In mathematics, an oper is a principal connection, or in more elementary terms a type of differential operator. They were first defined and used by Vladimir Drinfeld and Vladimir Sokolov to study how the KdV equation and related integrable PDEs correspond to algebraic structures known as Kac–Moody algebras. Their modern formulation is due to Drinfeld and Alexander Beilinson.
History Opers were first defined, although not named, in a 1981 Russian paper by Drinfeld and Sokolov on Equations of Korteweg–de Vries type, and simple Lie algebras. They were later generalized by Drinfeld and Beilinson in 1993, later published as an e-print in 2005.
Formulation
Abstract Let G {\displaystyle G} be a connected reductive group over the complex plane C {\displaystyle \mathbb {C} } , with a distinguished Borel subgroup B = B G ⊂ G {\displaystyle B=B_{G}\subset G} . Set N = [ B , B ] {\displaystyle N=[B,B]} , so that H = B / N {\displaystyle H=B/N} is the Cartan group. Denote by n < b < g {\displaystyle {\mathfrak {n}}<{\mathfrak {b}}<{\mathfrak {g}}} and h = b / n {\displaystyle {\mathfrak {h}}={\mathfrak {b}}/{\mathfrak {n}}} the corresponding Lie algebras. There is an open B {\displaystyle B} -orbit O {\displaystyle \mathbf {O} } consisting of vectors stabilized by the radical N ⊂ B {\displaystyle N\subset B} such that all of their negative simple-root components are non-zero. Let X {\displaystyle X} be a smooth curve. A G-oper on X {\displaystyle X} is a triple ( F , ∇ , F B ) {\displaystyle ({\mathfrak {F}},\nabla ,{\mathfrak {F}}_{B})} where F {\displaystyle {\mathfrak {F}}} is a principal G {\displaystyle G} -bundle, ∇ {\displaystyle \nabla } is a connection on F {\displaystyle {\mathfrak {F}}} and F B {\displaystyle {\mathfrak {F}}_{B}} is a B {\displaystyle B} -reduction of F {\displaystyle {\mathfrak {F}}} , such that the one-form ∇ / F B {\displaystyle \nabla /{\mathfrak {F}}_{B}} takes values in O F B {\displaystyle \mathbf {O} _{{\mathfrak {F}}_{B}}} .
Example Fix X = P 1 = C P 1 {\displaystyle X=\mathbb {P} ^{1}=\mathbb {CP} ^{1}} the Riemann sphere. Working at the level of the algebras, fix g = s l ( 2 , C ) {\displaystyle {\mathfrak {g}}={\mathfrak {sl}}(2,\mathbb {C} )} , which can be identified with the space of traceless 2 × 2 {\displaystyle 2\times 2} complex matrices. Since P 1 {\displaystyle \mathbb {P} ^{1}} has only one (complex) dimension, a one-form has only one component, and so an s l ( 2 , C ) {\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )} -valued one form is locally described by a matrix of functions
A ( z ) = ( a ( z ) b ( z ) c ( z ) − a ( z ) ) {\displaystyle A(z)={\begin{pmatrix}a(z)&b(z)\\c(z)&-a(z)\end{pmatrix}}}
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