7 / E8 E8 / E7 ie / K ie ie2 / ie2 ie3 / K ie4 ie4 / K ie3 ie5 / K ie6 ie6 / K ie5 ie7 / K ie8 ie8 / K ie7 (4.5.)
21 Check that theta is a Cartan involution for su 2 C. Query(theta "CartanInvolution"); (4.5.2) Check that q commutes with s. DGequal(ComposeTransformations(theta sigma) ComposeTransformations(sigma theta)); (4.5.3) We calculate the restriction of q to su 2. A := convert(theta DGMatrix)[..8..8]; 0 A := 0 K (4.5.4) The restriction to su 2 is the desired Cartan involution. su2 > Theta := LinearTransformation(su2 su2 A); Q := e / e e2 / K e2 e3 / e4 e4 / e3 e5 / e6 e6 / e5 e7 / e8 e8 / e7 (4.5.5) Check this result.
22 su2 > Query(Theta "CartanInvolution"); (4.5.6) The steps --5 are implemented with the command CartanInvolution. Theta := CartanInvolution(CSA RSD PosRts); Q := e / e e2 / K e2 e3 / e4 e4 / e3 e5 / e6 e6 / e5 e7 / e8 e8 / e7 (4.5.7) Highlighted Commands CartanDecomposition CartanInvolution CartanSubalgebra Complexify ComposeTranformations DGRe DGIm DGequal InverseTransformation Killing PositiveRoots SimpleLieAlgebraData SplitAndCompactForms RootSpaceDecomposition LinearTransformation References. A. Cap and Jan Slovak Parabolic Geometrics I - Background and General Theory Mathematical Surveys and Monographs 54 American Mathematical Society (2009). Release Notes This worksheet was compiled with Maple 7 and DG release USU available by request from ian.anderson@usu.edu Author Ian M. Anderson Department of Mathematics and Statistics Utah State University December 8 204
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