Adjoint
English
Etymology
From Template:Der, from late 19th C; see also Template:M. Template:Doublet.
In the case of category theory (which brings together concepts from numerous fields), the term is often confounded with Template:M and the relationship is called an Template:M. The origin of any particular usage may therefore be uncertain.
Pronunciation
Adjective
- Template:Lb Template:Non-gloss
- Template:Lb That is related to another functor by an adjunction.
- Template:Lb Having a relationship of the nature of an adjoint (adjoint curve); sharing multiple points with.
Usage notes
The adjoint operator, or Hermitian transpose, of an operator generalises the concept of transpose conjugate of a matrix. (See Template:Pedia)
In the case of an Template:W representation of a Lie group, the representation in question describes the group's elements as linear transformations of its Lie algebra, itself considered as a vector space. The representation is obtained by differentiating ("linearising") the group action of conjugation (i.e., differentiating the function x → gxg-1 for each element g).
The Template:W representation of a Lie algebra is the differential of the adjoint representation of a Lie group at the identity element of the group.
In relation to functors in category theory (and therefore in numerous fields of mathematics), the term is often synonymous with Template:M and the functors are said to be related by an Template:M. Functors may be left or right adjoint (adjunct).
Synonyms
Derived terms
Related terms
Translations
- Esperanto: Template:T
- German: Template:T+
- Spanish: Template:T+
Noun
- Template:Lb The transpose of the cofactor matrix of a given square matrix.
- Template:Lb Transpose conjugate.
- Template:Lb Hermitian conjugate.
- Template:Lb A functor related to another functor by an adjunction.
- Template:Lb A curve A such that any point of a given curve C of multiplicity r has multiplicity at least r–1 on A. Sometimes the multiple points of C are required to be ordinary, and if this condition is not satisfied the term sub-adjoint is used.
- An assistant to someone who holds a position in the military or civil service.
- An assistant mayor of a French commune.
Derived terms
Translations
- German: Template:T
- Spanish: Template:T, Template:T
Template:Rfc Template:Trans-top
- Bulgarian: Template:T+check, Template:T+check
- Slovak: Template:T-check