Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero
Assume that is an algebraically closed field with characteristic zero. The Racah algebra ℜ is the unital associative -algebra defined by generators and relations in the following way. The generators are A, B, C, D, and the relations assert that [A, B]=[B, C]=[C, A]=2D and that each of [A, D]+AC−B...
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2020
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| Цитувати: | Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero. Hau-Wen Huang and Sarah Bockting-Conrad. SIGMA 16 (2020), 018, 17 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859863072189972480 |
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| author | Huang, Hau-Wen Bockting-Conrad, Sarah |
| author_facet | Huang, Hau-Wen Bockting-Conrad, Sarah |
| citation_txt | Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero. Hau-Wen Huang and Sarah Bockting-Conrad. SIGMA 16 (2020), 018, 17 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Assume that is an algebraically closed field with characteristic zero. The Racah algebra ℜ is the unital associative -algebra defined by generators and relations in the following way. The generators are A, B, C, D, and the relations assert that [A, B]=[B, C]=[C, A]=2D and that each of [A, D]+AC−BA, [B, D]+BA−CB, [C, D]+CB−AC is central in ℜ. In this paper, we discuss the finite-dimensional irreducible ℜ-modules in detail and classify them up to isomorphism. To do this, we apply an infinite-dimensional ℜ-module and its universal property. We additionally give the necessary and sufficient conditions for A, B, C to be diagonalizable on finite-dimensional irreducible ℜ-modules.
|
| first_indexed | 2025-12-17T12:04:17Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 018, 17 pages
Finite-Dimensional Irreducible Modules
of the Racah Algebra at Characteristic Zero
Hau-Wen HUANG † and Sarah BOCKTING-CONRAD ‡
† Department of Mathematics, National Central University, Chung-Li 32001, Taiwan
E-mail: hauwenh@math.ncu.edu.tw
‡ Department of Mathematical Sciences, DePaul University, Chicago, Illinois, USA
E-mail: sarah.bockting@depaul.edu
Received November 12, 2019, in final form March 16, 2020; Published online March 24, 2020
https://doi.org/10.3842/SIGMA.2020.018
Abstract. Assume that F is an algebraically closed field with characteristic zero. The
Racah algebra < is the unital associative F-algebra defined by generators and relations in
the following way. The generators are A, B, C, D and the relations assert that [A,B] =
[B,C] = [C,A] = 2D and that each of [A,D]+AC−BA, [B,D]+BA−CB, [C,D]+CB−AC
is central in <. In this paper we discuss the finite-dimensional irreducible <-modules in detail
and classify them up to isomorphism. To do this, we apply an infinite-dimensional <-module
and its universal property. We additionally give the necessary and sufficient conditions for
A, B, C to be diagonalizable on finite-dimensional irreducible <-modules.
Key words: Racah algebra; quadratic algebra; irreducible modules; tridiagonal pairs; uni-
versal property
2020 Mathematics Subject Classification: 81R10; 16S37
1 Introduction
Throughout this paper, we adopt the following conventions. Let F denote an algebraically closed
field and let charF denote the characteristic of F. Let Z denote the set of integers and let N
denote the set of nonnegative integers. The bracket [ , ] stands for the commutator.
In this paper we consider the Racah algebra < over F defined by generators and relations as
follows. The generators are A, B, C, D and the relations assert that
[A,B] = [B,C] = [C,A] = 2D
and that each of
[A,D] +AC −BA, [B,D] +BA− CB, [C,D] + CB −AC
is central in <. The Racah algebra < is a universal analog of the original Racah algebras
which first appeared in [17]. In that paper, the original Racah algebras were used to establish
a link between representation theory and the quantum mechanical coupling of three angular
momenta. Since that time, the connections between the Racah algebras and many other areas
have been explored. We mention a few of them here. Their connections with the additive
double-affine Hecke algebra of type
(
C∨1 , C1
)
, the Bannai–Ito algebra, and the Lie algebras
su(2), su(1, 1) were investigated in [6, 9, 10, 14]. Their realizations via the Racah polynomials,
the intermediate Casimir operators, and the superintegrable models in two dimensions were
presented in [5, 7, 8, 9, 11]. For information concerning the higher rank Racah algebras, see [3, 4].
We now mention an error in the literature on Racah algebras. In [2], the authors considered
the finite-dimensional irreducible modules of the original Racah algebras when charF = 0.
mailto:hauwenh@math.ncu.edu.tw
mailto:sarah.bockting@depaul.edu
https://doi.org/10.3842/SIGMA.2020.018
2 H.-W. Huang and S. Bockting-Conrad
In [2, Lemma 5.6], it was claimed that the defining generators can be diagonalized on any
finite-dimensional irreducible module of the Racah algebras. This result was then used in their
classification of finite-dimensional irreducible modules of the Racah algebras in [2, Section 6]. It
turns out that [2, Lemma 5.6] is conditional. We give the following example to help illustrate
the issue arising in [2].
Example 1.1. It is routine to verify that there exists a five-dimensional <-module V that has
an F-basis {vi}4i=0 with respect to which the matrices representing A, B, C, D are 1
4 times
15 0 0 0 0
4 3 0 0 0
0 4 −1 0 0
0 0 4 3 0
0 0 0 4 15
,
15 −36 0 0 0
0 3 −6 0 0
0 0 −1 −6 0
0 0 0 3 −36
0 0 0 0 15
,
−9 36 0 0 0
−4 15 6 0 0
0 −4 23 6 0
0 0 −4 15 36
0 0 0 −4 −9
,
18 −54 0 0 0
6 −15 −3 0 0
0 2 0 3 0
0 0 −2 15 54
0 0 0 −6 −18
,
respectively. It follows that
[A,D] +AC −BA = [B,D] +BA− CB = [C,D] + CB −AC = 0
on the <-module V . For each of A, B, C, it is straightforward to verify that its minimal
polynomial on V is(
x+
1
4
)(
x− 3
4
)2(
x− 15
4
)2
.
Therefore none of A, B, C is diagonalizable on V . We now show that V is in fact irreducible.
