Let ,
, ... be operators. Then the
commutator of
and
is defined as
(1)
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Let ,
, ... be constants, then identities include
(2)
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(3)
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(4)
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(5)
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(6)
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(7)
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(8)
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Let and
be tensors. Then
(9)
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There is a related notion of commutator in the theory of groups. The commutator of two group elements and
is
, and two elements
and
are said to commute when their
commutator is the identity element. When the
group is a Lie group, the Lie
bracket in its Lie algebra is an infinitesimal
version of the group commutator. For instance, let
and
be square matrices, and let
and
be paths in the Lie group
of nonsingular matrices which satisfy
(10)
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(11)
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(12)
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then
(13)
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