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Affine Flag Varieties and Quantum Symmetric Pairs (Paperback)
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Affine Flag Varieties and Quantum Symmetric Pairs (Paperback)
Series: Memoirs of the American Mathematical Society
Expected to ship within 12 - 17 working days
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The quantum groups of finite and affine type $A$ admit geometric
realizations in terms of partial flag varieties of finite and
affine type $A$. Recently, the quantum group associated to partial
flag varieties of finite type $B/C$ is shown to be a coideal
subalgebra of the quantum group of finite type $A$. In this paper
the authors study the structures of Schur algebras and Lusztig
algebras associated to (four variants of) partial flag varieties of
affine type $C$. The authors show that the quantum groups arising
from Lusztig algebras and Schur algebras via stabilization
procedures are (idempotented) coideal subalgebras of quantum groups
of affine $\mathfrak{sl}$ and $\mathfrak{gl}$ types, respectively.
In this way, the authors provide geometric realizations of eight
quantum symmetric pairs of affine types. They construct monomial
and canonical bases of all these quantum (Schur, Lusztig, and
coideal) algebras. For the idempotented coideal algebras of affine
$\mathfrak{sl}$ type, the authors establish the positivity
properties of the canonical basis with respect to multiplication,
comultiplication and a bilinear pairing. In particular, the authors
obtain a new and geometric construction of the idempotented quantum
affine $\mathfrak{gl}$ and its canonical basis.
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