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- W4376645329 abstract "Abstract Lie subalgebras of $$ L = {mathfrak {g}}(!(x)!) times {mathfrak {g}}[x]/x^n{mathfrak {g}}[x] $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>=</mml:mo> <mml:mi>g</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mspace /> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mspace /> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>×</mml:mo> <mml:mi>g</mml:mi> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>x</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> <mml:mo>/</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mi>g</mml:mi> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>x</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , complementary to the diagonal embedding $$Delta $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>Δ</mml:mi> </mml:math> of $$ {mathfrak {g}}[![x]!] $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>[</mml:mo> <mml:mspace /> <mml:mo>[</mml:mo> <mml:mi>x</mml:mi> <mml:mo>]</mml:mo> <mml:mspace /> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> and Lagrangian with respect to some particular form, are in bijection with formal classical r -matrices and topological Lie bialgebra structures on the Lie algebra of formal power series $$ {mathfrak {g}}[![x]!] $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>[</mml:mo> <mml:mspace /> <mml:mo>[</mml:mo> <mml:mi>x</mml:mi> <mml:mo>]</mml:mo> <mml:mspace /> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> . In this work we consider arbitrary subspaces of L complementary to $$Delta $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>Δ</mml:mi> </mml:math> and associate them with so-called series of type ( n , s ) . We prove that Lagrangian subspaces are in bijection with skew-symmetric ( n , s ) -type series and topological quasi-Lie bialgebra structures on $$ {mathfrak {g}}[![x]!] $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>[</mml:mo> <mml:mspace /> <mml:mo>[</mml:mo> <mml:mi>x</mml:mi> <mml:mo>]</mml:mo> <mml:mspace /> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> . Using the classificaiton of Manin pairs we classify up to twisting and coordinate transformations all quasi-Lie bialgebra structures. Series of type ( n , s ) , solving the generalized classical Yang-Baxter equation, correspond to subalgebras of L . We discuss their possible utility in the theory of integrable systems." @default.
- W4376645329 created "2023-05-17" @default.
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- W4376645329 date "2023-05-16" @default.
- W4376645329 modified "2023-10-05" @default.
- W4376645329 title "Topological Manin pairs and $$(n,s)$$-type series" @default.
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- W4376645329 doi "https://doi.org/10.1007/s11005-023-01678-8" @default.
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