Exotic Bialgebras: Nontriangular Case S14

In this subsection we consider the matrix bialgebraS14. We obtain it by applying the RTT relation (4.223) for the caseR = R_S1,4, whenq²= 1 where:̸

R_S1,4 ≡ ( 0 0 0q 0 0 1 0 0 1 0 0 q0 0 0

) (4.330)

ThisR-matrix is given in [339].

The relations which follow from (4.223) and (4.330) whenq²= 1 are:̸

b²–c²= 0 , a²–d²= 0 (4.331)

ab=ba= 0 , ac=ca= 0

bd=db= 0 , cd=dc= 0.

In terms of the generatorsa,̃ b,̃ c,̃ d̃

̃b ̃c+ ̃c ̃b= 0 ̃a ̃d+ ̃d ̃a= 0 (4.332)

̃

ab̃=b̃ã= 0 ãc̃=c̃ã= 0

̃b ̃d= ̃d ̃b= 0 ̃c ̃d= ̃d ̃c= 0.

From the above relations it is clear that the PBW basis ofS14 is:

̃

a^kd̃^ℓ, b̃^kc̃^ℓ. (4.333)

4.8 Duality for Exotic Bialgebras 177

4.8.7.1 Dual Algebra

Let us denote bys14 the unknown yet dual algebra ofS14, and byA,̃ B,̃ C,̃ D̃ the four generators of s14. We define the pairing as (4.186):⟨Z,f⟩, Z = A,̃ B,̃ C,̃ D,̃ fis from (4.333). Explicitly, we obtain:

⟨ ̃A,f ⟩ = %(𝜕f

𝜕 ̃a) = { {{

k$_ℓ0 f =ã^kd̃^ℓ

0 f =b̃^kc̃^ℓ (4.334a)

⟨ ̃B, f⟩ = %(𝜕f

𝜕 ̃b) = { {{

0 f =ã^kd̃^ℓ

$k1$_ℓ0 f =b̃^kc̃^ℓ (4.334b)

⟨ ̃C, f⟩ = %(𝜕f

𝜕 ̃c) = { {{

0 f =ã^kd̃^ℓ

$k0$_ℓ1 f =b̃^kc̃^ℓ (4.334c)

⟨ ̃D, f⟩ = %(𝜕f

𝜕 ̃d) = { {{

$_ℓ1 f =ã^kd̃^ℓ

0 f =b̃^kc̃^ℓ (4.334d) We shall need (as in Section 4.8.2) the auxiliary operatorE:

⟨E,f⟩ = { {{

1 for f = 1_A

0 otherwise (4.335)

Using the above we obtain:

Proposition 10.The generatorsA,̃ B,̃ C,̃ D introduced above obey the following relations:̃

̃C= ̃D ̃B= – ̃B ̃D, [ ̃A, ̃D] = 0 (4.336)

̃A ̃B= ̃B ̃A= ̃D² ̃B= ̃B³= ̃B, EZ = ZE = 0 , Z=A,̃ B,̃ D̃,

$_U(A) =̃ Ã⊗1_U + 1_U ⊗ ̃A (4.337)

$_U(B) =̃ B̃⊗E+E⊗ ̃B

$_U(D) =̃ D̃⊗K+ 1_U ⊗ ̃D, K≡(–1)^Ã

$(E) =E⊗E

%_U(Z) = 0 , Z = A,̃ B,̃ D,̃ %_U(E) = 1 (4.338)

̃A, ̃B²and ̃D²are Casimir operators. The bialgebra s14is not a Hopf algebra.

For the Proof we refer to [50]. ◊

Corollary 3.The algebra generated by the generatorà is a sub-bialgebra ofs14. The algebras14^󸀠generated byB,̃ D̃ is a subalgebra ofs14, but is not a sub-bialgebra (cf.

