Last updates: 25 November 2011
Let be a Hopf algebra with an element
such that .
The dual of ,
is an algebra homomorphism since
is a commutative subalgebra of ,
the Bethe subalgebra of ,
is a "large" commutative subalgebra of .
Let be a Hopf algebra with an invertible element
for . The dual
of is a Hopf algebra. Fix a positive integer and an index set
be a set of representations of . Their matrix entries
On the the coproduct
so that is a matrix in
is a concise way of encoding the relations
which are satisfied by the
Let be the Hopf algebra given by
with comultiplication given by
Then the map
is a Hopf algebra homomorphism.
We really want a map ,
But it is "easy" to make maps .
For example, one can construct a map by
In the case of the Yangian or
is surjective and is generated by the elements of the center of
RTT and the quantum double
Let be a Hopf algebra,
be a representation of and
Notes and References
These notes are an attempt to work out and exposit a portion of the material in
[D, last paragraph of section 10, section 11, and the last paragraph of section 12].
These notes are retyped and extended version of part of the notes at
which were written in collaboration with N. Rojkovskaia. The section
§ 3 RTT algebras and the quantum double is taken from [RTF, § 3].
V.G. Drinfeld, Quantum Groups, Vol. 1 of Proccedings of the International Congress of Mathematicians (Berkeley, Calif., 1986). Amer. Math. Soc., Providence, RI, 1987, pp. 198–820.
V.G. Drinfelʹd, Almost cocommutative Hopf algebras, (Russian) Algebra i Analiz 1 (1989), 30–46; translation in
Leningrad Math. J. 1 (1990), 321–342.
R. Leduc and A. Ram,
A ribbon Hopf algebra approach to the irreducible representations of centralizer
algebras: The Brauer, Birman-Wenzl and type A Iwahori-Hecke algebras, Adv. Math.
125 (1997), 1-94.
N. Yu. Reshetikhin, L.A. Takhtadzhyan and L.D. Faddeev, Quantization of Lie groups and Lie algebras,
Leningrad Math. J. 1 (1990), 193-225.
N. Yu. Reshetikhin, Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links I, LOMI preprint no. E–4–87, (1987).