Twist deformations for generalized Heisenberg algebras

12 Pages·0.11 MB·

by  Vladimir D. Lyakhovsky

0 Twist deformations for generalized Heisenberg
0
algebras
0
2
n Vladimir D. Lyakhovsky 1
a
J Theoretical Department, St. Petersburg State University, 198904,

5 St. Petersburg, Russia
2

]
A Abstract
Q
Multidimensional Heisenberg algebras, whose creation
.
h a+ and annihilation a− operators are the n-dimensional
t
a vectors, can be injected into simple Lie algebras g. It is
m
demonstrated that the spectrum of their deformations can
[ be investigated using chains of extended Jordanian twists

1 applied to U(g). In the case of U(sl(N)) (for N > 5 ) the
v two-dimensional Heisenberg subalgebras H have nine de-
6
formed costructures connected by four ”internal” and ”ex-
3
1 ternal” twists composing the commutativeediagram.
1
0
0
1 Introduction
0
/
h
t A Hopf algebra A(m,∆,ǫ,S) can be transformed [1] by an in-
a
m vertible twisting element F ∈ A ⊗ A, into a twisted algebra
: A (m,∆ ,ǫ,S ), that has the same multiplication and counit but
v F F F
i the twisted coproduct and the antipode given by
X
ar ∆ (a) = F∆(a)F−1,S (a) = vS(a)v−1,v = f(1)S(f(2)),a ∈ A.
F F i i
(1)
X
The twisting element has to satisfy the equations
F (∆⊗id)(F) = F (id⊗∆)(F), (ǫ⊗id)(F) = (id⊗ǫ)(F) = 1.
12 23
(2)
1
E-mail address: [email protected]
1
Let A and B be the universal enveloping algebras: A = U(l) ⊂
B = U(g) with l ⊂ g. If U(l) is the minimal subalgebra on which
F is completely defined as F ∈ U(l) ⊗ U(l) then l is called the
carrier algebra for F.
The first and well known example of the twist that was written
in the explicit form [2] corresponds to the carrier subalgebra B(2)
with generators H and E, [H,E] = E. It is called the Jordanian
twist and has the twisting element
Φ = eH⊗σ, σ = ln(1+E). (3)
J
For the extended Jordanian twists suggested in [3] the car-
rier subalgebra L is four-dimensional:
[H,E] = E, [H,A] = αA, [H,B] = βB,
(4)
[A,B] = E, [E,A] = [E,B] = 0, α+β = 1.
The corresponding twisting element contains the Jordanian fac-
tor:
F = Φ Φ (5)
E(α,β) E(α,β) J
and the extension
Φ = exp{A⊗Be−βσ}. (6)
E(α,β)
This twist defines the deformed Hopf algebras L with the
E(α,β)
costructure
∆ (H) = H ⊗e−σ +1⊗H −A⊗Be−(β+1)σ,
E(α,β)
∆ (A) = A⊗e−βσ +1⊗A,
E(α,β) (7)
∆ (B) = B ⊗eβσ +eσ ⊗B,
E(α,β)
∆ (E) = E ⊗eσ +1⊗E.
E(α,β)
Ingeneralthecompositionoftwotwistsisnotatwist. Butthere
are some important examples of the opposite behavior. When L is
a subalgebra L ⊂g there may exist several pairs of generators of
2
the type (A,B) arranged so that the Jordanian twist can acquire
several similar extensions [3]. This demonstrates that some twist-
ings can be applied successively to the initial Hopf algebra even in
the case when their carrier subalgebras are nontrivially linked. In
the universal enveloping algebrasfor classical Lie algebras there ex-
iststhepossibility toconstruct systematically thespecial sequences
of twists called chains [4]:
F ≡ F F ...F . (8)
Bp≺0 Bp Bp−1 B0
The factors F = Φ Φ of the chain are the twisting elements
B E J
k k k
of the extended Jordanian twists for the initial Hopf algebra A .
0
Here the extensions {Φ ,k = 0,...p−1} contain the fixed set of
E
k
normalized factors Φ = exp{A ⊗ Be−βσ} , the so called full
E(α,β)
set. It was proved that in the classical Lie algebras that conserve
symmetric invariant forms such chains can be made maximal and
proper. This means that forthe algebrasU(A ), U(B ) and U(D )
n n n
there exist chains F that cannot be reduced to a chain for a
Bp≺0
simple subalgebra and their full sets of extensions are the maximal
sets in the sense described below.