(4.337b,c)). It has the following PBW basis:

̃B, ̃B², ̃D ̃B, ̃D ̃B², ̃D^ℓ, ℓ= 0, 1, 2, ... (4.339)

where we use the conventionD̃⁰= 1_U. ◊

4.8.7.2 Regular Representation

We start with the study of the right regular representation of the subalgebras14^󸀠. For this we use the right multiplication table:

̃B ̃B² ̃D ̃B ̃D ̃B² ̃D^2k ̃D^2k+1

̃B ̃B² ̃B ̃D ̃B² ̃D ̃B ̃B ̃D ̃B

̃D – ̃D ̃B ̃D̃B² – ̃B ̃B² ̃D^2k+1 ̃D^2k+2

From the above table follows that there is a four-dimensional subspace spanned by

̃B, ̃B², ̃D ̃B, ̃D ̃B². It is reducible and decomposes into four one-dimensional representa-tions spanned by:

v_:,:󸀠 =B̃+:B̃²–:^󸀠D̃B̃+::^󸀠D̃B̃² (4.340) The action ofB,̃ D̃on these vectors is:

̃Bv_:,:󸀠 =:v_:,:󸀠, ̃Dv_:,:󸀠 =:^󸀠v_:,:󸀠 (4.341) The value of the CasimirsB̃²,D̃²on these vectors is 1.

The quotient of the RRR by the above submodules has the following multiplication table:

̃D^2k ̃D^2k+1

̃B 0 0

̃D ̃D^2k+1 ̃D^2k+2

This representation is reducible. It contains an infinite set of nested submodulesVⁿ⊃ Vⁿ⁺¹,n= 0, 1, ..., whereVⁿis spanned byD̃^n+ℓ,ℓ= 0, 1, .... Correspondingly there is an infinite set of one-dimensional irreducible factor-modulesFⁿ≡Vⁿ/Vⁿ⁺¹(generated by

̃Dⁿ), which are all isomorphic to the trivial representation since the generators ̃B, ̃Dact as zero on them. Thus there are five irreps arising from the RRR ofs14^󸀠:

– one-dimensional trivial

– four one-dimensional with both CasimirsB̃²,D̃²having value 1

4.8 Duality for Exotic Bialgebras 179

Turning to the algebras14 we note that it inherits the representation structure of its subalgebras14^󸀠. On the representations (4.340) the CasimirÃhas the value 1. However, on the one-dimensional irrepsFⁿthe CasimirÃhas no fixed value. Thus, the list of the irreps arising from the RRR ofs14 is:

– one-dimensional with Casimir values,, 0, 0 forA,̃ B̃²,D̃², respectively,,∈ ℂ – four one-dimensional with all CasimirsA,̃ B̃²,D̃²having value 1

4.8.7.3 Weight Representations

Here we study weight representations, first w.r.t. D, as in (4.317). The resulting̃ representation ofs14^󸀠is three-dimensional:

v₀,B ṽ ₀,B̃²v₀. (4.342) It is reducible and contains one one-dimensional and one two-dimensional irrep:

– one-dimensional+∈ ℂ:

w₀= (B̃²– 1_U)v₀, (4.343)

̃Bw₀= 0 , ̃Dw₀=+w₀, (4.344)

– two-dimensional with+=±1:

{v₀,v₁=B ṽ ₀} (4.345)

̃B(v₀ v₁)=(v₁

v₀), D̃(v₀

v₁)=+(v₀

–v₁) (4.346) Turning to the algebras14 we note that it inherits the representation structure of its subalgebras14^󸀠. On the one-dimensional irrep (4.343) the CasimirÃhas no fixed value sinceB̃ is trivial, and [A,̃ D] = 0. On the two-dimensional irrep (4.345) the Casimir̃ Ã has the value 1 sinceÃB̃=B.̃

Thus, there are the following irreps ofs14 which are obtained as weight irreps of the generatorD:̃

– one-dimensional with Casimir values,, 0,+²forA,̃ B̃²,D̃², respectively,,,+∈ ℂ – two two-dimensional with all CasimirsA,̃ B̃²,D̃²having the value 1

Next we consider weight representations w.r.t.B:̃

̃B v₀=-v₀, (4.347)

with-∈ ℂ. FromB̃³ =B̃follows that-= 0,±1. Acting with the generators we obtain the following representation vectors:v_ℓ=D̃^ℓv₀. We have thatDṽ _ℓ=v_ℓ+1.

