A bi-Hamiltonian nature of the Gaudin algebras

DOI10.1016/J.AIM.2022.108805arXiv2105.01020MaRDI QIDQ6366728

Publication date: 3 May 2021

Abstract: Let

m a t h f r a k q

be a Lie algebra over a field

m a t h b b K

and

p, i l d e p i n m a t h b b K [t]

two different normalised polynomials of degree at least 2. As vector spaces both quotient Lie algebras

m a t h f r a k q [t] / (p)

and

m a t h f r a k q [t] / (i l d e p)

can be identified with

. If

m a t h r m d e g, (p - i l d e p)

is at most 1, then the Lie brackets

[,,,,]_{p}

,

[,,,,]_{i l d e p}

induced on

W

by

p

and

i l d e p

, respectively, are compatible. By a general method, known as the Lenard-Magri scheme, we construct a subalgebra

Z = Z (p, i l d e p) s u b s e t m a t h c a l S (W)^{m a t h f r a k q c d o t 1}

such that

{Z, Z}_{p} = {Z, Z}_{i l d e p} = 0

. If

m a t h r m t r . d e g, m a t h c a l S (m a t h f r a k q)^{m a t h f r a k q} = m a t h r m i n d, m a t h f r a k q

and

m a t h f r a k q

has the codim-

2

property, then

m a t h r m t r . d e g, Z

takes the maximal possible value, which is

((n - 1) d i m m a t h f r a k q) / 2 + ((n + 1) m a t h r m i n d, m a t h f r a k q) / 2

. If

m a t h f r a k q = m a t h f r a k g

is semisimple, then

Z

contains the Hamiltonians of a suitably chosen Gaudin model. Therefore, in a non-reductive case, we obtain a completely integrable generalisation of Gaudin models.

Mathematics Subject Classification ID

Infinite-dimensional Lie (super)algebras (17B65) Applications of Lie algebras and superalgebras to integrable systems (17B80) Poisson algebras (17B63)

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