To construct a maximal proper chain for a classical Lie alge-
bra the sequence A ≡ A ⊃ A ⊃ ... ⊃ A ⊃ A of Hopf
0 1 p−1 p
subalgebras is to be fixed. For example in the case of U(sl(N))
the corresponding sequence is: U(sl(N)) ⊃ U(sl(N −2)) ⊃ ... ⊃
U(sl(N−2k)).... In each element of the sequence the initial root
λk must be chosen. Here λk = e −e for each sl(M) (the roots are
0 0 1 2
written in the standard e-basis). For the initial root let us form
the set π of constituent roots,
k
π = λ′,λ′′|λ′ +λ′′ = λk; λ′ +λk,λ′′ +λk¬ ∈ Λ (9)
k 0 0 0 A
n o
(here Λ is the root system of A ). For each element λ′ ∈ π one
A 0 k
can choose such an element λ′′ ∈ π that λ′ + λ′′ = λk. So, π is
k 0 k
naturally decomposed as
π = π′ ∪ π′′, π′ = {λ′}, π′′ = {λ′′}. (10)
k k k k k
3
In these terms the factors F of the chain (8) are fixed as
B
k
follows:
F = Φ Φ (11)
B E J
k k k
with
Φ = exp{H ⊗σk}, σk = ln(1+L ); (12)
Jk λk0 0 0 λk0
ΦEk = ΦEλ′ = exp{Lλ′ ⊗Lλk0−λ′e−21σ0k} (13)
λ′∈π′ λ′∈π′
Yk Yk
(here L is the generator associated to the root λ).
λ
2 Deformed Heisenberg algebras
One of the characteristic features of the extension Φ in the ordi-
E
nary extended Jordanian twists is that it connects the Heisenberg
subalgebras H and H with [A,B] = E,α+β = 1 and different
J EJ
deformed costructures (see [5]):
∆ (A) = A⊗eασ +1⊗A,
J
Φ : ∆ (B) = B ⊗eβσ +1⊗B, −→
E  J 
 ∆J(E) = E ⊗eσ +1⊗E,  (14)
∆ (A) = A⊗e−βσ +1⊗A,
EJ 
∆ (B) = B ⊗eβσ +eσ ⊗B,
 EJ 
 ∆EJ(E) = E ⊗eσ +1⊗E. 
A chainof twists (8) F ≡ F F ...F : A −→ A
Bp≺0 Bp Bp−1 B0 Bp≺0
contains p + 1 Jordanian factors Φ . Their product Φ ≡
Jk Jp≺0
Φ is also a twisting element for the universal enveloping
k Jk−1
algebra A, it produces the multijordanian deformation Φ :
Q Jp≺0
A −→ A . This means that the product Φ ≡ Φ with
Jp≺0 Ep≺0 k Ek
Φ = ( Φ )Φ ( Φ )−1 is in turn a twisting element
Ek m>k Jm Ek m>k Jm b Q b
for A , Φ : A −→ A .
b Jp≺Q0 Ep≺0 Jp≺0Q Bp≺0
In [6] it was shown that Φ plays the role of an extension for
b Ep≺0
the multijordanian twists Φ . It was proved that there are the
Jbp≺0
4
subalgebras in A that are carriers for Φ and whose costructures
Ep≺0
are shifted from one possible deformed ”state” to the other by the
b
twists Φ .
Ep≺0
Here we consider a certain type of such subalgebras, namely the
b
multidimensional Heisenberg subalgebras H(2,N−4) in U (sl(N))
generated by the 2 × 2-block of central and 2 × (N − 4) pairs of
f
creation and annihilation operators:
E E ... E E E
13 14 1,N−2 1,N−1 1N
E E ... E E E
23 24 2,N−2 2,N−1 2N
E E
3,N−1 3N
. . (15)
. .
. .
E E
N−3,N−1 N−3,N
E E
N−2,N−1 N−2,N
We shall normalize the Cartan elements as H = 1/2(E −E ),
i,k ii kk
use the standard gl(N)-basis {E } , and σ = ln(1+E).