Let W denote a nonzero <-submodule of V . We will show that W = V . Observe that the
element B has exactly three eigenvalues on V , namely 15
4 , 3
4 , −1
4 . A direct calculation yields
that the corresponding eigenspaces are each of dimension 1 and are spanned by
v0, 3v0 + v1, 27v0 + 12v1 + 8v2, (1.1)
respectively. Since W is nonzero, at least one of 15
4 , 3
4 , −1
4 is an eigenvalue of B on W . There-
fore W contains at least one of the elements listed in (1.1). Observe that the <-module V is
generated by v0. Thus, if v0 ∈W then W = V . If 3v0 + v1 ∈W then
v0 =
2
9
(
A− 2D − 9
4
)
(3v0 + v1) ∈W
and hence W = V . If 27v0 + 12v1 + 8v2 ∈W then
3v0 + v1 = − 1
18
(
A+ 2D − 11
4
)
(27v0 + 12v1 + 8v2) ∈W
and hence W = V . Therefore W = V . Since the <-module V is irreducible, we now have
a counterexample to [2, Lemma 5.6].
Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero 3
In light of the above example, we see that the finite-dimensional irreducible <-modules are
not yet completely classified. The goal of this paper is to provide such a classification. The
idea of our classification comes from [13]. We mention that a similar issue arises in the case of
the Bannai–Ito algebra BI [12] which is addressed by the first author in [16]. The result [15,
Theorem 5.4] reveals that the Racah algebra < is isomorphic to an F-subalgebra of BI. As an
application of [16] and this result, the lattices of <-submodules of finite-dimensional irreducible
BI-modules are classified in [14].
The outline of this paper is as follows. In Section 2 we state our classification of finite-
dimensional irreducible <-modules in Theorem 2.5. In Section 3 we display an infinite-dimensio-
nal <-module and describe its universal property. In Section 4 we give necessary and sufficient
conditions for the irreducibility of finite-dimensional <-modules. In Section 5 we study the
isomorphism classes of finite-dimensional irreducible <-modules. In Section 6 we give our proof
of Theorem 2.5.
2 Statement of results
In this section we more formally introduce the Racah algebra < and state the main result of
the paper which gives a classification of the finite-dimensional irreducible modules of the Racah
algebra <. This main result will be proved later in Section 6.
Definition 2.1 ([1, Definition 3.1]). The Racah algebra < is the unital associative F-algebra
defined by generators and relations in the following way. The generators are A, B, C, D. The
relations state that
[A,B] = [B,C] = [C,A] = 2D (2.1)
and that each of
[A,D] +AC −BA, [B,D] +BA− CB, [C,D] + CB −AC
is central in <.
It follows from the above definition that the element A + B + C is also central in <. For
notational convenience, we let
α = [A,D] +AC −BA, (2.2)
β = [B,D] +BA− CB, (2.3)
γ = [C,D] + CB −AC, (2.4)
δ = A+B + C. (2.5)
Lemma 2.2.
(i) The Racah algebra < is generated by the elements A, B, C.
(ii) The Racah algebra < is generated by the elements A, B, δ.
Proof. (i) By (2.1) the element D can be expressed in terms of A, B. Hence (i) follows from
Definition 2.1. (ii) By (2.5) the element C can be expressed in terms of A, B, δ. Hence (ii)
follows from (i). �
Lemma 2.3. The F-algebra < has a presentation given by generators A, B, α, β, δ and relations
A2B − 2ABA+BA2 − 2AB − 2BA = 2A2 − 2Aδ + 2α, (2.6)
AB2 − 2BAB +B2A− 2AB − 2BA = 2B2 − 2Bδ − 2β, (2.7)
αA = Aα, βA = Aβ, δA = Aδ,
αB = Bα, βB = Bβ, δB = Bδ, αδ = δα, βδ = δβ.
4 H.-W. Huang and S. Bockting-Conrad
Proof. We know from Lemma 2.2(ii) that A, B, α, β, δ generate <. Observe that C = δ−A−B
by (2.5) and D = 1
2 [A,B] by (2.1). The result can now be obtained by either using these two
facts to eliminate C, D from the presentation of < given in Definition 2.1 or by using D = 1
2 [A,B]
to eliminate D from the presentation of < given in [1, Proposition 3.4]. �
In the following proposition, we assert the existence of certain finite-dimensional <-modules
and describe the actions of the generators of < on these modules. A reader familiar with the
theory of tridiagonal pairs will immediately recognize the form of the matrices representing A
and B as precisely those given in Terwilliger’s 2001 seminal work on tridiagonal pairs [18,
Theorem 3.2].
Proposition 2.4. For any a, b, c ∈ F and any d ∈ N there exists a (d+1)-dimensional <-module
Rd(a, b, c) satisfying each of the following conditions:
(i) There exists an F-basis {vi}di=0 for Rd(a, b, c) with respect to which the matrices represen-
ting A and B are
θ0 0
1 θ1
1 θ2
. . .
. . .
0 1 θd
,
θ∗0 ϕ1 0
θ∗1 ϕ2
θ∗2
. . .
. . . ϕd
0 θ∗d
,
respectively, where
θi =
(
a+ d
2 − i
)(
a+ d
2 − i+ 1
)
, 0 ≤ i ≤ d,
θ∗i =
(
b+ d
2 − i
)(
b+ d
2 − i+ 1
)
, 0 ≤ i ≤ d,
ϕi = i(i− d− 1)
(
a+ b+ c+ d
2 − i+ 2
)(
a+ b− c+ d
2 − i+ 1
)
, 1 ≤ i ≤ d.
(ii) The elements α, β, δ act on Rd(a, b, c) as scalar multiplication by
(c− b)(c+ b+ 1)
(
a− d
2
)(
a+ d
2 + 1
)
,
(a− c)(a+ c+ 1)
(
b− d
2
)(
b+ d
2 + 1
)
,
d
2
(
d
2 + 1
)
+ a(a+ 1) + b(b+ 1) + c(c+ 1),
respectively.
Proof. Using Lemma 2.3, this result can be verified through routine, though tedious, compu-
tations. �
In order to state our main result more succinctly, we will use the following conventions and
definitions. Let d ∈ N and let P = Pd denote the set of all (a, b, c) ∈ F3 that satisfy
a+ b+ c+ 1,−a+ b+ c, a− b+ c, a+ b− c 6∈
{
d
2 − i
∣∣ i = 1, 2, . . . , d
}
.
