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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(${\mathbf{θ}}$, ${\mathbf{ρ}}$)

Parameters

$\mathrm{θ}$    - a transformation, defining a Cartan involution on a semi-simple Lie algebra $\mathrm{𝔤}$

$\mathrm{ρ}$    - a representation of $\mathrm{𝔤}\mathit{.}$

Description

 • Let $\mathrm{𝔤}$ be a semi-simple Lie algebra with Killing form and Cartan involution By definition, the inner product is positive-definite and satisfies, by the Jacobi identity, This situation generalizes to any representation space of $\mathrm{𝔤}$. Specifically, there always exists on a positive-definite inner product such that

for all  and .

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 which satisfies (*).

Examples

 > with(DifferentialGeometry): with(LieAlgebras):

Example 1.

We illustrate the command PositiveDefiniteMetricOnRepresentationSpace for the standard representation for $\mathrm{sl}\left(3\right)$. We use SimpleLieAlgebraData and DGsetup to initialize this Lie algebra.

 ${\mathrm{LD}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{-}{2}{}{\mathrm{e8}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{-}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e2}}\right]$ (2.1)
 > DGsetup(LD);
 ${\mathrm{Lie algebra: sl3}}$ (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);
 ${\mathrm{frame name: V}}$ (2.3)
 V > rho := StandardRepresentation(sl3, representationspace = V);
 ${\mathrm{ρ}}{:=}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{rrr}{1}& {0}& {0}\\ {0}& {0}& {0}\\ {0}& {0}& {-}{1}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{rrr}{0}& {0}& {0}\\ {0}& {1}& {0}\\ {0}& {0}& {-}{1}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{rrr}{0}& {1}& {0}\\ {0}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e4}}{,}\left[\begin{array}{rrr}{0}& {0}& {1}\\ {0}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e5}}{,}\left[\begin{array}{rrr}{0}& {0}& {0}\\ {1}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e6}}{,}\left[\begin{array}{rrr}{0}& {0}& {0}\\ {0}& {0}& {1}\\ {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e7}}{,}\left[\begin{array}{rrr}{0}& {0}& {0}\\ {0}& {0}& {0}\\ {1}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e8}}{,}\left[\begin{array}{rrr}{0}& {0}& {0}\\ {0}& {0}& {0}\\ {0}& {1}& {0}\end{array}\right]\right]\right]$ (2.4)
 sl3 > theta := SimpleLieAlgebraProperties(sl3)["CartanInvolution"];
 ${\mathrm{θ}}{:=}\left[\left[{\mathrm{e1}}{,}{-}{\mathrm{e1}}\right]{,}\left[{\mathrm{e2}}{,}{-}{\mathrm{e2}}\right]{,}\left[{\mathrm{e3}}{,}{-}{\mathrm{e5}}\right]{,}\left[{\mathrm{e4}}{,}{-}{\mathrm{e7}}\right]{,}\left[{\mathrm{e5}}{,}{-}{\mathrm{e3}}\right]{,}\left[{\mathrm{e6}}{,}{-}{\mathrm{e8}}\right]{,}\left[{\mathrm{e7}}{,}{-}{\mathrm{e4}}\right]{,}\left[{\mathrm{e8}}{,}{-}{\mathrm{e6}}\right]\right]$ (2.5)

We find that the standard Euclidean metric on is the $\mathrm{θ}$-compatible metric.

 sl3 > PositiveDefiniteMetricOnRepresentationSpace(theta, rho);
 ${\mathrm{dx1}}{}{\mathrm{dx1}}{+}{\mathrm{dx2}}{}{\mathrm{dx2}}{+}{\mathrm{dx3}}{}{\mathrm{dx3}}$ (2.6)

Example 2.

In this example, we consider the adjoint representation for $\mathrm{sl}\left(3\right)$.

 sl3 > DGsetup([y1, y2, y3, y4, y5, y6, y7, y8], W);
 ${\mathrm{frame name: W}}$ (2.7)
 V > chi := Adjoint(sl3, representationspace = W);
 ${\mathrm{χ}}{:=}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{rrrrrrrr}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {1}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {2}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {-}{1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {-}{2}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {-}{1}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{rrrrrrrr}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {-}{1}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {2}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {-}{1}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {-}{2}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{rrrrrrrr}{0}& {0}& {0}& {0}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {-}{1}& {0}& {0}& {0}\\ {-}{1}& {1}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {-}{1}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e4}}{,}\left[\begin{array}{rrrrrrrr}{0}& {0}& {0}& {0}& {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {1}\\ {-}{2}& {-}{1}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {-}{1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e5}}{,}\left[\begin{array}{rrrrrrrr}{0}& {0}& {-}{1}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {1}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {1}& {-}{1}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e6}}{,}\left[\begin{array}{rrrrrrrr}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {-}{1}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {1}& {0}\\ {-}{1}& {-}{2}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e7}}{,}\left[\begin{array}{rrrrrrrr}{0}& {0}& {0}& {-}{1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {-}{1}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {2}& {1}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {1}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e8}}{,}\left[\begin{array}{rrrrrrrr}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {-}{1}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {1}& {0}& {0}& {0}\\ {1}& {2}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]\right]$ (2.8)

 sl3 > Q := PositiveDefiniteMetricOnRepresentationSpace(theta, chi);
 ${Q}{:=}{2}{}{\mathrm{dy1}}{}{\mathrm{dy1}}{+}{\mathrm{dy1}}{}{\mathrm{dy2}}{+}{\mathrm{dy2}}{}{\mathrm{dy1}}{+}{2}{}{\mathrm{dy2}}{}{\mathrm{dy2}}{+}{\mathrm{dy3}}{}{\mathrm{dy3}}{+}{\mathrm{dy4}}{}{\mathrm{dy4}}{+}{\mathrm{dy5}}{}{\mathrm{dy5}}{+}{\mathrm{dy6}}{}{\mathrm{dy6}}{+}{\mathrm{dy7}}{}{\mathrm{dy7}}{+}{\mathrm{dy8}}{}{\mathrm{dy8}}$ (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);
 ${12}{}{\mathrm{dy1}}{}{\mathrm{dy1}}{+}{6}{}{\mathrm{dy1}}{}{\mathrm{dy2}}{+}{6}{}{\mathrm{dy2}}{}{\mathrm{dy1}}{+}{12}{}{\mathrm{dy2}}{}{\mathrm{dy2}}{+}{6}{}{\mathrm{dy3}}{}{\mathrm{dy3}}{+}{6}{}{\mathrm{dy4}}{}{\mathrm{dy4}}{+}{6}{}{\mathrm{dy5}}{}{\mathrm{dy5}}{+}{6}{}{\mathrm{dy6}}{}{\mathrm{dy6}}{+}{6}{}{\mathrm{dy7}}{}{\mathrm{dy7}}{+}{6}{}{\mathrm{dy8}}{}{\mathrm{dy8}}$ (2.10)