On point-local deformations of minimal extensions of non-serial Dynkin diagrams
In this paper we study point-local deformations of Tits quadratic forms of finite graphs. We describe all P-limiting numbers of minimal extensions ofnon-serial Dynkin dyagrams in the case when these extensions are neither usual neither extended Dynkin diagrams.
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| author | Bondarenko, V. M. Chervyakov, I. V. Pereguda, Yu. M. Bondarenko, V. M. Chervyakov, I. V. Pereguda, Yu. M. |
| author_facet | Bondarenko, V. M. Chervyakov, I. V. Pereguda, Yu. M. Bondarenko, V. M. Chervyakov, I. V. Pereguda, Yu. M. |
| author_institution_txt_mv | [
{
"author": "V. M. Bondarenko",
"institution": null
},
{
"author": "I. V. Chervyakov",
"institution": null
},
{
"author": "Yu. M. Pereguda",
"institution": null
}
] |
| author_sort | Bondarenko, V. M. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2018-01-23T12:10:05Z |
| description | In this paper we study point-local deformations of Tits quadratic forms of finite graphs. We describe all P-limiting numbers of minimal extensions ofnon-serial Dynkin dyagrams in the case when these extensions are neither usual neither extended Dynkin diagrams. |
| first_indexed | 2026-08-04T01:03:14Z |
| format | Article |
| fulltext |
Збiрник праць Iн-ту математики НАН України 2015, том 12, № 3, 47–55
УДК 512.64+519.17
V. M. Bondarenko 1, I. V. Chervyakov 2,
Yu. M. Pereguda 3
1 , 2 (Institute of mathematics of NAN of Ukraine, Kyiv)
3 (Zhytomyr Military Institute of the State University of
Telecommunications, Zhytomyr, Ukraine)
1 vit-bond@imath.kiev.ua, 3 pereguda.juli@rambler.ru
On point-local deformations of minimal
extensions of non-serial Dynkin
diagrams
In this paper we study point-local deformations of Tits quadratic forms of
finite graphs. We describe all P -limiting numbers of minimal extensions of
non-serial Dynkin dyagrams in the case when these extensions are neither
usual neither extended Dynkin diagrams.
1. Introduction. Let
f(z) = f(z1, . . . , zn) :=
n∑
i=1
fiiz
2
i +
∑
i<j
fijzizj
be a qudratic form over the field of real numbers R. By the definition from
[1], a quadratic form of the form
f (s)(z, t) = tfssz
2
s +
∑
i ̸=s
fiiz
2
i +
∑
i<j
fijzizj with fss ̸= 0
where t is a parameter running R, is called the local deformation of f(z)
with respect to zs or the s-deformation of f(z).
c⃝ Institute of mathematics of NAN of Ukraine, 2015
48 V. M. Bondarenko, et al.
Let now fss > 0. Denote by F
(s)
+ the set of all a ∈ R such that the
quadratic form f (s)(z, a) is positive definite, and put F
(s)
− = R \ F
(s)
+ .
Obviously, F
(s)
− ̸= ∅ (since f (s)(z, 0) is not positive definite), and if
F
(s)
− ̸= R then m
(s)
f = supF
(s)
− ∈ F
(s)
− is called the P -limiting number
of f(z) for zs or the s-th P -limiting number of f(z). In the case F
(s)
− = R
we put m
(s)
f = ∞. Concerning general properties of P -limiting numbers
see in [1, 2, 3]. Deformations considered above were called point-local
deformations of f(z) in [3]. This paper is devoted to study of point-local
deformations of the Tits quadratic form of quivers.
2. Minimal extensions of graphs. Elsewhere in the paper all
graphs are finite and non-oriented. Sets of vertices and edges of a graph
X are denoted by X0 and X1, respectively. A graph G = (G0, G1) is said
to be a minimal extension of a graph Q = (Q0, Q1) if G0 = S0 ∪ d with
d /∈ Q0 and G1 = Q1 ∪ (d, j) with j ∈ Q0. The vertex d is said to be the
added vertex of G. In this case we write G = Q ∪ (d, j). If Q is a Dynkin
diagram, the most interesting from the point of view of deformations (as
we can see in the next section) is the case when the graph G is neither
an usual not an extended Dynkin diagram. We call such G an essential
minimal extension of Q.