We define an action of the abelian group {±1}3 on P by
(a, b, c)(−1,1,1) = (−a− 1, b, c), (a, b, c)(1,−1,1) = (a,−b− 1, c),
(a, b, c)(1,1,−1) = (a, b,−c− 1)
for all (a, b, c) ∈ P. We let P/{±1}3 denote the set of the {±1}3-orbits of P. For (a, b, c) ∈ P,
let [a, b, c] denote the {±1}3-orbit of P that contains (a, b, c). We are now ready to state the
classification of finite-dimensional irreducible <-modules.
Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero 5
Theorem 2.5. Assume that F is algebraically closed with charF = 0. Let d denote a nonnegative
integer. Let M denote the set of all isomorphism classes of irreducible <-modules that have
dimension d+ 1. Then there exists a bijection R : P/{±1}3 →M given by
[a, b, c] 7→ the isomorphism class of Rd(a, b, c)
for all [a, b, c] ∈ P/{±1}3.
We will give a proof of Theorem 2.5 in Section 6.
3 An infinite-dimensional <-module and its universal property
In this section we introduce an infinite-dimensional <-module and its universal property in order
to prove Theorem 2.5. For convenience the following conventions are used throughout the rest
of this paper. We let a, b, c, ν denote any scalars in F. We define the following families of
parameters associated with a, b, c, ν:
θi =
(
a+ ν
2 − i
)(
a+ ν
2 − i+ 1
)
for all i ∈ Z, (3.1)
θ∗i =
(
b+ ν
2 − i
)(
b+ ν
2 − i+ 1
)
for all i ∈ Z, (3.2)
φi = i(i− ν − 1)
(
a− b+ c− ν
2 + i
)(
a− b− c− ν
2 + i− 1
)
for all i ∈ Z, (3.3)
ϕi = i(i− ν − 1)
(
a+ b+ c+ ν
2 − i+ 2
)(
a+ b− c+ ν
2 − i+ 1
)
for all i ∈ Z, (3.4)
ζ = (c− b)(c+ b+ 1)
(
a− ν
2
)(
a+ ν
2 + 1
)
, (3.5)
ζ∗ = (a− c)(a+ c+ 1)
(
b− ν
2
)(
b+ ν
2 + 1
)
, (3.6)
η = ν
2
(
ν
2 + 1
)
+ a(a+ 1) + b(b+ 1) + c(c+ 1). (3.7)
Proposition 3.1. There exists an <-module Mν(a, b, c) satisfying each of the following condi-
tions:
(i) There exists an F-basis {mi}∞i=0 for Mν(a, b, c) with respect to which the matrices repre-
senting A and B are
θ0 0
1 θ1
1 θ2
· ·
· ·
0 · ·
,
θ∗0 ϕ1 0
θ∗1 ϕ2
θ∗2 ·
· ·
· ·
0 ·
,
respectively.
(ii) The elements α, β, δ act on Mν(a, b, c) as scalar multiplication by ζ, ζ∗, η, respectively.
Proof. Using Lemma 2.3, this result can be verified through routine computations. �
Throughout the rest of this paper we will let {mi}∞i=0 denote the F-basis for Mν(a, b, c) from
Proposition 3.1(i). The following result is an immediate consequence of Proposition 3.1(i).
Lemma 3.2. mj+1 =
j∏
h=i
(A− θh)mi for any i, j ∈ N with i ≤ j.
Shortly we will describe the <-module Mν(a, b, c) in an alternate way. To aid us in this
endeavor, we first recall a Poincaré–Birkhoff–Witt basis for <.
6 H.-W. Huang and S. Bockting-Conrad
Lemma 3.3 ([1, Theorem 5.1]). The elements
AiDjBkαrδsβt for all i, j, k, r, s, t ∈ N
form an F-basis of <.
Let Iν(a, b, c) denote the left ideal of < generated by the elements
B − θ∗0, (3.8)
(B − θ∗1)(A− θ0)− ϕ1, (3.9)
α− ζ, β − ζ∗, δ − η. (3.10)
We now consider certain cosets of </Iν(a, b, c).
Lemma 3.4. For each n ∈ N, each of the following holds:
(i) BAn + Iν(a, b, c) is an F-linear combination of Ai + Iν(a, b, c) for all 0 ≤ i ≤ n,
(ii) DAn + Iν(a, b, c) is an F-linear combination of Ai + Iν(a, b, c) for all 0 ≤ i ≤ n+ 1,
(iii) Dn + Iν(a, b, c) is an F-linear combination of Ai + Iν(a, b, c) for all 0 ≤ i ≤ n.
Proof. (i) We proceed by induction on n. Since Iν(a, b, c) contains the element (3.8), the
statement holds for n = 0. Since Iν(a, b, c) contains both of the elements (3.8) and (3.9), the
statement holds for n = 1. Now suppose n ≥ 2. Right multiplying each side of (2.6) by An−2
yields that
A2BAn−2 − 2ABAn−1 +BAn − 2ABAn−2 − 2BAn−1 = 2An − 2An−1δ + 2An−2α.
Since Iν(a, b, c) contains each of the elements listed in (3.10), it follows that BAn is congruent
to
2ABAn−1 + 2ABAn−2 −A2BAn−2 + 2BAn−1 + 2An − 2ηAn−1 + 2ζAn−2 (3.11)
modulo Iν(a, b, c). By the inductive hypothesis, the element (3.11) is congruent to an F-linear
combination of Ai, for all 0 ≤ i ≤ n, modulo Iν(a, b, c). Therefore (i) follows.
(ii) Observe that DAn = 1
2
(
ABAn − BAn+1
)
by (2.1). In light of this fact, the result now
follows from Lemma 3.4(i).
