## English

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### Etymology

In the case of category theory (which brings together concepts from numerous fields), the term is often confounded with adjunct and the relationship is called an adjunction. The origin of any particular usage may therefore be uncertain.

### Pronunciation

1. (mathematics) Used in certain contexts, in each case involving a pair of transformations, one of which is, or is analogous to, conjugation (either inner automorphism or complex conjugation).
2. () That is related to another functor by an adjunction.
3. (geometry, of one curve to another curve) Having a relationship of the nature of an adjoint (adjoint curve); sharing multiple points with.
• 1933, H. F. Baker, Principles of Geometry, 2010, Volume 5, page 103,
The sets A + A0, B + B0, together, form the complete intersection, with f = 0, of a composite adjoint curve of order m + k, consisting of the adjoint curve of order m through A + B, together with the non-adjoint curve ω = 0; and the set B + B0 consists of p points, and lies on i + j adjoint φ-curves of f = 0.
• 1963, Julian Lowell Coolidge, A History of Geometrical Methods, page 205,
As we have stated before, a curve ${\displaystyle f'}$ is adjoint to a curve ${\displaystyle f}$ if it have at least the multiplicity ${\displaystyle r_{i}-1}$ at each point where ${\displaystyle f}$ has the multiplicity ${\displaystyle r_{i}}$. A first polar ${\displaystyle \sum _{i}y_{i}\left(\partial f/\partial x_{i}\right)=0}$ is an example of an adjoint curve.
• 2016, Eugene Wachspress, Rational Bases and Generalized Barycentrics: Applications to Finite Elements and Graphics, page 216,
This imposes n(n - 3)/2 conditions on the n-gon adjoint curve.

#### Usage notes

The adjoint operator, or Hermitian transpose, of an operator generalises the concept of transpose conjugate of a matrix. (See Hermitian adjoint on Wikipedia.Wikipedia:Hermitian adjoint)

In the case of an adjoint 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 xgxg-1 for each element g).

The adjoint 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 adjunct and the functors are said to be related by an adjunction. Functors may be left or right adjoint (adjunct).

#### Synonyms

• (mathematics): adjunct (in certain contexts)

### Noun

1. (mathematics) The transpose of the cofactor matrix of a given square matrix.
2. (mathematics, linear algebra, of a matrix) Transpose conjugate.
3. (mathematics, analysis, of an operator) Hermitian conjugate.
4. () A functor related to another functor by an adjunction.
5. (geometry, algebraic geometry) 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.
6. An assistant mayor of a French commune.