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LieAlgebras[Representation] - define a representation of a Lie algebra
LieAlgebras[ApplyRepresentation] - apply a representation of a Lie algebra to a vector in the Lie algebra
Calling Sequences
Representation(A, V, M)
ApplyRepresentation(rho, x, y)
Parameters
A - a Maple name or string, the name of a defined Lie Algebra g
V - a Maple name or string, the name of the defined frame used as the vector space V for the representation
M - a list of matrices [M_1, M_2, M_3, ...] each defining a linear transformation from V to V
rho - a representation of a Lie algebra g on a vector space V, as constructed with the Representation command
x - a vector in g
y - (optional) a vector in V
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Description
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Let g be a Lie algebra and V a vector space. A representation of g on V is a linear transformation rho: g -> gl(V) (where gl(V) is the Lie algebra of all linear transformations on V) such that rho([x, y]) = [rho(x), rho(y)] (*) for all x, y in g. Here [x, y] denotes the Lie bracket of two vectors in g while [rho(x), rho(y)] denotes the commutator of two linear transformations in gl(V).
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The command Representation(A, V, M) creates a data structure which defines the representation rho : g -> gl(V) such that rho(e_i) = M_i.
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The command Query can be used to verify that the map rho does in fact satisfy the representation property (*).
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If x = c_1e_1 + c_2e_2 + ..., then ApplyRepresentation(rho, x) returns the matrix rho(x) = c_1M1 + c_2M2 + .... If y is vector in the representation space V, then Representation(rho, x, y) returns the vector rho(x)(y) in V.
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Examples
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Example 1.
First we retrieve a Lie algebra from the DifferentialGeometry Library and initialize it.
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![L := Retrieve("Winternitz", 1, [4, 7], Alg1)](/support/helpjp/helpview.aspx?si=5605/file05829/math156.png)
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![_DG([["LieAlgebra", Alg1, [4]], [[[2, 3, 1], 1], [[1, 4, 1], 2], [[2, 4, 2], 1], [[3, 4, 2], 1], [[3, 4, 3], 1]]])](/support/helpjp/helpview.aspx?si=5605/file05829/math159.png)
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The center of Alg1 is trivial and therefore the adjoint representation is faithful. The command Adjoint will return the list of adjoint matrices for the Lie algebra Alg1.
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Alg1 >
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![[]](/support/helpjp/helpview.aspx?si=5605/file05829/math180.png)
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Alg1 >
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To define a representation of Alg1 using the matrices M, we shall need to define a representation space V.
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Alg1 >
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![DGsetup([x1, x2, x3, x4], V)](/support/helpjp/helpview.aspx?si=5605/file05829/math201.png)
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V >
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| (2.4) |
We can verify that this is really a representation of Alg1 with the Query command.
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Alg1 >
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Apply the representation to some vectors in Alg1:
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Alg1 >
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Alg1 >
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| (2.7) |
Apply the linear transformations to some vectors in V:
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Alg1 >
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![_DG([["vector", V, []], [[[1], 2]]])](/support/helpjp/helpview.aspx?si=5605/file05829/math256.png)
| (2.8) |
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V >
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![_DG([["vector", V, []], [[[1], 2], [[2], -3], [[3], -3]]])](/support/helpjp/helpview.aspx?si=5605/file05829/math263.png)
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Example 2.
The infinitesimal automorphisms or derivations of a Lie algebra g define a matrix Lie algebra which is automatically a representation. In this case we can take g to be the representation space.
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V >
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![L := Retrieve("Winternitz", 1, [3, 2], Alg2)](/support/helpjp/helpview.aspx?si=5605/file05829/math277.png)
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![_DG([["LieAlgebra", Alg2, [3]], [[[1, 3, 1], 1], [[2, 3, 1], 1], [[2, 3, 2], 1]]])](/support/helpjp/helpview.aspx?si=5605/file05829/math280.png)
| (2.10) |
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V >
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The Derivations command calculates the derivations on the Lie algebra Alg2.
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Alg2 >
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We calculate the structure equations for this matrix algebra and initialize the result.
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Alg2 >
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![_DG([["LieAlgebra", Alg2a, [4, table( [ ] )]], [[[1, 3, 3], 1], [[1, 4, 4], 1], [[2, 4, 3], 1]]])](/support/helpjp/helpview.aspx?si=5605/file05829/math311.png)
| (2.12) |
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Alg2 >
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| (2.13) |
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Alg2a >
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| (2.14) |
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Alg2a >
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| (2.15) |
Example 3.
