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Lie algebra–valued differential form

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In differential geometry, a Lie-algebra-valued form is a differential form with values in a Lie algebra. Such forms have important applications in the theory of connections on a principal bundle as well as in the theory of Cartan connections.

Formal definition

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A Lie-algebra-valued differential -form on a manifold, , is a smooth section of the bundle , where is a Lie algebra, is the cotangent bundle of and denotes the exterior power.

Wedge product

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The wedge product of ordinary, real-valued differential forms is defined using multiplication of real numbers. For a pair of Lie algebra–valued differential forms, the wedge product can be defined similarly, but substituting the bilinear Lie bracket operation, to obtain another Lie algebra–valued form. For a -valued -form and a -valued -form , their wedge product is given by

where the 's are tangent vectors. The notation is meant to indicate both operations involved. For example, if and are Lie-algebra-valued one forms, then one has

The operation can also be defined as the bilinear operation on satisfying

for all and .

Some authors have used the notation instead of . The notation , which resembles a commutator, is justified by the fact that if the Lie algebra is a matrix algebra then is nothing but the graded commutator of and , i. e. if and then

where are wedge products formed using the matrix multiplication on .

Operations

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Let be a Lie algebra homomorphism. If is a -valued form on a manifold, then is an -valued form on the same manifold obtained by applying to the values of : .

Similarly, if is a multilinear functional on , then one puts[1]

where and are -valued -forms. Moreover, given a vector space , the same formula can be used to define the -valued form when

is a multilinear map, is a -valued form and is a -valued form. Note that, when

giving amounts to giving an action of on ; i.e., determines the representation

and, conversely, any representation determines with the condition . For example, if (the bracket of ), then we recover the definition of given above, with , the adjoint representation. (Note the relation between and above is thus like the relation between a bracket and .)

In general, if is a -valued -form and is a -valued -form, then one more commonly writes when . Explicitly,

With this notation, one has for example:

.

Example: If is a -valued one-form (for example, a connection form), a representation of on a vector space and a -valued zero-form, then

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Forms with values in an adjoint bundle

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Let be a smooth principal bundle with structure group and . acts on via adjoint representation and so one can form the associated bundle:

Any -valued forms on the base space of are in a natural one-to-one correspondence with any tensorial forms on of adjoint type.

See also

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Notes

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  1. ^ S. Kobayashi, K. Nomizu. Foundations of Differential Geometry (Wiley Classics Library) Volume 1, 2. Chapter XII, § 1.}}
  2. ^ Since , we have that
    is

References

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