(iii) We proceed by induction on n. The statement holds trivially for n = 0. Now suppose
that n ≥ 1. By the inductive hypothesis, Dn = DDn−1 is congruent to an F-linear combination
of
DAi, 0 ≤ i ≤ n− 1, (3.12)
modulo Iν(a, b, c). By Lemma 3.4(ii) each of the elements listed in (3.12) is an F-linear combi-
nation of Ak, for all 0 ≤ k ≤ n, modulo Iν(a, b, c). Therefore the result follows. �
Lemma 3.5. The F-vector space </Iν(a, b, c) is spanned by
Ai + Iν(a, b, c) for all i ∈ N.
Proof. By Lemma 3.3, the F-vector space </Iν(a, b, c) is spanned by
AiDjBkαrδsβt + Iν(a, b, c) for all i, j, k, r, s, t ∈ N. (3.13)
Since Iν(a, b, c) contains the elements listed in (3.8) and (3.10), each of the elements listed
in (3.13) can be expressed as an F-linear combination of
AiDj + Iν(a, b, c) for all i, j ∈ N.
The result now follows from these facts along with Lemma 3.4(iii). �
Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero 7
We are now ready to give our second description of Mν(a, b, c).
Theorem 3.6. There exists a unique <-module homomorphism
Φ: </Iν(a, b, c)→Mν(a, b, c)
that sends 1 + Iν(a, b, c) to m0. Moreover, Φ is an isomorphism.
Proof. Consider the <-module homomorphism Ψ: < → Mν(a, b, c) that sends 1 to m0. By
Proposition 3.1(i), the elements (3.8) and (3.9) are in the kernel of Ψ. By Proposition 3.1(ii),
the elements listed in (3.10) are also in the kernel of Ψ. Hence Iν(a, b, c) is contained in the kernel
of Ψ. It follows that Ψ induces an <-module homomorphism Φ: </Iν(a, b, c)→Mν(a, b, c) that
maps 1 + Iν(a, b, c) to m0. Observe that Φ is the unique <-module homomorphism with the
desired property since </Iν(a, b, c) is generated by 1 + Iν(a, b, c) as an <-module.
By Lemma 3.2 the homomorphism Φ sends
i−1∏
h=1
(A− θh) + Iν(a, b, c) (3.14)
to mi for all i ∈ N. Since {mi}∞i=0 are linearly independent, the cosets (3.14) are linearly
independent. Combining this with Lemma 3.5, we see that the cosets (3.14) are an F-basis for
</Iν(a, b, c). Therefore Φ is an isomorphism. �
As a consequence of Theorem 3.6, the <-module Mν(a, b, c) satisfies the following universal
property.
Proposition 3.7. If V is an <-module which has a vector v ∈ V satisfying
Bv = θ∗0v,
(B − θ∗1)(A− θ0)v = ϕ1v,
αv = ζv, βv = ζ∗v, δv = ηv,
then there exists a unique <-module homomorphism Mν(a, b, c)→ V that sends m0 to v.
For the rest of the present paper, we will consider the case ν = d. Define Nd(a, b, c) to be the
A-cyclic F-subspace of Md(a, b, c) generated by the element md+1.
Lemma 3.8. Nd(a, b, c) is an <-submodule of Md(a, b, c) with the F-basis {mi}∞i=d+1.
Proof. Recall from Lemma 3.2 that
(A− θi)mi = mi+1 for all i ≥ d+ 1.
It follows from this fact that {mi}∞i=d+1 is an F-basis for Nd(a, b, c).
We now show that Nd(a, b, c) is an <-submodule of Md(a, b, c). By Proposition 3.1(i),
(B − θ∗i )mi = ϕimi−1 for all i ≥ d+ 1.
By (3.4), the scalar ϕd+1 = 0 when ν = d. Hence Nd(a, b, c) is B-invariant. By Proposi-
tion 3.1(ii), the element δ acts on Nd(a, b, c) as scalar multiplication by η. It now follows from
Lemma 2.2(ii) that Nd(a, b, c) is an <-submodule of Md(a, b, c). �
Recall the <-module Rd(a, b, c) from Proposition 2.4. In the sequel we display how the <-
module Rd(a, b, c) is connected to Mν(a, b, c). For convenience we let {vi}di=0 denote the F-basis
for Rd(a, b, c) from Proposition 2.4(i) in the rest of this paper.
8 H.-W. Huang and S. Bockting-Conrad
Lemma 3.9. There exists a unique <-module isomorphism
Md(a, b, c)/Nd(a, b, c)→ Rd(a, b, c)
that sends mi +Nd(a, b, c) to vi for all 0 ≤ i ≤ d.
Proof. By Lemma 3.8, Md(a, b, c)/Nd(a, b, c) is a (d+1)-dimensional <-module with the F-basis
{mi +Nd(a, b, c)}di=0. (3.15)
Observe that the matrices representing A and B with respect to the F-basis {vi}di=0 for Rd(a, b, c)
are identical with the matrices representing A and B with respect to the F-basis (3.15) for
Md(a, b, c)/Nd(a, b, c) by Propositions 2.4(i) and 3.1(i). By Propositions 2.4(ii) and 3.1(ii), the
actions of δ on Rd(a, b, c) andMd(a, b, c)/Nd(a, b, c) are scalar multiplication by the same scalar η.
In light of these comments, the result now follows from Lemma 2.2(ii). �
Proposition 3.10. Suppose that V is an <-module which has a vector v ∈ V satisfying
d∏
i=0
(A− θi)v = 0. (3.16)
If there is an <-module homomorphism Md(a, b, c)→ V that sends m0 to v, then there exists an
<-module homomorphism Rd(a, b, c)→ V that sends v0 to v.
Proof. Let % denote the <-module homomorphism Md(a, b, c) → V that sends m0 to v. By
Lemma 3.2, we have
md+1 =
d∏
i=0
(A− θi)m0.
Combining this with (3.16), we see that md+1 is in the kernel of %. Therefore Nd(a, b, c)
is contained in the kernel of %. By Lemma 3.8, there exists an <-module homomorphism
Md(a, b, c)/Nd(a, b, c) → V that sends m0 + Nd(a, b, c) to v. The result follows from this fact
along with Lemma 3.9. �
4 Conditions for the irreducibility of Rd(a, b, c)
In this section, we derive the necessary and sufficient conditions for Rd(a, b, c) to be irreducible
in terms of the parameters a, b, c, d. Throughout this section, we let
wi =
i−1∏
h=0
(A− θd−h)v0, 0 ≤ i ≤ d. (4.1)
Lemma 4.1. If the <-module Rd(a, b, c) is irreducible, then each of the following holds:
(i) charF = 0 or charF > d,
(ii) a+ b+ c+ 1, a+ b− c 6∈
{
d
2 − i
∣∣ i = 1, 2, . . . , d
}
.