If Gamma is a Lie algebra of vector fields on a manifold M, then the isotropy subalgebra at a given point admits a natural representation, defined by the Lie bracket, on the tangent space of M. To illustrate the definition of this representation, we first obtain a Lie algebra of vector fields from the DifferentialGeometry Library with the Retrieve command. Then we use the IsotropySubalgebra from the GroupActions package to calculate the isotropy subalgebra and its representation.
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Alg2a >
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![DGsetup([x, y], "M")](/support/helpjp/helpview.aspx?si=5605/file05829/math360.png)
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M >
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![GAMMA := Retrieve("Gonzalez-Lopez", 1, [7], manifold = "M")](/support/helpjp/helpview.aspx?si=5605/file05829/math364.png)
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![[_DG([["vector", "M", []], [[[1], 1]]]), _DG([["vector", "M", []], [[[2], 1]]]), _DG([["vector", "M", []], [[[1], x], [[2], y]]]), _DG([["vector", "M", []], [[[1], y], [[2], -x]]]), _DG([["vector", "M", []], [[[1], x^2-y^2], [[2], 2*x*y]]]), _DG([["vector", "M", []], [[[1], 2*x*y], [[2], y^2-x^2]]])]](/support/helpjp/helpview.aspx?si=5605/file05829/math367.png)
| (2.16) |
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M >
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![IsoAlg, IsoRep := IsotropySubalgebra(GAMMA, [x = 1, y = 2], output = ["Vector", "Representation"])](/support/helpjp/helpview.aspx?si=5605/file05829/math371.png)
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| (2.17) |
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M >
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![_DG([["LieAlgebra", Alg3, [4]], [[[1, 3, 1], 1], [[1, 3, 3], -2/5], [[1, 3, 4], -4/5], [[1, 4, 2], -1], [[1, 4, 3], 4/5], [[1, 4, 4], -2/5], [[2, 3, 2], 1], [[2, 3, 3], -4/5], [[2, 3, 4], 2/5], [[2, 4, 1], 1], [[2, 4, 3], -2/5], [[2, 4, 4], -4/5]]])](/support/helpjp/helpview.aspx?si=5605/file05829/math385.png)
| (2.18) |
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M >
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Alg3 >
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![rho3 := [[e1, Matrix(2, 2, {(1, 1) = 2/5, (1, 2) = 4/5, (2, 1) = -4/5, (2, 2) = 2/5})], [e2, Matrix(2, 2, {(1, 1) = 4/5, (1, 2) = -2/5, (2, 1) = 2/5, (2, 2) = 4/5})], [e3, Matrix(2, 2, {(1, 1) = 1, (1, 2) = 0, (2, 1) = 0, (2, 2) = 1})], [e4, Matrix(2, 2, {(1, 1) = 0, (1, 2) = 1, (2, 1) = -1, (2, 2) = 0})]]](/support/helpjp/helpview.aspx?si=5605/file05829/math396.png)
| (2.19) |
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Alg3 >
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| (2.20) |
Example 4.
Here is 4 dimensional faithful representation of an indecomposable 4-dimensional Lie algebra with a center.
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Alg3 >
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![L4 := LieAlgebraData([[x2, x4] = x1, [x3, x4] = x3], [x1, x2, x3, x4], Alg4)](/support/helpjp/helpview.aspx?si=5605/file05829/math413.png)
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![_DG([["LieAlgebra", Alg4, [4, table( [ ] )]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]])](/support/helpjp/helpview.aspx?si=5605/file05829/math416.png)
| (2.21) |
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M >
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| (2.22) |
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Alg4 >
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![M4 := [Matrix([[0, 0, 0, 0], [0, 0, 0, -1], [0, 0, 0, 0], [0, 0, 0, 0]]), Matrix([[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -1], [0, 0, 0, 0]]), Matrix([[0, 0, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]), Matrix([[-1, 0, 0, 0], [0, 0, -1, 0], [0, 0, 0, 0], [0, 0, 0, 0]])]](/support/helpjp/helpview.aspx?si=5605/file05829/math427.png)
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| (2.23) |
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Alg4 >
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![DGsetup([x1, x2, x3, x4], V)](/support/helpjp/helpview.aspx?si=5605/file05829/math434.png)
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V >
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Alg4 >
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| (2.25) |
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