Centralizer construction of the Yangian of the queer Lie superalgebra

Maxim Nazarov; Alexander Sergeev

doi:10.1007/0-8176-4478-4_17

Centralizer construction of the Yangian of the queer Lie superalgebra

Maxim Nazarov, Alexander Sergeev

Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Other chapter contribution

Abstract

Consider the complex matrix Lie superalgebra $ \mathfrak{g}\mathfrak{l}_{\left. N \right|N} $Unknown control sequence '\mathfrak' with the standard generators E ij where i, j = ±1, . . . , ± N. Define an involutive automorphism ¿ of $ \mathfrak{g}\mathfrak{l}_{\left. N \right|N} $Unknown control sequence '\mathfrak' by ¿(E ij) = E -i,-j . The queer Lie superalgebra qN is the fixed point subalgebra in $ \mathfrak{g}\mathfrak{l}_{\left. N \right|N} $Unknown control sequence '\mathfrak' relative to ¿. Consider the twisted polynomial current Lie superalgebra
$ \mathfrak{g} = \left\{ {X\left( t \right) \in \mathfrak{g}\mathfrak{l}_{\left. N \right|N} \left[ t \right]:\eta \left( {X\left( t \right)} \right) = X\left( { - t} \right)} \right\} $Unknown control sequence '\mathfrak'
. The enveloping algebra U( $ \mathfrak{g} $Unknown control sequence '\mathfrak' ) of the Lie superalgebra g has a deformation, called the Yangian of qN. For each M = 1,2, . . . , denote by A N M the centralizer of qM ¿ q N+M in the associative superalgebra U(q N+M ). In this article we construct a sequence of surjective homomorphisms U(qN) ¿ A N 1 ¿ A N 2 ¿ . . . . We describe the inverse limit of the sequence of centralizer algebras A N 1 , A N 2 , . . . in terms of the Yangian of qN.

Original language	English
Title of host publication	Studies in Lie Theory: Dedicated to A. Joseph on his Sixtieth Birthday
Subtitle of host publication	Progress in Mathematics
Pages	417-441
Number of pages	25
Volume	243
Edition	Part II
ISBN (Electronic)	978-0-8176-4478-9
DOIs	https://doi.org/10.1007/0-8176-4478-4_17
Publication status	Published - 2006

Keywords

Algebra

Access to Document

10.1007/0-8176-4478-4_17

http://arxiv.org/abs/math.RT/0404086

Cite this

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title = "Centralizer construction of the Yangian of the queer Lie superalgebra",

abstract = "Consider the complex matrix Lie superalgebra $ \mathfrak{g}\mathfrak{l}_{\left. N \right|N} $Unknown control sequence '\mathfrak' with the standard generators E ij where i, j = ±1, . . . , ± N. Define an involutive automorphism ¿ of $ \mathfrak{g}\mathfrak{l}_{\left. N \right|N} $Unknown control sequence '\mathfrak' by ¿(E ij) = E -i,-j . The queer Lie superalgebra qN is the fixed point subalgebra in $ \mathfrak{g}\mathfrak{l}_{\left. N \right|N} $Unknown control sequence '\mathfrak' relative to ¿. Consider the twisted polynomial current Lie superalgebra$ \mathfrak{g} = \left\{ {X\left( t \right) \in \mathfrak{g}\mathfrak{l}_{\left. N \right|N} \left[ t \right]:\eta \left( {X\left( t \right)} \right) = X\left( { - t} \right)} \right\} $Unknown control sequence '\mathfrak'. The enveloping algebra U( $ \mathfrak{g} $Unknown control sequence '\mathfrak' ) of the Lie superalgebra g has a deformation, called the Yangian of qN. For each M = 1,2, . . . , denote by A N M the centralizer of qM ¿ q N+M in the associative superalgebra U(q N+M ). In this article we construct a sequence of surjective homomorphisms U(qN) ¿ A N 1 ¿ A N 2 ¿ . . . . We describe the inverse limit of the sequence of centralizer algebras A N 1 , A N 2 , . . . in terms of the Yangian of qN. ",

keywords = "Algebra",

author = "Maxim Nazarov and Alexander Sergeev",

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doi = "10.1007/0-8176-4478-4_17",