ij i,j=1,...N
First we shall study the properties of H (2,N −4) twisted by
F
the factors of the 2-chain:
f
F = Φ Φ Φ Φ (16)
B1≺0 E1 J1 E0 J0
with
Φ = exp(H ⊗σ ),
Jk−1 k,N−k+1 k,N−k+1
ΦEk−1 = exp sN=−kk+1Ek,s ⊗Es,N−k+1e−12σk,N−k+1 (17)
= N−(cid:16)k Φ . (cid:17)
r=k+P1 Ek−1(r)
Q
To visualize the resulting deformations the following notations will
5
be used for the costructures:
P0(L) ≡ L⊗1+1⊗L,
Pi±(L) ≡ L⊗e±12σi,N+1−i +1⊗L,
Ri(L) ≡ L⊗e21σi,N+1−i +eσi,N+1−i ⊗L,
Ti(L) ≡ L⊗eσi,N+1−i +1⊗L,
T++(L) ≡ L⊗e21σ1,N+21σ2,N−1 +1⊗L,
T∓±(L) ≡ L⊗e∓12σ1,N±21σ2,N−1 +1⊗L,
(18)
TRi (L) ≡ L⊗e12σ1,N+21σ2,N−1 +eσi,N+1−i ⊗L,
S2− ≡ −E2,r ⊗E1,N−1e−12σ2,N−1,
S2+ ≡ E2,N ⊗Er,N−1e12σ1,N−12σ2,N−1,
S1− ≡ −E1,r ⊗E2,Ne−12σ1,N,
S1+ ≡ E1,N−1 ⊗Er,Ne−21σ1,N+12σ2,N−1,
i = 1,2; r = 3,...,N −2.
In these terms the transformations performed in H by the factors
of the canonical extended Jordanian twist (use the formulas (4)-(7)
with α = β = 1/2 ) can be written as
P0(A) P0(E) P+(A) T (E)
Φ : −→ . (19)
J P0(B) P+(B)
( ) ( )
P+(A) T (E) P−(A) T (E)
Φ : −→ . (20)
E P+(B) R(B)
( ) ( )
Let us perform the 2-Jordanian twisting in H(2,N−4), Φ :
J J
1 2
H(2,N − 4) −→ H (2,N − 4). Here the deformed coproducts
J J
1 2 f
will be presented by the schemes analogous to (19) and (20). In
f f
terms of (18) the costructure of H (2,N − 4) will acquire the
J J
1 2
6
f
following form:
P+(E ) ... P+(E ) T++(E ) T (E )
1 13 1 1,N−2 1,N−1 1 1N
P+(E ) ... P+(E ) T (E ) T++(E )
2 23 2 2,N−2 2 2,N−1 2N
P+(E ) P+(E )
2 3,N−1 1 3N
. .
. .
. .
P+(E ) P+(E )
2 N−2,N−1 1 N−2,N
(21)
As we have mentioned above any Heisenberg subalgebra with the
costructure {P+,T,P+} can be twisted by the corresponding ex-
tension Φ . In (21) such are the triples P+(E ), T (E ),
E i ir i i,N−i+1
P+(E ) . Obviously the deformationns performed by the ex-
i r,N−i+1
tensions ΦEi−1(or) = exp Ei,r ⊗Er,N−i+1e−21σi,N−i+1 in these triples
do not touch the gener(cid:16)ators other than {E ,E (cid:17), E ,E }.
1,r 2,r r,N−1 r,N
Thus to study the deformations induced on H(2,N − 4) by the
chain of twists (16) it is sufficient to focus the attention on the
f
behavior of the subalgebra H ≡ H(2,1),
f f
E E E
1r 1,N−1 1N
E E E
2r 2,N−1 2N (22)
E E
r,N−1 rN
Starting with its 2-Jordaniandeformation H one can apply any
J J
1 0
of the two extensions: ΦEi−1(r) = exp Ei,fr ⊗Er,N−i+1e−12σi,N−i+1
(i = 1,2). Notice that in the chain (16(cid:16)) the extension Φ com(cid:17)-
E1(r)
mutes not only with all the other extension factors Φ but also
Ei−1(r)
with Φ .We get three twisted ”states”:
J
1
H Φ←E0−(r) H Φ−E→1(r) H
E J J J J E J J
0 1 0 1 0 1 1 0
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f f f
The corresponding coalgebras are transformed as follows
P− T++ T P+ T++ T
1 1 1 1
P+ +S− T T++ P+ T T++
 2 1 2  ←−  2 2 
 P2+ +S1+ R1   P2+ P1+ 
 P1+ +S2− T++  T1 
P− T T++
−→  2 2 
 R2 P1+ +S2+ 
  (23)
Now we shall show that among the twists for U (sl(N)) one can
find those that perform further deformations of the states (23).
Their carrier subalgebras are not included in the H, so with re-
spect to H these twists must be treated as external. Consider
f
again the chain (16). According to the ”matreshka” effect (see [4])
f
after the first two twists Φ and Φ the generators in the subalge-
J E
0 0
bra U (sl(N −2)) will acquire trivial coproducts. Consequently
E J
0 0
when the first three factors Φ Φ Φ are applied the generators
J E J
1 0 0
{E ,E } return to the state P+. The twisting factor that per-
2,r r,N−1 2
forms this transition appears when we drag the Jordanian twisting
element Φ to the right:
J
1
Φ Φ Φ = Φ Φ Φ Φ Φ =
J1 E0 J0 J1 E0(2) E0(N−1) E0(N−2≺3) J0
Φ Φ Φ Φ−1Φ Φ Φ = (24)
J1 E0(2) E0(N−1) J1 J1 E0(N−2≺3) J0
Φ Φ Φ
E0(2,N−1) E0(N−2≺3) J1J0.
