We construct the inverse Shapovalov form of a simple complex quantum group from its universal R-matrix based on a generalized Nagel-Moshinsky approach to lowering operators. We establish a connection between this algorithm and the ABRR equation for dynamical twist.
REFERENCES
1.
N. N.
Shapovalov
, “On a bilinear form on the universal enveloping algebra of a complex semisimple Lie algebra
,” Funk. Anal.
6
, 65
–70
(1972
).2.
J. C.
Jantzen
, Lectures on Quantum Groups
, Graduate Studies in Mathematics
(AMS
, Providence, RI
, 1996
), Vol. 6
.3.
A.
Alekseev
and A.
Lachowska
, “Invariant ∗-product on coadjoint orbits and the Shapovalov pairing
,” Comment. Math. Helv.
80
, 795
–810
(2005
).4.
V.
Drinfeld
, “Quantum groups
,” in Proceedings of the International Congress of Mathematicians, Berkeley, 1986
, edited byA. V.
Gleason
(AMS
, Providence
, 1987
), pp. 798
–820
.5.
C.
de Concini
and V. G.
Kac
, “Representations of quantum groups at roots of 1. Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989)
,” in Progress in Mathematics
(Birkhäuser
, 1990
), Vol. 92
, pp. 471
–506
.6.
S.
Levendorskiy
and J.
Soibelman
, “Quantum Weyl group and multiplicative formula for R-matrix of simple Lie algebra
,” Funct. Anal. Appl.
25
, 143
–145
(1991
).7.
S.
Khoroshkin
and V.
Tolstoy
, “The universal R-matrix for quantum untwisted affine Lie algebras
,” Funct. Anal. Appl.
26
, 69
–71
(1992
).8.
J. G.
Nagel
and M.
Moshinsky
, “Operators that lower or raise the irreducible vector spaces of Un−1 contained in an irreducible vector space of Un
,” J. Math. Phys.
6
, 682
–694
(1965
).9.
A. I.
Molev
, “Gelfand-Tsetlin bases for classical Lie algebras
,” in Handbook of Algebra
(Elsevier/North-Holland
, Amsterdam
, 2006
), Vol. 4
, pp. 109
–170
.10.
T.
Ashton
and A.
Mudrov
, “R-matrix and Mickelsson algebras for orthosymplectic quantum groups
,” J. Math. Phys.
56
, 081701
(2015
).11.
P.
Etingof
and O.
Schiffmann
, Lectures on the Dynamical Yang-Baxter Equation, Quantum Groups and Lie Theory
, London Mathematical Society Lecture Note Series, Durham, 1999
(Cambridge University Press
, 2001
), Vol. 290
.12.
D.
Arnaudon
, E.
Buffenoir
, E.
Ragoucy
, and P.
Roche
, “Universal solutions of quantum dynamical Yang-Baxter equations
,” Lett. Math. Phys.
44
(3
), 201
–214
(1998
).13.
V.
Chari
and A.
Pressley
, A Guide to Quantum Groups
(Cambridge University Press
, Cambridge
, 1995
).© 2016 Author(s).
2016
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