Drawing the diagrams for Yinyang Wuxing and Bagua as McKay quivers Jin Yun Guo

2025-05-03 0 0 347.42KB 12 页 10玖币

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Drawing the diagrams for Yinyang, Wuxing and

Bagua as McKay quivers

Jin Yun Guo

School of mathematics and Statistics

Hunan Normal University

Abstract

In this paper, we draw diagrams of yinyang, wuxing and bagua as the

McKay quivers of the groups of order 2, of order 5 and of order 23.

In [12], McKay established the correspondence among the ﬁnite subgroups

of SL(2,C), the simple complex Li algebras and Klein singularities. Now McKay

correspondence is a much studied ﬁeld in mathematics. The ﬁnite subgroups

of SL(2,C) are related to the ﬁnite subgroup of the rotation group SO(3,R) of

real space of dimension 3. The 3-dimensional geometric objects invariant under

the ﬁnite subgroup of SO(3,R) are exactly the Platoic solids. McKay conclude

his paper [12] with the words ”Would not the Greeks appreciate the result that

the simple Lie algebra may be derived from the Platonic solids?”

Yinyang(阴阳)[14], wuxing(五行)[2] and bagua(八卦)[10] are conceptual frame-

works in ancient Chinese thoughts in oberserving and analyzing the world. They

are used even today in many aspects such as practising tranditional Chinese

medicine, Fengshui, and so on. There are diagrams illustrating the meaning for

yinyang and wuxing.

Returning arrow quiver and covering of quiver are two constructions we used

in constructing algebras in higher representation theory [5, 9, 6]. We recover

Taijitu for yinyang and the diagrams for wuxing using the McKay quivers for the

groups of order 2 and of order 5, repsectiely, we also draw a diagram for bagua

using the McKay quiver of order 23. Our approach gives interpretations to some

assertions in Daodejing and in Yijing, from the point of view of representation

theory.

Quivers are important realizations of certain categories. By expressing the

diagrams of yinyang, wuxing and bagua as quivers, we may rediscover the math-

ematical idea hide behind the old Chinese thoughts, under the point of view of

category.

arXiv:2210.11920v2 [math.HO] 1 Mar 2023

1 Preliminary.

In 1979, John McKay introduced McKay quiver in [12] for ﬁnite groups and their

representations, and discovered the relationship between the Dynkin diagrams

and the McKay quivers of ﬁnite subgroups of SL(2,C).

Let kbe an algebraically closed ﬁeld of characteristics zero and let Vbe a

vector space of dimension nover k. Consider a ﬁnite subgroup Gof the general

linear group GL(n, k) = GL(V), Vis naturally a representation of G. Assume

that the irreducible representations of Gare S1, . . . , Sm, and assume that we

have a decomposition

V⊗Si=

j=1

ai,j Sj,

of the tensor product V⊗Sifor i= 1,2,· · · , m.The McKay quiver QV(G) of G

is deﬁned as the quiver with the vertices S1,· · · , Sm, and there are ai,j arrows

from Sito Sj.

In 2011, we prove the following theorems illustration the two constructions

of McKay quivers in [4]:

Theorem 1. (Theorem 1.2 in [4]) Assume that Gis a ﬁnite subgroup of

GL(n, k)and that N=G∩SL(n, k). If every irreducible representation of N

is extendible, then the McKay quiver of Gis a regular covering of the McKay

quiver of Nwith G/N as the group of the automorphisms.

The groups we considered are abelian, so the irreducible representations of

its subgroups are extendible.

Theorem 2. (Theorem 3.1 of [4]) Assume that Gis a ﬁnite subgroup of

GL(n, k). Embed GL(n, k)naturally into SL(n+ 1, k). Then the McKay quiver

of Gin SL(n+ 1, k)is obtained as the returning arrow quiver of the McKay

quiver of Gin GL(n, k).

Regarding the McKay quiver QV(G) of G⊂GL(V) as the bound quivers

for the skew group algebra of Gover exterior algebra of V[7], then non-zero

paths in the algebra has ﬁnite length. For each maximal nonzero path pin the

algebra, adding an arrow, called ”returing arrow”, from the ending vertex of p

to the starting vertex of p. In this way, we get the returning arrow quiver of

QV(G) [4, 9].

In this paper, we ﬁrst draw the diagrams of generating and of overcoming for

wuxing as the McKay quivers of cyclic group of order 5, directly from deﬁnition.

We also show how to construct the McKay quivers of groups of order 2 and of

order 23, using Theorems 1 and 2, to get the diagram (Taijitu) for yinyang and

a diagram for bagua，respectively.

2 Wuxing.

Wuxing (”ﬁve processes” or ”ﬁve phases”, also translated as ”ﬁve elements”)

refers to a ﬁvefold conceptual scheme that is found throughout traditional Chi-

nese thought. These ﬁve phases are wood (mu), ﬁre (huo), earth (tu), metal

(jin), and water (shui); they are regarded as dynamic, interdependent modes or

aspects of the universe’s ongoing existence and development [15]. The origins

of wuxing trace back to Shang dynasty(1600-1046 B.C.E.), and was developed

to its present form in Han dynasty((206 B. C. E.220 C. E.). The doctrine of

wuxing is discribed in two diagrams, a diagram of generating or creation (生),

and a diagram of overcoming or destruction (克). We now draw the McKay quiv-

ers of the cyclic group of order 5 as the diagram of generating and the diagram

of overcoming of wuxing.

水

金

生

土

生

火

生

木

生

Figure 1: Wuxing–the diagram of generating

The cyclic group G五行of order 5 is generated by an element aof order

5. It has 5 irreducible representations, all of dimension 1. The values of the

characters of these representations are the ﬁfth roots of the unit. The irreducible

representations are determined by the value of their characters on the generator

aof G五行. Denote by Stthe irreducible representation whose character takes

value e2tπi

5on afor 0 ≤t≤4. Write“金”=S0,“水”=S1,“木”=S2,“火

”=S3,“土”=S4.

Take V五行=S1and regard G五行as a subgroup of GL(1, k) = GL(V五行). Write

“生”for tensoring with V生=S1. We get the McKay quiver QV生(G五行)(see

Figure 1) by direct calculation, which is the diagram of generating of wuxing.

Now take V五行=S2, write “克”for tensoring with V克=S2. We get the

McKay quiver QV克(G五行) (see Figure 2), this is the diagram of overcoming of

wuxing when Figure 1 is the diagram of generating of wuxing.

水

金

土克

火

克

木

克

Figure 2: Wuxing–the diagram of overcoming

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DrawingthediagramsforYinyang,WuxingandBaguaasMcKayquiversJinYunGuoSchoolofmathematicsandStatisticsHunanNormalUniversityAbstractInthispaper,wedrawdiagramsofyinyang,wuxingandbaguaastheMcKayquiversofthegroupsoforder2,oforder5andoforder23.In[12],McKayestablishedthecorrespondenceamongthenitesubgroupsofS...

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Drawing the diagrams for Yinyang Wuxing and Bagua as McKay quivers Jin Yun Guo

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