Further we consider first the case-²= 1. Then we apply the relationD̃²B̃=B̃tov_ℓ and we get:

̃D² ̃Bv_ℓ= (–1)^ℓ-v_ℓ+2= ̃Bv_ℓ= (–1)^ℓ-v_ℓ

from which follows that we have to identifyv_ℓ+2withv_ℓ. Thus the representation is given as follows:

Further we consider the case-= 0. This representation is reducible. It contains an infinite set of nested submodulesVⁿ⊃ Vⁿ⁺¹,n= 0, 1, ..., whereVⁿis spanned by

̃D^n+ℓv₀,ℓ= 0, 1, .... Correspondingly there is an infinite set of one-dimensional irredu-cible factor-modulesFⁿ ≡ Vⁿ/Vⁿ⁺¹(generated byD̃ⁿv₀), which are all isomorphic to the trivial representation since the generatorsB,̃ D̃act as zero on them.

Turning to the algebras14 we note that it inherits the representation structure of its subalgebras14^󸀠, with the value of the CasimirÃbeing not fixed ifB̃acts trivially, and being 1, ifB̃acts non trivially.

Thus, there are the following irreps ofs14 which are obtained as weight irreps of the generatorB:̃

– one-dimensional with Casimir values,, 0, 0 forA,̃ B̃²,D̃², respectively,,∈ ℂ – two two-dimensional with all CasimirsA,̃ B̃²,D̃²having the value 1

4.8.7.4 Representations ofs14 onS14

Here we shall study the representations ofs14 obtained by the use of its right regular representation (RRR) on the dual bialgebraS14. The RRR is defined as in (4.320). For the generators ofs14 we have:

4.8 Duality for Exotic Bialgebras 181

For the action on the basis ofS14 we use formula (4.322). We obtain:

0R(A)ãⁿd̃^k= (n+k)ãⁿd̃^k, 0R(A)b̃ⁿc̃^k= (n+k)b̃ⁿc̃^k (4.350) 0R(B)ãⁿd̃^k=$k0$n1b̃+$n0$k1c̃, 0R(B)b̃ⁿc̃^k=$k0$n1ã+$n0$k1d̃

0R(D)ã^kd̃^ℓ= (–1)^ℓ+1ℓ ̃a^k+1d̃^ℓ–1+ (–1)^ℓkã^k–1d̃^ℓ+1 0_R(D)b̃^kc̃^ℓ= (–1)^ℓℓ ̃b^k+1c̃^ℓ–1+ (–1)^ℓ+1kb̃^k–1c̃^ℓ+1

We see that similarly to Section 4.8.6.4 the CasimirÃacts as the length of the elements ofS14, that is, (4.321) holds. Thus, also here we classify the irreps by the value,Aof the CasimirÃwhich runs over the non-negative integers. For fixed,Athe basis of the corresponding representations is spanned by the elementsf such thatℓ(f) =,A. The dimension of each such representation is:

dim (,_A) ={ {{

2(,A+ 1) for ,A≥1

1 for ,A= 0 (4.351)

The classification goes as follows:

– ,A= 0

This is the one-dimensional trivial representation spanned by 1_A. – ,A= 1

This representation is four-dimensional spanned by the four generatorsa,̃ b,̃ c,̃ d̃ ofS14. It decomposes in two two-dimensional isomorphic to each other irreps with basis vectors as in (4.325) – this is due to the fact that the action (4.349b,c) is the same as the action (4.321). The value of the CasimirsB̃²,D̃²is 1.

