On quasitriangular Hopf algebras related to the Borel subalgebra of sl(2)

4 Pages·0.05 MB·English

by  A. A. Vladimirov

ON QUASITRIANGULAR HOPF ALGEBRAS
RELATED TO THE BOREL SUBALGEBRA OF sl
2
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A.A.VLADIMIROV
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1 Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,

n Dubna, Moscow region, 141980, Russia
a
J
0 E-mail: [email protected]
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v
1 ABSTRACT
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1 Explicit isomorphism is established between quasitriangular Hopf algebras studied
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recently by O.Ogievetsky and by the present author.
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In my recent papers [1, 2] a regular method has been proposed for constructing
e
h quasitriangular Hopf algebras (actually, quantum doubles) out of invertible matrix
v: solutions R of the quantum Yang-Baxter equation
i
X
r R12R13R23 = R23R13R12. (1)
a
As an illustration, the R-matrix of Jordanian type [3, 4]
1 1 −1 1
0 1 0 1
  (2)
0 0 1 −1
 
 0 0 0 1 
 
 
has been shown [2, 5] to produce, among others, a quasitriangular Hopf algebra with
generators {v,h} obeying the relations
[v,h] = 2sinhh,
∆(v) = eh ⊗v +v ⊗e−h, ∆(h) = h⊗1+1⊗h, (3)
±1 ±1
S (h) = −h, S (v) = −v ∓2sinhh.
Its universal R-matrix is found [5] to be
h⊗1+1⊗h
R = exp (sinhh⊗v−v ⊗sinhh) . (4)
(sinh(h⊗1+1⊗h) )
After submitting the paper [5] I have learned from O.Ogievetsky that in paper [6],
using quite different approach (which allows classification of all Hopf structures on the
Lie algebra [x,h] = 2h), he obtained a quasitriangular Hopf algebra {τ,σ} defined by
[τ,σ] = 2(1−eσ),
∆(τ) = τ ⊗eσ +1⊗τ , ∆(σ) = σ ⊗1+1⊗σ, (5)
S(τ) = −τe−σ, S−1(τ) = −τe−σ +2(1−e−σ), S±1(σ) = −σ,
with universal R-matrix
σ σ
R = exp( ⊗τ) exp(−τ ⊗ ). (6)
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Since the Hopf algebra (3), due to a reparametrization
sinhh
v = x =⇒ [x,h] = 2h, (7)
h
evidently belongs to the class considered by Ogievetsky, it is only natural to look for
an isomorphism between the Hopf algebras (3) and (5).
Such an isomorphism really exists and can be fixed by
τ = −e−hv, σ = −2h. (8)
Relations (5) are readily obtained from (3) and (8). As a by-product, we come to an
alternative form of the R-matrix (4):
R = exp(h⊗e−hv) exp(−e−hv ⊗h). (9)
In [2, 5] (see also [7]), the following quasitriangular Hopf algebra {b,g,v,h} (the
quantum double of {v,h}) was also considered:
[g,b] = [h,b] = 2sinhg, [g,v] = [h,v] = −2sinhh,
[b,v] = 2(coshg)v+2(coshh)b, [g,h] = 0,
∆(b) = eg ⊗b+b⊗e−g, ∆(v) = eh ⊗v +v ⊗e−h, (10)
±1
∆(g) = g ⊗1+1⊗g, ∆(h) = h⊗1+1⊗h, S (g) = −g,
±1 ±1 ±1
S (h) = −h, S (b) = −b±2sinhg, S (v) = −v ∓2sinhh.
The (anti)duality relations between {b,g} and {v,h} are:
< 1,b >=< 1,g >=< v,1 >=< h,1 >= 0,
< 1,1 >=< h,b >=< v,g >= 1, (11)
< v,b >= −1, < h,g >= 0.
Universal R-matrix looks like [5]
g ⊗1+1⊗h
R = exp (sinhg ⊗v +b⊗sinhh) . (12)
(sinh(g ⊗1+1⊗h) )
In terms of {τ,σ} and their antiduals {µ,ν} the same quantum double is characterized
by the relations
[τ,σ] = [τ,ν] = 2(1−eσ), [µ,σ] = [µ,ν] = 2(1−eν),
[τ,µ] = 2(µ−τ), [σ,ν] = 0,
∆(τ) = τ ⊗eσ +1⊗τ , ∆(σ) = σ ⊗1+1⊗σ,
∆(µ) = µ⊗eν +1⊗µ, ∆(ν) = ν ⊗1+1⊗ν, (13)
S(τ) = −τe−σ, S−1(τ) = −τe−σ +2(1−e−σ),
S(µ) = −µe−ν , S−1(µ) = −µe−ν +2(1−e−ν),
±1 ±1
S (σ) = −σ, S (ν) = −ν
with antiduality conditions
< 1,µ >=< 1,ν >=< τ,1 >=< σ,1 >= 0,
< 1,1 >= 1, < τ,ν >= 2, < σ,µ >= −2, (14)
< τ,µ >=< σ,ν >= 0
and universal R-matrix [6]
ν σ
R = exp( ⊗τ) exp(−µ⊗ ). (15)
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Apparent similarity of (15) and (6), as well as (12) and (4), is due to ’selfduality’ [5, 6]
of both {τ,σ}- and {v,h}-algebras.
Hopf algebras (10) and (13) are isomorphic by
µ = e−gb, ν = −2g, τ = −e−hv, σ = −2h. (16)
In particular, this produces an alternative representation for R (12):
R = exp(g ⊗e−hv) exp(e−gb⊗h). (17)
Surprisingly, it appears problematic to derive (12) directly from (17) by the mere use
of Baker-Campbell-Hausdorff formula.
Acknowledgments
The present work was supported by the Heisenberg-Landau program, 1993.
I am very grateful to Prof. J.Wess for organizing my visit to Munich where this
research was carried out. I appreciate stimulating discussions with O.Ogievetsky,
A.Kempf, M.Pillin and R.Engeldinger.
References
[1] A.A.Vladimirov, Mod.Phys.Lett.A 8 (1993) 1315.
[2] A.A.Vladimirov, Z.Phys.C 58 (1993) 659.
[3] V.V.Lyubashenko, Usp.Mat.Nauk 41 Vol.5 (1986) 185;
Engl.transl.: Russ.Math.Surveys 41 (1988) 153.
[4] E.E.Demidov, Yu.I.Manin, E.E.Mukhin and D.V.Zhdanovich,
Progr.Theor.Phys. Suppl. 102 (1990) 203.
[5] A.A.Vladimirov, Mod.Phys.Lett.A 8 (1993) 2573.
[6] O.Ogievetsky, Max-Planck-Institut preprint MPI-Ph/92-99.
[7] Cˇ.Burdik and P.Hellinger, Charles Univ. preprint PRA-HEP-93/2.