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LieAlgebras[PositiveDefiniteMetricOnRepresentationSpace] - find a positive-definite inner product on a representation space which is compatible with a Cartan involution

Calling Sequences

     PositiveDefiniteMetricOnRepresentationSpace(θ, ρ)

Parameters

     θ    - a transformation, defining a Cartan involution θ:𝔤 𝔤 on a semi-simple Lie algebra 𝔤

     ρ    - a representation ρ: 𝔤  glV of 𝔤.

  

 

Description

 Examples

Description

• 

 Let 𝔤 be a semi-simple Lie algebra with Killing form B and Cartan involution θ.  By definition, the inner product Qy, z =By, θz is positive-definite and satisfies, by the Jacobi identity, Qx, y, z + Qy,θx,z = 0.  This situation generalizes to any representation space V of 𝔤. Specifically, there always exists on V a positive-definite inner product Q such that

Qxu, v + Qu, θx v =0  for all x  𝔤 and u, v  V.   (*) 

This inner product is unique (apart from an overall factor) when the representation V is irreducible.

• 

The calling sequence PositiveDefiniteMetricOnRepresentationSpace(theta, rho) returns the most general quadratic form Q on V which satisfies (*).

 Examples

with(DifferentialGeometry): with(LieAlgebras):

 

Example 1.

We illustrate the command PositiveDefiniteMetricOnRepresentationSpace for the standard representation for sl3. We use SimpleLieAlgebraData and DGsetup to initialize this Lie algebra.

LD := SimpleLieAlgebraData("sl(3)", sl3);

LD:=e1,e3=e3,e1,e4=2e4,e1,e5=e5,e1,e6=e6,e1,e7=2e7,e1,e8=e8,e2,e3=e3,e2,e4=e4,e2,e5=e5,e2,e6=2e6,e2,e7=e7,e2,e8=2e8,e3,e5=e1e2,e3,e6=e4,e3,e7=e8,e4,e5=e6,e4,e7=e1,e4,e8=e3,e5,e8=e7,e6,e7=e5,e6,e8=e2

(2.1)

DGsetup(LD);

Lie algebra: sl3

(2.2)

 

For Lie algebras created by the SimpleLieAlgebraData command, the standard representation and Cartan involution can be obtained from the commands StandardRepresentation and SimpleLieAlgebraProperties. First we define our representation space, the representation and the Cartan involution.

sl3 > 

DGsetup([x1, x2, x3], V);

frame name: V

(2.3)
V > 

rho := StandardRepresentation(sl3, representationspace = V);

ρ:=e1,100000001,e2,000010001,e3,010000000,e4,001000000,e5,000100000,e6,000001000,e7,000000100,e8,000000010

(2.4)
sl3 > 

theta := SimpleLieAlgebraProperties(sl3)["CartanInvolution"];

θ:=e1,e1,e2,e2,e3,e5,e4,e7,e5,e3,e6,e8,e7,e4,e8,e6

(2.5)

 

We find that the standard Euclidean metric on V is the θ-compatible metric.

sl3 > 

PositiveDefiniteMetricOnRepresentationSpace(theta, rho);

dx1dx1+dx2dx2+dx3dx3

(2.6)

 

Example 2.

In this example, we consider the adjoint representation for sl3.

sl3 > 

DGsetup([y1, y2, y3, y4, y5, y6, y7, y8], W);

frame name: W

(2.7)
V > 

chi := Adjoint(sl3, representationspace = W);

χ:=e1,0000000000000000001000000002000000001000000001000000002000000001,e2,0000000000000000001000000001000000001000000002000000001000000002,e3,0000100000001000110000000000010000000000000000000000000000000010,e4,0000001000000000000000012100000000000000000010000000000000000000,e5,0010000000100000000000000000000011000000000100000000000100000000,e6,0000000000000001000000000010000000000010120000000000000000000000,e7,0001000000000000000000000000000000000100000000002100000000100000,e8,0000000000000100000100000000000000000000000000000000100012000000

(2.8)

 

sl3 > 

Q := PositiveDefiniteMetricOnRepresentationSpace(theta, chi);

Q:=2dy1dy1+dy1dy2+dy2dy1+2dy2dy2+dy3dy3+dy4dy4+dy5dy5+dy6dy6+dy7dy7+dy8dy8

(2.9)

 

Apart from a numerical factor this coincides with the metric defined by the product of the Killing form and the matrix defining the Cartan involution.

V > 

K := Killing(sl3):

W > 

J := Tools:-DGinfo(theta, "JacobianMatrix"):

sl3 > 

convert(-K.J, DGtensor, [["cov_bas", "cov_bas"],[]], W);

12dy1dy1+6dy1dy2+6dy2dy1+12dy2dy2+6dy3dy3+6dy4dy4+6dy5dy5+6dy6dy6+6dy7dy7+6dy8dy8

(2.10)

See Also

DifferentialGeometry

LieAlgebras

CartanInvolution

Killing

SimpleLieAlgebraData

SimpleLieAlgebraProperties