交错仿射A(1)1模块：表示理论与Virasoro的连接
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本研究论文深入探讨了分数阶仿射李代数A(1)1的表示理论及其与Virasoro代数表示理论之间的相互关系。在该领域的一个重要进展是，作者将仿射Kac模块定义为Wakimoto模块中的特定有限生成子模块。仿射Kac模块在数学物理学中扮演着关键角色，它们与经典的Kac模块概念相似，但适应了分数阶背景。
研究者提出了交错A(1)1模块的猜想，并通过详细的实例给出了证明，这展示了对新类型模块结构的理解和探索。这些交错模块不仅丰富了A(1)1代数的表示理论，而且可能有潜在的应用于量子场论或其他物理模型中的实际意义。
论文进一步拓展了Goddard-Kent-Olive（GKO）余子结构在仿射Kac模块和交错模块中的适用性，这是一种在代数簇上构建非平凡模块结构的有效工具。通过构建精确的函子，作者将A(1)1模块关联类别与对应于Virasoro代数的类别连接起来。这一函子在字符层面上的作用，不仅反映了Mukhi-Panda残差公式的精髓，还在不同代数结构间的相互转化中起到了桥梁作用。
更为具体地，论文分析了Verma模的分解，得出了所有不可约A(1)1模块字符的明确表达式。这对于理解这些模块的基本性质和结构至关重要。同时，研究者还重新审视了Malikov-Feigin-Fuchs向量的构造，揭示了它们在A(1)1代数框架下的具体表现。最后，Fuchs-Astashkevich定理被成功地从Virasoro代数推广到了A(1)1，这是一个重要的理论突破。
这篇开放获取的文章发表在《核物理学B》杂志上，对于理解和应用分数阶仿射李代数的表示理论学者来说，提供了宝贵的洞察和新的研究方向。整个研究展示了理论数学与实际物理问题之间的紧密联系，为未来的研究工作开辟了新的可能性。
8 J. Rasmussen / Nuclear Physics B 950 (2020) 114865
such that all t ∈ Q
+
are covered. Accordingly, we refer to levels of the form (3.1)simply as
fractional.
For fractional le
vel, following (3.2), the Verma module characters are invariant under simul-
taneous translations of r, s by multiples of p, p
such that (r + p)(s + p
) > 0or r + p >
0, s + p
= 0:
χ
j
r+p ,s +p
(q, z) =
χ
j
r,s
(q, z), ∈ Z. (3.5)
Contrarily, the affine Kac characters (2.42)are all distinct.
3.2. Ve r m a modules
For t ∈ Q
+
and j
r,s
as in (2.37), the submodule structure of the Verma module V
j
r,s
depends
critically on whether or not r is a multiple of p. Indeed, for r not a multiple of p, the submodule
structure is described by a Loewy diagram of the form [46,44,47,3]
r/∈ pZ:
•
•
→
•
→
•
→
•
→
•
→
•
→
•
→
•
→
•
→
•
→
•
→
•
→
α
0
β
0
α
−1
β
−1
α
−2
β
−2
α
−3
β
1
α
1
β
2
α
2
β
3
α
3
...
...
(3.6)
whereas for r a nonzero multiple of p, it is described by a Loewy diagram of the form
r ∈ pZ
×
: •
→
•
→
•
→
•
→
•
→
•
→
γ
1
γ
2
γ
3
γ
4
γ
5
γ
6
...
(3.7)
Each node in these diagrams represents a Verma module of highest weight α
i
, β
i
or γ
i
, where
an arrow connecting two such modules, V
μ
→ V
μ
, indicates that the target module, V
μ
, is a
submodule of the initial module, V
μ
. Moreover , the submodules in (3.6)or (3.7)are all generated
by singular vectors, where the maximal dimension of the space of singular vectors of any given
weight is one.
Every Ve
rma module of the form V
j
r,s
appears as a submodule of exactly one of the (p + 1)p
distinct Verma modules
V
j
ρ,s
0
, 0 ≤ ρ ≤ p, 0 ≤ s
0
≤ p
− 1, (3.8)
where it is noted that j
0,s
0
= j
−p,−(p
−s
0
)
= j
p,p
+s
0
and j
p,s
0
are indeed of the form (2.37). The
submodule structure of V
j
r,s
thus follows from the Loewy diagram, (3.6)or (3.7), of the corre-
sponding ambient Verma module (3.8); one merely has to identify the ambient Loewy diagram
and locate the position of the Verma module V
j
r,s
in that diagram. To see this in action, we first
recall the submodule structures of the Verma modules (3.8).
For ρ = r
0
, where as usual 1 ≤ r
0
≤ p − 1, the Loewy diagram is of the form (3.6) with
α
i
(r
0
,s
0
) = j
r
0
+2ip,s
0
= j
r
0
,s
0
+ ip,
β
i
(r
0
,s
0
) = j
−r
0
+2ip,s
0
= j
r
0
,s
0
− r
0
+ ip, i ∈ Z.
(3.9)
For ρ = δp, where δ = 0, 1, the Loewy diagram can be obtained from a Loewy diagram of the
form (3.6)by imposing
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