Proof. Suppose that there is an integer i, with 1 ≤ i ≤ d, such that ϕi = 0. By Proposition 2.4,
the F-subspace W of Rd(a, b, c) spanned by {vh}dh=i is invariant under A, B, δ. It follows from
Lemma 2.2(ii) that W is an <-submodule of Rd(a, b, c), a contradiction to the irreducibility of
Rd(a, b, c). Therefore ϕi 6= 0 for all 1 ≤ i ≤ d, which is equivalent to (i) and (ii) by (3.4). �
Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero 9
Lemma 4.2. The elements {wi}di=0 form an F-basis for Rd(a, b, c).
Proof. It follows from Proposition 2.4(i) that
vi =
i−1∏
h=0
(A− θh)v0, 0 ≤ i ≤ d.
Comparing this with (4.1), the result now follows. �
Proposition 4.3. The <-module Rd(a, b, c) is isomorphic to the <-module Rd(−a − 1, b, c).
Moreover, the matrices representing A and B with respect to the F-basis {wi}di=0 for Rd(a, b, c)
are
θd 0
1 θd−1
1 θd−2
. . .
. . .
0 1 θ0
,
θ∗0 φ1 0
θ∗1 φ2
θ∗2
. . .
. . . φd
0 θ∗d
, (4.2)
respectively.
Proof. By Proposition 2.4(i), there exists an F-basis {ui}di=0 for Rd(−a − 1, b, c) with respect
to which the matrices representing A and B are equal to the matrices displayed in (4.2). By
Lemma 4.2, it suffices to show that there is an <-module homomorphism Rd(a, b, c)→ Rd(−a−1,
b, c) that sends wi to ui for all 0 ≤ i ≤ d.
Observe that Bu0 = θ∗0u0 and a direct calculation yields that
(B − θ∗1)(A− θ0)u0 = ϕ1u0.
By Proposition 2.4(ii), the elements α, β, δ act on Rd(−a − 1, b, c) as scalar multiplication
by ζ, ζ∗, η, respectively. According to Proposition 3.7, there exists a unique <-module ho-
momorphism Md(a, b, c) → Rd(−a − 1, b, c) that sends m0 to u0. By inspecting the matrix
representing A given in (4.2) we see that
d∏
i=0
(A− θi)u0 = 0.
Hence there exists a <-module homomorphism
Rd(a, b, c)→ Rd(−a− 1, b, c)
that maps v0 to u0 by Proposition 3.10. It now follows from (4.1) that this homomorphism
sends wi to ui for all 0 ≤ i ≤ d. The result follows. �
Lemma 4.4. If the <-module Rd(a, b, c) is irreducible, then each of the following holds:
(i) charF = 0 or charF > d,
(ii) a+ b+ c+ 1,−a+ b+ c, a− b+ c, a+ b− c 6∈
{
d
2 − i
∣∣ i = 1, 2, . . . , d
}
.
Proof. By Proposition 4.3, the <-module Rd(a, b, c) is isomorphic to Rd(−a − 1, b, c). Hence
the result follows by applying Lemma 4.1 to both Rd(a, b, c) and Rd(−a− 1, b, c). �
10 H.-W. Huang and S. Bockting-Conrad
Shortly we will show that the converse of Lemma 4.4 is also true. To aid us in doing so, we
establish the following notation. We define
R =
d∏
h=1
(B − θ∗h),
Si =
d−i∏
h=1
(A− θd−h+1), 0 ≤ i ≤ d.
It follows from Proposition 2.4(i) that Rv is a scalar multiple of v0 for all v ∈ Rd(a, b, c). Thus,
for any integers i, j with 0 ≤ i, j ≤ d, there exists a unique Lij ∈ F such that
RSivj = Lijv0. (4.3)
By examining Proposition 2.4(i) further, we see that
Lij = 0, 0 ≤ i < j ≤ d, (4.4)
Lij = (θi − θj−1)Li,j−1 + Li−1,j−1, 1 ≤ j ≤ i ≤ d. (4.5)
It follows from Proposition 4.3 that
Li0 =
i∏
h=1
(θ∗0 − θ∗d−h+1)
d−i∏
h=1
φh, 0 ≤ i ≤ d. (4.6)
Solving the recurrence relation (4.5) with the initial conditions (4.4) and (4.6) yields that
Lij =
(
d− i+ j
j
)(
i
j
)(
d
j
)−1 i−j∏
h=1
(θ∗0 − θ∗d−h+1)
d−i∏
h=1
φh
j∏
h=1
ϕh, 0 ≤ j ≤ i ≤ d. (4.7)
Theorem 4.5. The <-module Rd(a, b, c) is irreducible if and only if both of the following con-
ditions hold:
(i) charF = 0 or charF > d,
(ii) a+ b+ c+ 1,−a+ b+ c, a− b+ c, a+ b− c 6∈
{
d
2 − i
∣∣ i = 1, 2, . . . , d
}
.
Proof. (⇒) This is immediate from Lemma 4.4.
(⇐) To see the irreducibility of Rd(a, b, c), we assume that W is a nonzero <-submodule of
Rd(a, b, c) and show that W = Rd(a, b, c). Pick a nonzero vector w ∈ W . Since W is invariant
under A and B, it follows that
RSiw ∈W, 0 ≤ i ≤ d. (4.8)
Since {vi}di=0 is an F-basis for Rd(a, b, c), there exist aj ∈ F, for 0 ≤ j ≤ d, such that
w =
d∑
j=0
ajvj .