We consider minimal extensions of the non-serial Dynkin diagtams, i. e.
the diagrams E6, E7, E8:
t t t
t
1 2 3 4 5
6
t t
t t t
t
1 2 3 4 5 6
7
t t t
On point-local deformations 49
t t t
t
1 2 3 4 5 6 7
8
t t t t
Directly from the definitions we have the following statement.
Proposition 1 An extension G = E ∪ (0, j) of a Dynkin diagram E = Ei
(i = 6, 7, 8) is essential if and only if one of the following condition holds:
1) E = E6, j ̸= 1, 5, 6; 2) E = E7, j ̸= 1, 6; 3) E = E8, j ̸= 7.
3. Formulation of the main results. By the definition (see
[4]) the Tits quadratic form of a graph Q = (Q0, Q1) is the following
integral quadratic form:
qQ(z) = qQ(z1, z2, . . . , zn) :=
∑
i∈Q0
z2i −
∑
{i−j}∈Q1
zizj .
It is well-known that qQ(z) is positive definite if and only if the graph Q
is a disjoint union of Dynkin diagrams (see [4]). All P -limiting numbers
of such quadratic forms are describes in [2]; they are rational numbers
belonging to [0, 1).
Note that formally it is more convenient to say about P -limiting
numbers of a graph Q instead of the quadratic form qQ(z). As in [2],
by the P -limiting number of a vertex i ∈ Q0 we mean the i-th P -limiting
number of qQ(z), and we write m
(i)
Q instead of m(i)
qQ(z).
We consider minimal extensions G of Dynkin diagtams. If G is an
extended Dynkin diagram then by Theorem 1 in [3] the P -limiting number
of the added vertex of G is equal to 1. Therefore, the most interesting is
the case of essential extensions. Note that in this case by the same theorem
the P -limiting numbers of the added vertices belong to (1,∞).
Theorem 1 Let G = E6 ∪ (0, j) be an essential minimal extension of the
Dynkin diagram E6. Then the P -limiting number of the added vertex 0 is
50 V. M. Bondarenko, et al.
the following:
m
(0)
G =
{
1 2
3 , if j = 2, 4;
3, if j = 3.
Theorem 2 Let G = E7 ∪ (0, j) be an essential minimal extension of the
Dynkin diagram E7. Then the P -limiting number of the added vertex 0 is
the following:
m
(0)
G =
3, if j = 2;
6, if j = 3;
3 3
4 , if j = 4;
2, if j = 5;
1 3
4 , if j = 7.
Theorem 3 Let G = E8 ∪ (0, j) be an essential minimal extension of the
Dynkin diagram E8. Then the P -limiting number of the added vertex 0 is
the following:
m
(0)
G =
2, if j = 1;
7, if j = 2;
15, if j = 3;
10, if j = 4;
6, if j = 5;
3, if j = 6;
4, if j = 8.
4. Proofs of the theorems. It follows from [1] – [3] that the P -
limiting number m(0)
G of a graph G = Ei∪ (0, j) is the root of the equation
(linear with respect to t) |Aj
i (t)| = 0, where Aj
i (t) is the symmetric matrix
of the quadratic form 2q(0)G (z, t). The determinant |Aj
i (t)| is denoted by
Dj
i (t). In particular cases, we have:
A2
6(t) =
2t 0 −1 0 0 0 0
0 2 −1 0 0 0 0
−1 −1 2 −1 0 0 0
0 0 −1 2 −1 0 −1
0 0 0 −1 2 −1 0
0 0 0 0 −1 2 0
0 0 0 −1 0 0 2
On point-local deformations 51
D2
6(t) = 6t− 10;
A3
6(t) =
2t 0 −1 0 0 0 0
0 2 −1 0 0 0 0
−1 −1 2 −1 0 0 0
0 0 −1 2 −1 0 −1
0 0 0 −1 2 −1 0
0 0 0 0 −1 2 0
0 0 0 −1 0 0 2
D3
6(t) = 6t− 18;
A2
7(t) =
2t 0 −1 0 0 0 0 0
0 2 −1 0 0 0 0 0
−1 −1 2 −1 0 0 0 0
0 0 −1 2 −1 0 0 −1
0 0 0 −1 2 −1 0 0
0 0 0 0 −1 2 −1 0
0 0 0 0 0 −1 2 0
0 0 0 −1 0 0 0 2
D2
7(t) = 4t− 12;
A3
7(t) =
2t 0 −1 0 0 0 0 0
0 2 −1 0 0 0 0 0
−1 −1 2 −1 0 0 0 0
0 0 −1 2 −1 0 0 −1
0 0 0 −1 2 −1 0 0
0 0 0 0 −1 2 −1 0
0 0 0 0 0 −1 2 0
0 0 0 −1 0 0 0 2
D3
7(t) = 4t− 24;
