nLab
Lie algebra representation
Redirected from "Lie action".
Contents
Context
Representation theory
representation theory
geometric representation theory
Ingredients
representation, 2-representation, ∞-representation
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group, ∞-group
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group algebra, algebraic group, Lie algebra
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vector space, n-vector space
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affine space, symplectic vector space
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action, ∞-action
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module, equivariant object
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bimodule, Morita equivalence
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induced representation, Frobenius reciprocity
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Hilbert space, Banach space, Fourier transform, functional analysis
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orbit, coadjoint orbit, Killing form
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unitary representation
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geometric quantization, coherent state
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socle, quiver
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module algebra, comodule algebra, Hopf action, measuring
Geometric representation theory
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D-module, perverse sheaf,
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Grothendieck group, lambda-ring, symmetric function, formal group
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principal bundle, torsor, vector bundle, Atiyah Lie algebroid
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geometric function theory, groupoidification
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Eilenberg-Moore category, algebra over an operad, actegory, crossed module
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reconstruction theorems
Lie theory
∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
Contents
Idea
The representation/action of a Lie algebra on a vector space .
Definition
A Lie algebra representation is a Lie algebra homomorphism to the endomorphism Lie algebra of .
This means equivalently that
is a bilinear map such that for all and we have the Lie action property:
(1)
Properties
In terms of string diagrams / Jacobi diagrams
In string diagram-notation for Lie algebra objects internal to tensor categories, the Lie action property (1) looks as follows:
Here the last line shows the equivalence to the Jacobi identity on the Lie algebra object itself in the case that the Lie action is the adjoint action.
In the language of Jacobi diagrams this is called the STU-relation, and is the reason behind the existence of Lie algebra weight systems in knot theory.
References
See also
Last revised on May 1, 2021 at 08:19:41.
See the history of this page for a list of all contributions to it.