It now follows from (4.3) that
RSiw =
d∑
j=0
Lijaj
v0, 0 ≤ i ≤ d. (4.9)
Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero 11
Recall the parameters {φi}i∈Z and {ϕi}i∈Z from (3.3) and (3.4), respectively. It follows from
our assumptions (i) and (ii) that the scalars ϕi 6= 0 and φi 6= 0 for all 1 ≤ i ≤ d. Let L denote the
(d+ 1)× (d+ 1) matrix, indexed by 0, 1, . . . , d, with (i, j)-entry given by Lij for all 0 ≤ i, j ≤ d.
By (4.4), the square matrix L is lower triangular. By (4.7), the diagonal entries of L are
Lii =
d−i∏
h=1
φh
i∏
h=1
ϕi, 0 ≤ i ≤ d,
which we know to be nonzero. Therefore the matrix L is nonsingular. Since w is nonzero at
least one of {aj}dj=0 is nonzero. Hence there exists an integer i with 0 ≤ i ≤ d such that
d∑
j=0
Lijaj 6= 0. (4.10)
Combining (4.10) with (4.8) and (4.9), we find that v0 ∈ W . Since the <-module Rd(a, b, c) is
generated by v0, it follows that W = Rd(a, b, c) and so Rd(a, b, c) is irreducible. �
5 The isomorphism class of the <-module Rd(a, b, c)
In Proposition 4.3, we showed that the <-module Rd(a, b, c) is isomorphic to the <-module
Rd(−a− 1, b, c). In this section, we discuss the isomorphism class of Rd(a, b, c) in further detail.
Proposition 5.1. The <-module Rd(a, b, c) is isomorphic to the <-module Rd(a, b,−c− 1).
Proof. By Proposition 2.4(i), there are F-bases for Rd(a, b, c) and Rd(a, b,−c− 1) with respect
to which the matrices representing A and B are the same. By Proposition 2.4(ii), the actions
of δ on Rd(a, b, c) and Rd(a, b,−c−1) are both scalar multiplication by the same scalar η. Hence
Rd(a, b, c) is isomorphic to Rd(a, b,−c− 1) by Lemma 2.2(ii). �
Proposition 5.2. If the <-module Rd(a, b, c) is irreducible, then Rd(a, b, c) is isomorphic to the
<-module Rd(a,−b− 1, c).
Proof. By Proposition 2.4(i), there is an F-basis {ui}di=0 for Rd(a,−b − 1, c) with respect to
which the matrices representing A and B are
θ0 0
1 θ1
1 θ2
. . .
. . .
0 1 θd
,
θ∗d φd 0
θ∗d−1 φd−1
θ∗d−2
. . .
. . . φ1
0 θ∗0
, (5.1)
respectively. Since the <-module Rd(a, b, c) is irreducible, it follows from Theorem 4.5 that
φi 6= 0 for all 1 ≤ i ≤ d. Thus we may set
v =
d∑
i=0
i∏
h=1
θ∗0 − θ∗d−h+1
φd−h+1
ui.
A direct calculation yields that Bv = θ∗0v and
(B − θ∗1)(A− θ0)v = ϕ1v.
12 H.-W. Huang and S. Bockting-Conrad
By Proposition 2.4(ii), the elements α, β, δ act on Rd(a,−b − 1, c) as scalar multiplication
by ζ, ζ∗, η, respectively. According to Proposition 3.7, there exists a unique <-module ho-
momorphism Md(a, b, c) → Rd(a,−b − 1, c) that maps m0 to v. By inspecting the matrix
representing A given in (5.1), we see that
d∏
i=0
(A− θi)v = 0.
Hence there exists an <-module homomorphism
Rd(a, b, c)→ Rd(a,−b− 1, c) (5.2)
that sends v0 to v by Proposition 3.10. Since the <-module Rd(a, b, c) is irreducible, the <-
module Rd(a,−b−1, c) is also irreducible by Theorem 4.5. Therefore (5.2) is an isomorphism. �
We end this section with a simple combination of Propositions 4.3, 5.1, and 5.2.
Theorem 5.3. If the <-module Rd(a, b, c) is irreducible, then Rd(a, b, c) is isomorphic to each
of the <-modules Rd(−a− 1, b, c), Rd(a,−b− 1, c) and Rd(a, b,−c− 1).
6 The proof of Theorem 2.5
Theorems 4.5 and 5.3 indicate that the map R in Theorem 2.5 is well-defined. In this section,
we shall show that R is a bijection.
Lemma 6.1. Assume that F is algebraically closed. If V is a finite-dimensional irreducible
<-module, then each central element of < acts on V as scalar multiplication.
Proof. This result follows from applying Schur’s lemma to <. �
Lemma 6.2. For any i ∈ Z, each of the following hold:
(i) θi+1 + θi−1 = 2(θi + 1),
(ii) θi+1θi−1 = θi(θi − 2).
Proof. The result can be routinely verified using (3.1). �
Theorem 6.3. Assume that F is algebraically closed with charF = 0. Let d denote a nonnegative
integer. If V is a (d+ 1)-dimensional irreducible <-module, then there exist a, b, c ∈ F such that
the <-module Rd(a, b, c) is isomorphic to V .
Proof. Given any scalar κ ∈ F, we define
ϑi(κ) = (κ− i)(κ− i+ 1) for all i ∈ Z.
Since charF = 0, for any distinct integers i, j, the scalars ϑi(κ) and ϑj(κ) are equal if and only
if i+ j = 2κ+ 1. In particular {ϑi(κ)}−∞i=0 contains infinitely many values.
Since F is algebraically closed, we may choose a scalar κ ∈ F such that ϑ0(κ) is an eigenvalue
of A on V . Since V is of dimension d+ 1, there are at most d+ 1 distinct eigenvalues of A on V .
Thus, there exists an integer j ≤ 0 such that ϑj(κ) is an eigenvalue of A but ϑj−1(κ) is not an
eigenvalue of A on V . Set
a = κ− j − d
2 .
Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero 13
Similarly, there exists a scalar λ ∈ F and an integer k ≤ 0 such that ϑk(λ) is an eigenvalue of B
but ϑk−1(λ) is not an eigenvalue of B in V . We set
b = λ− k − d
2 .