52 V. M. Bondarenko, et al.
A4
7(t) =
2t 0 −1 0 0 0 0 0
0 2 −1 0 0 0 0 0
−1 −1 2 −1 0 0 0 0
0 0 −1 2 −1 0 0 −1
0 0 0 −1 2 −1 0 0
0 0 0 0 −1 2 −1 0
0 0 0 0 0 −1 2 0
0 0 0 −1 0 0 0 2
D4
7(t) = 4t− 15;
A5
7(t) =
2t 0 −1 0 0 0 0 0
0 2 −1 0 0 0 0 0
−1 −1 2 −1 0 0 0 0
0 0 −1 2 −1 0 0 −1
0 0 0 −1 2 −1 0 0
0 0 0 0 −1 2 −1 0
0 0 0 0 0 −1 2 0
0 0 0 −1 0 0 0 2
D5
7(t) = 4t− 8;
A7
7(t) =
2t 0 −1 0 0 0 0 0
0 2 −1 0 0 0 0 0
−1 −1 2 −1 0 0 0 0
0 0 −1 2 −1 0 0 −1
0 0 0 −1 2 −1 0 0
0 0 0 0 −1 2 −1 0
0 0 0 0 0 −1 2 0
0 0 0 −1 0 0 0 2
D7
7(t) = 4t− 7;
On point-local deformations 53
A1
8(t) =
2t −1 0 0 0 0 0 0 0
−1 2 −1 0 0 0 0 0 0
0 −1 2 −1 0 0 0 0 0
0 0 −1 2 −1 0 0 0 −1
0 0 0 −1 2 −1 0 0 0
0 0 0 0 −1 2 −1 0 0
0 0 0 0 0 −1 2 −1 0
0 0 0 0 0 0 −1 2 0
0 0 0 −1 0 0 0 0 2
D1
8(t) = 2t− 4;
A2
8(t) =
2t −1 0 0 0 0 0 0 0
−1 2 −1 0 0 0 0 0 0
0 −1 2 −1 0 0 0 0 0
0 0 −1 2 −1 0 0 0 −1
0 0 0 −1 2 −1 0 0 0
0 0 0 0 −1 2 −1 0 0
0 0 0 0 0 −1 2 −1 0
0 0 0 0 0 0 −1 2 0
0 0 0 −1 0 0 0 0 2
D2
8(t) = 2t− 14;
A3
8(t) =
2t −1 0 0 0 0 0 0 0
−1 2 −1 0 0 0 0 0 0
0 −1 2 −1 0 0 0 0 0
0 0 −1 2 −1 0 0 0 −1
0 0 0 −1 2 −1 0 0 0
0 0 0 0 −1 2 −1 0 0
0 0 0 0 0 −1 2 −1 0
0 0 0 0 0 0 −1 2 0
0 0 0 −1 0 0 0 0 2
D3
8(t) = 2t− 30;
