nLab
tautological equivariant line bundle
Contents
Context
Bundles
bundles
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covering space
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retractive space
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fiber bundle, fiber ∞-bundle
numerable bundle
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principal bundle, principal ∞-bundle
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associated bundle, associated ∞-bundle
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vector bundle, 2-vector bundle, (∞,1)-vector bundle
real, complex/holomorphic, quaternionic
topological, differentiable, algebraic
with connection
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bundle of spectra
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natural bundle
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equivariant bundle
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
Contents
Idea
A tautological equivariant line bundle is an equivariant tautological line bundle over a projective G-space.
Definition
Let be a finite group (or maybe a compact Lie group) and let be a -linear representation over some topological ground field , with its projective G-space.
Then the corresponding tautological equivariant line bundle is the -fiber bundle which is associated to the canonical -principal bundle over projective G-space:
and equipped with the induced -action through (which passes to the quotient spaces since the -multiplication action commutes with it, by linearity).
Here
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is the group of units of ;
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denotes the quotient space of a product space by the diagonal action;
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and is equipped with the dual -action
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so that, for ,
Last revised on November 22, 2020 at 18:21:46.
See the history of this page for a list of all contributions to it.