– Each representation for fixed,A ≥ 2 is reducible and decomposes in two iso-morphic representations: one built on the basisã^kd̃^ℓ, and the other built on the basisb̃^kc̃^ℓ, each of dimension,A+ 1. Thus, for,A ≥2 we shall consider only the representations built on the basisã^kd̃^ℓ. These representations are also reducible and they all decompose in one-dimensional irreps. Further, the action ofB̃is zero, thus, we speak only about the action ofD.̃

– ,_A= 2n,n= 1, 2, ...

For fixednthe representation decomposes into 2n+ 1 one-dimensional irreps. On one of these, which is spanned by the element:

w₀=

n

∑

k=0

(n

k) ̃a^2n–2kd̃^2k, (4.352)

the generator D̃ acts by zero. The rest of the irreps are enumerated by the parameters:±,4, where4= 2, 4, ..., 2n=,_A, and are spanned by the vectors:

u^±₄ =u₀±4u₁, (4.353) u₀=

n

∑

k=0

!kã^2n–2kd̃^2k, !0= 1 ,

u₁=

n–1

∑

k=0

"kã^2n–2k–1d̃^2k+1, "0= 1 ,

on whichD̃acts by:

0_R(D)̃ u^±₄ =±4u^±₄ (4.354) which follows from:

0R(D)̃ (u₀

u₁)=(4²u₁

u₀ ) (4.355)

Note that the value of the CasimirD̃²is equal to4². The coefficients!_k,"_kdepend on4and are fixed from the two recursive equations which follow from (4.355):

4²"_k= 2(n–k)!_k– 2(k+ 1)!_k+1, k= 0, ...,n– 1

!_k= (2k+ 1)"_k– (2n– 2k+ 1)"_k–1, k= 0, ...,n, (4.356) where we set"–1 ≡0,"n≡0.

– ,A= 2n+ 1,n= 1, 2, ...

For fixednthe representation is (2n+ 2)-dimensional and decomposes into 2n+ 2 irreps which are enumerated by two parameters:±,4, where4= 1, 3, 5, ..., 2n+ 1 = ,A, and are spanned by the vectors:

w^±₄ =w₀±4w₁, (4.357)

w₀=

n

∑

k=0

!^󸀠_kã^2n–2k+1d̃^2k, !^󸀠₀= 1 , w₁=

n

∑

k=0

"^󸀠_kã^2n–2kd̃^2k+1, "^󸀠₀= 1 ,

on whichD̃acts by (4.354). Note that the value of the CasimirD̃²is equal to4². The coefficients!^󸀠_k,"^󸀠_k are fixed from the two recursive equations which follow from (4.354):

4²"^󸀠_k= (2n– 2k+ 1)!^󸀠_k– 2(k+ 1)!^󸀠_k+1, k= 0, ...,n;

!^󸀠_k= (2k+ 1)"^󸀠_k– 2(n–k+ 1)"^󸀠_k–1, k= 0, ...,n, (4.358) where we set!^󸀠n+1≡0,"^󸀠–1≡0.

4.8 Duality for Exotic Bialgebras 183

To summarize the list of irreps ofs14 onS14 is:

– one-dimensional trivial

– two two-dimensional with all CasimirsA,̃ B̃²,D̃²having the value 1

– one-dimensional enumerated byn= 1, 2, ..., which for fixednhave Casimir values 2n, 0, 0 forA,̃ B̃²,D̃², respectively

– pairs of one-dimensional irreps enumerated byn= 1, 2, ...4= 2, 4, ... ..., 2n, which have Casimir values 2n, 0,4²forA,̃ B̃²,D̃², respectively.

– pairs of one-dimensional irreps enumerated byn= 1, 2, ...;4= 1, 3, ... ..., (2n+ 1), which have Casimir values 2n+ 1, 0,4²forA,̃ B̃²,D̃², respectively

Finally, we note in the irreps ofs14 onS14 all Casimirs can take only non-negative integer values.

Im Dokument Vladimir K. Dobrev Invariant Differential Operators (Seite 188-195)