RTT algebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 25 November 2011

Bethe subalgebras

Let U be a Hopf algebra with an element UU such that (Δid)() =1323. The dual of U, U*= {:U | is linear}, has product given by (12) (a)=(1 2)Δ(a) . The function Ψ: U* U (id)() is an algebra homomorphism since Ψ(1 2) = (1 2id) () = (1 2id) (Δid) () = (1 2id) (1323) = (1id)() (2id)() = Ψ(1) Ψ(2). Since C0 = {U* | (xy) = (yx)} is a commutative subalgebra of U*, the Bethe subalgebra of U, Ψ(C0) = {(id) () | C0) }, where C0 = {U* | (xy) = (yx)}, is a "large" commutative subalgebra of U.

RTT algebras

Let U be a Hopf algebra with an invertible element =r arbr UU such that Δ(a) -1 =Δop(a), for aU. The dual U* of U is a Hopf algebra. Fix a positive integer n and an index set T^. Let { ρλ:U Mn() | λT^ } be a set of representations of U. Their matrix entries ρijλ :U are elements of U*.

On the ρijλ the coproduct Δ: U*U* U* has values Δ ( ρijλ ) = k=1n ρikλ ρkjλ ,since ρijλ (u1u2) = k=1n ρikλ (u1) ρkjλ (u2) , for u1,u2U. Let (λ,μ) = (ρλρμ ) () and T(λ) =( ρijλ ) so that T(λ) is a matrix in Mn(U*). Then T(λ)id = i,j,k tijλ (Eij Ekk), idT(μ) = i,k,l tklμ (Eii Ekl), and (λ,μ) = i,j,k,l ρijλ (ar) ρklμ (br) (Eij Ekl).

Since (λ,μ) (T(λ)id) (idT(μ)) = i,j,k,l ,x,y ρixλ (ar) txjλ ρkyμ (br) tylμ (Eil Ekl), and (idT(μ)) (T(λ)id) (λ,μ) = i,j,k,l ,α,β tkβμ tiαλ ραjλ (as) ρβlλ (bs), the equation (idT(μ)) (T(λ)id) (λ,μ) = (λ,μ) (T(λ)id) (idT(μ)) is a concise way of encoding the relations ( x,y ρixλ (ar) ρkyμ (br) ρxjλ ρylμ ) (a) = x,y,a ρixλ (ar) ρkyμ (br) ρxjλ (a(1)) ρylμ (a(2)) = a ρijλ (ar a(1) ) ρklμ (br a(2) ) = ( ρijλ ρklμ ) (Δ(a) ) = ( ρijλ ρklμ ) ( Δop(a) ) = a ρijλ ( a(2) as ) ρklμ ( a(1) bs ) = α,β ρiαλ ( a(2) ) ραjλ (as) ρkβμ ( a(1) ) ρβlμ (bs) = ( α,β ρkβμ ρiαλ ραjλ (as) ρβlμ (bs) ) (a) which are satisfied by the ρijλ in U*.

Let B be the Hopf algebra given by generators tijλ , 1i,jn, λT^, and relations (idT(μ)) (T(λ)id) (λ,μ) = (λ,μ) (T(λ)id) (idT(μ)) with comultiplication given by Δ( tijλ ) = k=1n tikλ tklλ . Then the map B U* tijλ ρijλ is a Hopf algebra homomorphism.

We really want a map BU, not BU*. But it is "easy" to make maps U*U. For example, one can construct a map U*U by l(idl) () or l(idl) (21-1) or l(idl) (21) . In the case of the Yangian or Uq𝔤 the composition Φ:B U*U is surjective and kerΦ is generated by the elements of the center of B.

RTT and the quantum double

Let A be a Hopf algebra, D(A) =AA*coop the Drinfeld double. Let {es} be a basis of A, {es} the dual basis in A* so that = s es es . Let {fst} be the basis of D(A)* dual to the basis {eset} of D(A). Let 𝒯= s,t eset fts D(A) D(A)* +=s eses =21 and -=s S(es)es =-1 . Let π:D(A) Mn() be a representation of D(A) and R=(ππ) (), T=(πid) (𝒯), L± =(πid)( ±) so that L+= (πid) (21) and L-= (πid) (-1).

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 http://researchers.ms.unimelb.edu.au/~aram@unimelb/Notes2005/ygnsBonn2.08.05.pdf which were written in collaboration with N. Rojkovskaia. The section § 3 RTT algebras and the quantum double is taken from [RTF, § 3].


[D] 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. MR0934283

[Dr] V.G. Drinfelʹd, Almost cocommutative Hopf algebras, (Russian) Algebra i Analiz 1 (1989), 30–46; translation in Leningrad Math. J. 1 (1990), 321–342. MR1025154

[LR] 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. MR1427801.

[RTF] N. Yu. Reshetikhin, L.A. Takhtadzhyan and L.D. Faddeev, Quantization of Lie groups and Lie algebras, Leningrad Math. J. 1 (1990), 193-225.

[Re] N. Yu. Reshetikhin, Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links I, LOMI preprint no. E–4–87, (1987).

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