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In mathematics, a Rosati involution, named after Carlo Rosati, is an involution of the rational
endomorphism ring In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in ...
of an
abelian variety In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functi ...
induced by a polarization. Let A be an
abelian variety In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functi ...
, let \hat = \mathrm^0(A) be the
dual abelian variety In mathematics, a dual abelian variety can be defined from an abelian variety ''A'', defined over a field ''K''. Definition To an abelian variety ''A'' over a field ''k'', one associates a dual abelian variety ''A''v (over the same field), which i ...
, and for a\in A, let T_a:A\to A be the translation-by-a map, T_a(x)=x+a. Then each divisor D on A defines a map \phi_D:A\to\hat A via \phi_D(a)= _a^*D-D/math>. The map \phi_D is a polarization if D is
ample In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ...
. The Rosati involution of \mathrm(A)\otimes\mathbb relative to the polarization \phi_D sends a map \psi\in\mathrm(A)\otimes\mathbb to the map \psi'=\phi_D^\circ\hat\psi\circ\phi_D, where \hat\psi:\hat A\to\hat A is the dual map induced by the action of \psi^* on \mathrm(A). Let \mathrm(A) denote the
Néron–Severi group In algebraic geometry, the Néron–Severi group of a variety is the group of divisors modulo algebraic equivalence; in other words it is the group of components of the Picard scheme of a variety. Its rank is called the Picard number. It is nam ...
of A. The polarization \phi_D also induces an inclusion \Phi:\mathrm(A)\otimes\mathbb\to\mathrm(A)\otimes\mathbb via \Phi_E=\phi_D^\circ\phi_E. The image of \Phi is equal to \, i.e., the set of endomorphisms fixed by the Rosati involution. The operation E\star F=\frac12\Phi^(\Phi_E\circ\Phi_F+\Phi_F\circ\Phi_E) then gives \mathrm(A)\otimes\mathbb the structure of a formally real
Jordan algebra In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms: # xy = yx (commutative law) # (xy)(xx) = x(y(xx)) (). The product of two elements ''x'' and ''y'' in a Jordan al ...
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References

* *{{Citation , last1=Rosati , first1=Carlo , title=Sulle corrispondenze algebriche fra i punti di due curve algebriche. , language=Italian , doi=10.1007/BF02419717 , year=1918 , journal=Annali di Matematica Pura ed Applicata , volume=3 , issue=28 , pages=35–60, s2cid=121620469 , url=https://zenodo.org/record/2226998 Algebraic geometry