54 V. M. Bondarenko, et al.
A4
8(t) =
2t −1 0 0 0 0 0 0 0
−1 2 −1 0 0 0 0 0 0
0 −1 2 −1 0 0 0 0 0
0 0 −1 2 −1 0 0 0 −1
0 0 0 −1 2 −1 0 0 0
0 0 0 0 −1 2 −1 0 0
0 0 0 0 0 −1 2 −1 0
0 0 0 0 0 0 −1 2 0
0 0 0 −1 0 0 0 0 2
D4
8(t) = 2t− 20;
A5
8(t) =
2t −1 0 0 0 0 0 0 0
−1 2 −1 0 0 0 0 0 0
0 −1 2 −1 0 0 0 0 0
0 0 −1 2 −1 0 0 0 −1
0 0 0 −1 2 −1 0 0 0
0 0 0 0 −1 2 −1 0 0
0 0 0 0 0 −1 2 −1 0
0 0 0 0 0 0 −1 2 0
0 0 0 −1 0 0 0 0 2
D5
8(t) = 2t− 12;
A6
8(t) =
2t −1 0 0 0 0 0 0 0
−1 2 −1 0 0 0 0 0 0
0 −1 2 −1 0 0 0 0 0
0 0 −1 2 −1 0 0 0 −1
0 0 0 −1 2 −1 0 0 0
0 0 0 0 −1 2 −1 0 0
0 0 0 0 0 −1 2 −1 0
0 0 0 0 0 0 −1 2 0
0 0 0 −1 0 0 0 0 2
D6
8(t) = 2t− 6;
On point-local deformations 55
A8
8(t) =
2t −1 0 0 0 0 0 0 0
−1 2 −1 0 0 0 0 0 0
0 −1 2 −1 0 0 0 0 0
0 0 −1 2 −1 0 0 0 −1
0 0 0 −1 2 −1 0 0 0
0 0 0 0 −1 2 −1 0 0
0 0 0 0 0 −1 2 −1 0
0 0 0 0 0 0 −1 2 0
0 0 0 −1 0 0 0 0 2
D8
8(t) = 2t− 8.
It follows directly from the quantities of determinants of the above
matrices the validity of Theorems 1 – 3.
References
[1] Bondarenko V. M., Pereguda Yu. M. On P-numbers of quadratic forms //
Proc. of Institute of Mathematics of NAN of Ukraine /Geometry, Topology,
and their Applications/. — 2009. — 6, no. 2. — P. 474 – 477.
[2] Bondarenko V. M., Bondarenko V. V., Pereguda Yu. M. Local deformations
of positive-definite quadratic forms // Ukrainian Math. J. — 2012. — 64,
no. 7. — P. 1019 – 1035 (in Russian).
[3] Bondarenko V. M. On types of local deformations of quadratic forms //
Algebra Discrete Math. — 2014. — 18, no. 2. — P. 163 – 170.
[4] Gabriel P. Unzerlegbare Darstellungen // Manuscripta Math. — 1972. —
6. — P. 71 – 103.
|
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| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
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| language | English |
| last_indexed | 2026-08-04T01:03:14Z |
| publishDate | 2015 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/bf/473bbea99df914eb847428bf9ef889bf.pdf |
| spelling | oai:trim.imath.kiev.ua:article-1562018-01-23T12:10:05Z On point-local deformations of minimal extensions of non-serial Dynkin diagrams Bondarenko, V. M. Chervyakov, I. V. Pereguda, Yu. M. Bondarenko, V. M. Chervyakov, I. V. Pereguda, Yu. M. In this paper we study point-local deformations of Tits quadratic forms of finite graphs. We describe all P-limiting numbers of minimal extensions ofnon-serial Dynkin dyagrams in the case when these extensions are neither usual neither extended Dynkin diagrams. Інститут математики НАН України 2015-04-23 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/156 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 12 No. 3 (2015): Analysis and Applications; 47–55 Сборник Трудов Института математики НАН Украины; Том 12 № 3 (2015): Анализ и приложения; 47–55 Збірник Праць Інституту математики НАН України; Том 12 № 3 (2015): Аналіз та застосування; 47–55 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/156/127 Авторське право (c) 2015 Праці Інституту математики НАН України |
| spellingShingle | Bondarenko, V. M. Chervyakov, I. V. Pereguda, Yu. M. Bondarenko, V. M. Chervyakov, I. V. Pereguda, Yu. M. On point-local deformations of minimal extensions of non-serial Dynkin diagrams |
| title | On point-local deformations of minimal extensions of non-serial Dynkin diagrams |
| title_full | On point-local deformations of minimal extensions of non-serial Dynkin diagrams |
| title_fullStr | On point-local deformations of minimal extensions of non-serial Dynkin diagrams |
| title_full_unstemmed | On point-local deformations of minimal extensions of non-serial Dynkin diagrams |
| title_short | On point-local deformations of minimal extensions of non-serial Dynkin diagrams |
| title_sort | on point-local deformations of minimal extensions of non-serial dynkin diagrams |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/156 |
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