Observe that under these settings, we have
θi = ϑi+j(κ) for all i ∈ Z, (6.1)
θ∗i = ϑi+k(λ) for all i ∈ Z. (6.2)
By Lemma 6.1, the element δ acts on V as scalar multiplication. Since F is algebraically closed,
there exists a scalar c ∈ F such that the action of δ on V is the scalar multiplication by
η = d
2
(
d
2 + 1
)
+ a(a+ 1) + b(b+ 1) + c(c+ 1).
To prove the theorem, it now suffices to show that there exists an <-module isomorphism from
Rd(a, b, c) into V .
Given any T ∈ < and θ ∈ F, we let
VT (θ) = {v ∈ V |Tv = θv}.
Pick any v ∈ VA(θ0). Applying each side of (2.6) to v and using Lemma 6.2 to simplify the
result, we obtain that
(A− θ−1)(A− θ1)Bv = 2(θ0(θ0 − η) + α)v. (6.3)
Left multiplying each side of (6.3) by (A− θ0), we obtain that
(A− θ−1)(A− θ0)(A− θ1)Bv = 0.
By (6.1), the scalar θ−1 is not an eigenvalue of A in V . Hence
(A− θ0)(A− θ1)Bv = 0.
In other words (A− θ1)Bv ∈ VA(θ0) and therefore VA(θ0) is invariant under (A− θ1)B. Since F
is algebraically closed, there exists an eigenvector u of (A − θ1)B in VA(θ0). Similarly, there
exists an eigenvector w of (B − θ∗1)A in VB(θ∗0). Define
ui =
i−1∏
h=0
(B − θ∗h)u for all i ∈ N, (6.4)
wi =
i−1∏
h=0
(A− θh)w for all i ∈ N. (6.5)
We now proceed by induction to show that
(A− θi)ui ∈ spanF{u0, u1, . . . , ui−1} for all i ∈ N. (6.6)
Since u is an eigenvector of (A − θ1)B in VA(θ0), the claim is true for i = 0, 1. Now suppose
that i ≥ 2. Applying each side of (2.7) to ui−2, we obtain that(
AB2 − 2BAB +B2A− 2AB − 2BA− 2B2 + 2ηB
)
ui−2 = −2βui−2. (6.7)
14 H.-W. Huang and S. Bockting-Conrad
By Lemma 6.1, the right-hand side of (6.7) is a scalar multiple of ui−2. Using the inductive
hypothesis, (6.4), and Lemma 6.2(i), we find that the left-hand side of (6.7) is equal to
(A− θi)ui (6.8)
plus an F-linear combination of u0, u1, . . . , ui−1. Combining the above results, the claim (6.6)
follows.
Next, we show that {ui}di=0 is an F-basis for V . Suppose on the contrary that there is an
integer h with 0 ≤ h ≤ d− 1 such that uh+1 is an F-linear combination of u0, u1, . . . , uh. Let W
denote the F-subspace of V spanned by u0, u1, . . . , uh. Since W is B-invariant by (6.4) and
A-invariant by (6.6), it follows that W is an <-submodule of V by Lemma 2.2(ii). Since V is
irreducible, this forces that W = V . By construction, W is of dimension at most d, a contradic-
tion. Therefore {ui}di=0 is an F-basis for V . By a similar argument, it follows that
(B − θ∗i )wi ∈ spanF{w0, w1, . . . , wi−1} for all i ∈ N (6.9)
and {wi}di=0 is an F-basis for V .
By (6.6), the matrix representing A with respect to the F-basis {ui}di=0 is upper triangular
with diagonal entries {θi}di=0. By the Cayley–Hamilton theorem, we have
d∏
i=0
(A− θi)w0 = 0. (6.10)
In other words Awd = θdwd by (6.5). Hence the matrix representing A with respect to the
F-basis {wi}di=0 is
θ0 0
1 θ1
1 θ2
. . .
. . .
0 1 θd
.
By (6.9), the matrix representing B with respect to the F-basis {wi}di=0 is upper triangular
with diagonal entries {θ∗i }di=0. We let {ϕ′i}di=1 denote its superdiagonal entries as follows
θ∗0 ϕ′1 ∗
θ∗1 ϕ′2
θ∗2
. . .
. . . ϕ′d
0 θ∗d
. (6.11)
Applying each side of (2.6) to wi−1, we obtain from the coefficients of wi that
ϕ′i+1 − 2ϕ′i + ϕ′i−1 = (θi − θi−1 + 2)θ∗i − (θi − θi−1 − 2)θ∗i−1 + 2(θi + θi−1 − η) (6.12)
for all 1 ≤ i ≤ d, where ϕ′0 and ϕ′d+1 are interpreted as zero. It is straightforward to verify
that {ϕi}di=1 also satisfy the recurrence relation (6.12). Since charF = 0, the corresponding
homogeneous recurrence relation
σi+1 − 2σi + σi−1 = 0, 1 ≤ i ≤ d,
Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero 15
with the initial values σ0 = 0 and σd+1 = 0, has the unique solution σi = 0 for all 0 ≤ i ≤ d+ 1.
Therefore ϕ′i = ϕi for all 1 ≤ i ≤ d.
Up to this point, we have shown that
Bw0 = θ∗0w0, (6.13)
(B − θ∗1)(A− θ0)w0 = ϕ1w0, (6.14)
δw0 = ηw0. (6.15)
Applying each side of (2.7) to w0 and using (6.14) to simplify the resulting equation, we find
that
βw0 = ζ∗w0. (6.16)
Using logic similar to what was used to show (6.14), we obtain (A − θ1)(B − θ∗0)u0 = ϕ1u0.
Now, by applying each side of (2.6) to u0 and using the above equation to simplify the resulting
equation, we find that αu0 = ζu0. It follows from Lemma 6.1 that
αw0 = ζw0. (6.17)
In view of (6.13)–(6.17), it follows from Proposition 3.7 that there exists a unique <-module
homomorphism Md(a, b, c) → V that sends m0 to w0. By Proposition 3.1(i), the entries
above the superdiagonal in (6.11) are zero. Combining the above <-module homomorphism
Md(a, b, c)→ V with (6.10), there is an <-module homomorphism
Rd(a, b, c)→ V (6.18)
that sends v0 to w0 by Proposition 3.10. Since the <-module V is irreducible, the homomor-
phism (6.18) is onto. Since Rd(a, b, c) and V are both of dimension d+ 1, it follows that (6.18)
is an isomorphism. The result follows. �
Lemma 6.4. The traces of A, B, C on the <-module Rd(a, b, c) are equal to d+ 1 times
a2 + a+
d(d+ 2)
12
, b2 + b+
d(d+ 2)
12
, c2 + c+
d(d+ 2)
12
,
respectively.