The factor ΦE0(N−2e≺3) is a twisting element for UJ1J0. The relation
(25) signifies that
Φ = exp E + 1E +E H ⊗
E0(2,N−1) 1,2 2 1,N−1 1,N−1 2,N−1
⊗Ee2,Ne−12(σ1,N+σ2,N(cid:16)−1(cid:16)) +E1,N−1 ⊗EN−1,Ne−21(σ1,N−σ2(cid:17),N−1)
(cid:19)
(25)
twiststhecostructureofU . Itisimportantthatforthe
E0(N−2≺3)J1J0
generators in (25) the coproducts in U and in U
J1J0 E0(N−2≺3)J1J0
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are the same. This means that the factor Φ can twist
E0(2,N−1)
directly the algebra U and the subalgebra H in it. The
J J J J
1 0 e 1 0
structure constants of this subalgebra are invariant with respect to
the renumbering (1 ⇀↽ 2,N −1 ⇀↽ N) of the indifces of generators.
Thus there exists the second external twisting factor
Φ = exp E + 1E +E H ⊗
E1(2,N−1) 2,1 2 2,N 2,N 1,N
⊗E1,Ne−1e−21(σ1,N+σ2,N−(cid:16)1)(cid:16)+E2,N ⊗EN,N−1e+21(σ1,N(cid:17)−σ2,N−1)
(cid:19)
(26)
that can be applied to H . One can find this twisting element
J J
1 0
directly considering the chain
f
F′ = Φ′ Φ Φ′ Φ (27)
B0≺1 E0 J0 E1 J1
where the maximal set of the constituent roots is used in the
extension Φ′ (that is for the root (e −e )). The algebra
E1 2 N−1
U = U can be considered as an intermediate deformed ob-
J J J J
1 0 0 1
ject for both chains (16) and (27).
When any of the external factors Φ and Φ are
E0(2,N−1) E1(2,N−1)
appliedtoH oneofthepairsofcreation-annihilationgenerators
J J
1 0 e e
retains the costructure P+:
f
P+ T−+ T P+ T++ T
1 1 1 1
P+ −S− T T P+ T T++
 2 1 2 R1  ←  2 2 
 P+ −S+ P+   P+ P+ 
2 1 1 2 1
 P1+ −S2− TR2  T1 
P+ T T+−
−→  2 2 
 P+ P+ −S+ 
2 1 2
  (28)
This shows that to any of them the corresponding extension Φ
Ei−1
can be applied. Returning to the sequence (28) we see that these
extensions remove the summands of the form S± so that the alter-
i
native extension also becomes applicable. As a result we get for
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the Heisenberg algebra H nine deformed costructures
f
P+ T++ T
1 1
∆ H = P+ T T++
J1J0  2 2 
(cid:16) (cid:17)  P2+ P1+ 
f P+ T−+ T
 1 1 
∆ H = P+ −S− T T
E0J1J0(cid:16) (cid:17)  2 1 P2+2−S1+R1P1+ 
e f  P1+ −S2− TR2 T1 
∆ H = P+ T T+−
E J J  2 2 
1 1 0(cid:16) (cid:17)  P2+ P1+ −S2+ 
e f  P1− T++ T1 
∆ H = P+ +S− T T++
E0J1J0  2 1 2 
(cid:16) (cid:17)  P2+ +S1+ R1 
f P− T−+ T
 1 1 
∆ H = P+ T T
E0E0J1J0(cid:16) (cid:17)  2 P2+2 R1R1 
e f  P1− TR2 T1 
∆ H = P− +S− T T+−
E E E J J  2 1 2 
1 0 1 1 0 (cid:16) (cid:17)  R2 +S1+ R1 
e f  P1+ +S2− T++ T1 
∆ H = P− T T++
E1J1J0  2 2 
(cid:16) (cid:17)  R2 P1+ +S2+ 
f P− +S− T−+ T
 1 2 1 
∆ H = P− T T
E1E0E0J1J0 (cid:16) (cid:17)  2 R22 R1 +RS12+ 
e f  P1+ TR2 T1 
∆ H = P− T T+−
E E J J  2 2 
1 1 1 0(cid:16) (cid:17)  R2 P1+ 
f
e  
Here is how these costructures are connected by the external
10