In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type
:(''Ax'', ''y'') = (''x'', ''By'').
Specifically, adjoint or adjunction may mean:
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Adjoint of a linear map, also called its transpose
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Hermitian adjoint
In mathematics, specifically in operator theory, each linear operator A on a Euclidean vector space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule
:\langle Ax,y \rangle = \langle x,A^*y \rangle,
wher ...
(adjoint of a linear operator) in functional analysis
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Adjoint endomorphism of a Lie algebra
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Adjoint representation of a Lie group
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL( ...
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Adjoint functors
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are k ...
in category theory
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Adjunction (field theory)
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
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Adjunction formula (algebraic geometry) In mathematics, especially in algebraic geometry and the theory of complex manifolds, the adjunction formula relates the canonical bundle of a variety and a hypersurface inside that variety. It is often used to deduce facts about varieties embedd ...
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Adjunction space In mathematics, an adjunction space (or attaching space) is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let ''X'' and ''Y'' be topological spaces, and let ''A'' be a subspace of ' ...
in topology
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Conjugate transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex c ...
of a matrix in linear algebra
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Adjugate matrix
In linear algebra, the adjugate or classical adjoint of a square matrix is the transpose of its cofactor matrix and is denoted by . It is also occasionally known as adjunct matrix, or "adjoint", though the latter today normally refers to a differ ...
, related to its inverse
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Adjoint equation
* The upper and lower adjoints of a
Galois connection In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the fu ...
in order theory
* The adjoint of a
differential operator with general polynomial coefficients
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Kleisli adjunction
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Monoidal adjunction
Suppose that (\mathcal C,\otimes,I) and (\mathcal D,\bullet,J) are two monoidal categories. A monoidal adjunction between two lax monoidal functors
:(F,m):(\mathcal C,\otimes,I)\to (\mathcal D,\bullet,J) and (G,n):(\mathcal D,\bullet,J)\to(\math ...
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Quillen adjunction
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Axiom of adjunction in set theory
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Adjunction (rule of inference)
Conjunction introduction (often abbreviated simply as conjunction and also called and introduction or adjunction) is a valid rule of inference of propositional logic. The rule makes it possible to introduce a conjunction into a logical proof. I ...
{{sia, mathematics
Mathematical terminology