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TY - CHAP

T1 - Centralizer construction of the Yangian of the queer Lie superalgebra

AU - Nazarov, Maxim

AU - Sergeev, Alexander

PY - 2006

Y1 - 2006

N2 - Consider the complex matrix Lie superalgebra $ \mathfrak{g}\mathfrak{l}_{\left. N \right|N} $Unknown control sequence '\mathfrak' with the standard generators E ij where i, j = ±1, . . . , ± N. Define an involutive automorphism ¿ of $ \mathfrak{g}\mathfrak{l}_{\left. N \right|N} $Unknown control sequence '\mathfrak' by ¿(E ij) = E -i,-j . The queer Lie superalgebra qN is the fixed point subalgebra in $ \mathfrak{g}\mathfrak{l}_{\left. N \right|N} $Unknown control sequence '\mathfrak' relative to ¿. Consider the twisted polynomial current Lie superalgebra$ \mathfrak{g} = \left\{ {X\left( t \right) \in \mathfrak{g}\mathfrak{l}_{\left. N \right|N} \left[ t \right]:\eta \left( {X\left( t \right)} \right) = X\left( { - t} \right)} \right\} $Unknown control sequence '\mathfrak'. The enveloping algebra U( $ \mathfrak{g} $Unknown control sequence '\mathfrak' ) of the Lie superalgebra g has a deformation, called the Yangian of qN. For each M = 1,2, . . . , denote by A N M the centralizer of qM ¿ q N+M in the associative superalgebra U(q N+M ). In this article we construct a sequence of surjective homomorphisms U(qN) ¿ A N 1 ¿ A N 2 ¿ . . . . We describe the inverse limit of the sequence of centralizer algebras A N 1 , A N 2 , . . . in terms of the Yangian of qN.

AB - Consider the complex matrix Lie superalgebra $ \mathfrak{g}\mathfrak{l}_{\left. N \right|N} $Unknown control sequence '\mathfrak' with the standard generators E ij where i, j = ±1, . . . , ± N. Define an involutive automorphism ¿ of $ \mathfrak{g}\mathfrak{l}_{\left. N \right|N} $Unknown control sequence '\mathfrak' by ¿(E ij) = E -i,-j . The queer Lie superalgebra qN is the fixed point subalgebra in $ \mathfrak{g}\mathfrak{l}_{\left. N \right|N} $Unknown control sequence '\mathfrak' relative to ¿. Consider the twisted polynomial current Lie superalgebra$ \mathfrak{g} = \left\{ {X\left( t \right) \in \mathfrak{g}\mathfrak{l}_{\left. N \right|N} \left[ t \right]:\eta \left( {X\left( t \right)} \right) = X\left( { - t} \right)} \right\} $Unknown control sequence '\mathfrak'. The enveloping algebra U( $ \mathfrak{g} $Unknown control sequence '\mathfrak' ) of the Lie superalgebra g has a deformation, called the Yangian of qN. For each M = 1,2, . . . , denote by A N M the centralizer of qM ¿ q N+M in the associative superalgebra U(q N+M ). In this article we construct a sequence of surjective homomorphisms U(qN) ¿ A N 1 ¿ A N 2 ¿ . . . . We describe the inverse limit of the sequence of centralizer algebras A N 1 , A N 2 , . . . in terms of the Yangian of qN.

KW - Algebra

U2 - 10.1007/0-8176-4478-4_17

DO - 10.1007/0-8176-4478-4_17

M3 - Other chapter contribution

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SP - 417

EP - 441

BT - Studies in Lie Theory: Dedicated to A. Joseph on his Sixtieth Birthday

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