Proof. To compute the traces of A, B on Rd(a, b, c), use Proposition 2.4(i). By (2.5), the trace
of C is equal to the trace of δ minus the trace of A+B on Rd(a, b, c). Use the above facts along
with Proposition 2.4(ii) to compute the trace of C on Rd(a, b, c). �
The following is a quick consequence of Theorems 5.3, 6.3 and Lemma 6.4.
Corollary 6.5. Assume that F is algebraically closed with charF = 0. Let V denote a (d+ 1)-
dimensional irreducible <-module. Let trA, trB, trC denote the traces of A, B, C on V ,
respectively. Then the <-module Rd(a, b, c) is isomorphic to V if and only if a, b, c are the roots of
x2 + x+
d(d+ 2)
12
=
trA
d+ 1
,
x2 + x+
d(d+ 2)
12
=
trB
d+ 1
,
x2 + x+
d(d+ 2)
12
=
trC
d+ 1
,
respectively.
16 H.-W. Huang and S. Bockting-Conrad
We are now ready to prove Theorem 2.5.
Proof of Theorem 2.5. By Theorems 4.5 and 5.3, the mapR is well-defined. By Theorem 6.3,
the map R is onto. Since any element of < has the same trace on the isomorphic finite-
dimensional <-modules, it follows from Lemma 6.4 that R is one-to-one. �
In Example 1.1 we showed a five-dimensional irreducible <-module on which none of A, B, C
is diagonalizable. Note that the <-module is isomorphic to R4
(
−1
2 ,−
1
2 ,−
1
2
)
. We finish this
paper with the necessary and sufficient conditions for A, B, C to be diagonalizable on finite-
dimensional irreducible <-modules.
Theorem 6.6. Assume that F is algebraically closed with charF = 0. For any a, b, c ∈ F
and d ∈ N satisfying the conditions (i) and (ii) of Theorem 4.5, the following statements are
equivalent:
(i) A (resp. B) (resp. C) is diagonalizable on Rd(a, b, c),
(ii) a (resp. b) (resp. c) is not in{
i−d−1
2
∣∣ i = 1, 2, . . . , 2d− 1
}
. (6.19)
Proof. With reference to Proposition 2.4(i), the minimal polynomial of A in Rd(a, b, c) is
d∏
i=0
(x− θi).
Thus A is diagonalizable on Rd(a, b, c) if and only if the scalars {θi}di=0 are mutually distinct.
A direct calculation yields that the latter holds if and only if a is not in (6.19). By a similar
argument B is diagonalizable on Rd(a, b, c) if and only if b is not in (6.19).
To derive the condition for C as diagonalizable on Rd(a, b, c), we consider the (d + 1)-
dimensional <-module Rd(b, c, a). By Theorem 4.5 the <-module Rd(b, c, a) is irreducible. By
Definition 2.1 or [1, Proposition 4.1] there exists a unique F-algebra automorphism % of < that
sends A, B, C, D to C, A, B, D, respectively. Let Rd(b, c, a)% denote the <-module obtained
by pulling back the <-module Rd(b, c, a) via %. Observe that C is diagonalizable on Rd(b, c, a)%
if and only if c is not in (6.19). Using Corollary 6.5 yields that Rd(b, c, a)% is isomorphic to the
<-module Rd(a, b, c). The result follows. �
Acknowledgements
The research of the first author is supported by the Ministry of Science and Technology of
Taiwan under the project MOST 106-2628-M-008-001-MY4.
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1 Introduction
2 Statement of results
3 An infinite-dimensional -module and its universal property
4 Conditions for the irreducibility of Rd(a,b,c)
5 The isomorphism class of the -module Rd(a,b,c)
6 The proof of Theorem 2.5
References
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| id | nasplib_isofts_kiev_ua-123456789-210592 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-17T12:04:17Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Huang, Hau-Wen Bockting-Conrad, Sarah 2025-12-12T10:33:31Z 2020 Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero. Hau-Wen Huang and Sarah Bockting-Conrad. SIGMA 16 (2020), 018, 17 pages 1815-0659 2020 Mathematics Subject Classification: 81R10; 16S37 arXiv:1910.11446 https://nasplib.isofts.kiev.ua/handle/123456789/210592 https://doi.org/10.3842/SIGMA.2020.018 Assume that is an algebraically closed field with characteristic zero. The Racah algebra ℜ is the unital associative -algebra defined by generators and relations in the following way. The generators are A, B, C, D, and the relations assert that [A, B]=[B, C]=[C, A]=2D and that each of [A, D]+AC−BA, [B, D]+BA−CB, [C, D]+CB−AC is central in ℜ. In this paper, we discuss the finite-dimensional irreducible ℜ-modules in detail and classify them up to isomorphism. To do this, we apply an infinite-dimensional ℜ-module and its universal property. We additionally give the necessary and sufficient conditions for A, B, C to be diagonalizable on finite-dimensional irreducible ℜ-modules. The research of the first author is supported by the Ministry of Science and Technology of Taiwan under the project MOST 106-2628-M-008-001-MY4. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero Article published earlier |
| spellingShingle | Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero Huang, Hau-Wen Bockting-Conrad, Sarah |
| title | Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero |
| title_full | Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero |
| title_fullStr | Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero |
| title_full_unstemmed | Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero |
| title_short | Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero |
| title_sort | finite-dimensional irreducible modules of the racah algebra at characteristic zero |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210592 |
| work_keys_str_mv | AT huanghauwen finitedimensionalirreduciblemodulesoftheracahalgebraatcharacteristiczero AT bocktingconradsarah finitedimensionalirreduciblemodulesoftheracahalgebraatcharacteristiczero |