Chương trình phân ban Đại số và Lý thuyết số trong hội nghị Khoa học Trường ĐH Khoa Học Tự Nhiên, ĐHQG-HCM Lần thứ XIII năm 2022

CHIỀU THỨ BẢY, 26/11/2022
Phòng: F203
Phiên 1: Chủ tọa PGS. TS. Mai Hoàng Biên

Phiên 2: Chủ tọa TS. Lê Văn Luyện

Phiên 3: Chủ tọa TS. Bùi Anh Tuấn

Le Hoang Mai
NOTES ON CHEN SIMPLE SEMIMODULES OVER LEAVITT PATH ALGEBRAS WITH COEFFICIENTS IN SEMIFIELD

In this paper, based essentially on Chen's paper, we construct Chen simple left semimodules over Leavitt path algebras with coefficients in the Boolean semifield and show that Leavitt path algebras of arbitrary graphs with coefficients in semifields are $J_s$-semisimple.

Nguyễn Thị Cẩm Tú and Lê Anh Vũ
COADJOINT ORBITS OF LIE GROUPS CORRESPONDING LIE(n, 2)

Lie(n, 2) denotes the class of all n-dimensional real solvable Lie algebras having 2- dimensional derived ideals. In this talk, we give a geometrical description of coadjoint orbits of Lie groups corresponding Lie(n, 2).

Tuyen T. M. Nguyen, Vu A. Le, and Tuan A. Nguyen,

Topology of foliations formed generic coadjoint orbits of lie groups corresponding to a class 7- dimensional solvable lie groups
We consider all connected and simply connected Lie groups which are corresponding to Lie algebras of dimension 7 such that the nilradical of them is 5-dimensional nilpotent Lie algebra g5,2 = span{X1, X2, X3, X4, X5 : [X1, X2] = X4, [X1, X3] = X5} of Dixmier. First, we give a geometric description of the maximal-dimensional orbits in the coadjoint representation of all considered Lie groups. Next, we prove that, for each considered group, the family of the generic coadjoint orbits forms a measurable foliation in the sense of Connes. Finally, the topological classification of all these foliations is also provided.

Nguyen Trong Tuan
CHARACTERIZING MAXIMAL SHIFTED INTERSECTING SET SYSTEM & SHORT INJECTIVE
PROOFS OF THE ERDÕS-KO-RADO AND HILTON-MILNER THEOREM

We give a canonical partition of shifted intersecting set system , from which one can obtain unified and elementary proofs of the Erdõs -Ko-Rado and Hilton -Milner Theorem, as well as a characterization of maximal shifted k-uniform intersecting set system over a set of n elements.

Chua Le
SKEW LINEAR GROUPS OF FINITE RANK

In this talk, we address the finite rank of skew linear groups. Let $D$ be a division ring and $n$ a positive integer. Suppose that $G$ is a subgroup of $GL_n(D)$. We prove that $G$ is locally generalized radical of finite rank if and only if $G$ has a normal solvable subgroup $N$ of finite rank such that $G/N$ is finite. As a corollary, $G$ has finite rank if and only if $G$ has a normal solvable subgroup $N$ of finite rank such that $G/N$ is finite provided $D$ is weakly locally finite. The result can be viewed as an extension of Platonov’s Theorem.

Tú Nguyễn Hoàng Huy
A GENERALIZATION OF THE CHEVALLEY THEOREM

Let $(R, \m)$ be Noetherian local ring. The Chevalley theorem state that if $R$ is complete and $(I_n )_{n \geq 0}$ a descending sequence of ideals of $R$ such that $\bigcap_{n \geq 0} I_n =0$. Then for all $n$ there is an interger $m=m(n)$ such that $I_m \subseteq \m^n$. This signifies that the natural topology of the complete local ring $R$ is weaker than any other $(I_n)$-topology of $R$ for which $R$ is a Hausdoff space. This resembles a classical property of compact spaces whereby a compact space possesses no Hausdorff topologies which are strictly weaker than the given topologyof the compact space. In order to generalize Lichtenbaum-Hartshorne vanishing theorem, P. Schenzel make a slight modification of Chevalley's theorem. Let $(R, \m)$ be local ring.
Set $ (N:_M \langle\m\rangle) := \{ x \in M : \m ^{n} x \in N \quad \text{for some} \quad
n \} $. Then $\bigcap_{n \geq 1} (I^n \widehat{M} :_{ \widehat{M})} \widehat{\m} )
=0$ if and only if for any integer $n$ there is an $s=s(n)$ such that $ (I^sM :_M \langle
\m \rangle) \subseteq I^nM$. Our paper is aim to generalize the Chevalley theorem.

Nam Sơn Trần
A LOOK AT EXPRESSING ELEMENTS IN A CERTAIN GROUP ALGEBRA
The main goal is to investigate the decomposition of the group algebra $FG$ and the unit group $(FG)^\ast$ of $FG$ into the center $\mathrm{Z}(FG)$ of $FG$, the linear span $[FG,FG]$ of additive commutators in $FG$, the Jacobson radical $J(FG)$ of $FG$, and the first derived subgroup $(FG)'$ of $FG$.

Huỳnh Việt Khánh
TORSION SUBGROUP OF THE WHITEHEAD GROUP OF FINITE DIMENSIONAL GRADED DIVISION ALGEBRAS

Let $E$ be a graded division ring which is finite dimensional over its center $T$. Let ${\\rm TK}_1(E)$ be the torsion subgroup of the abelian group $E^*/E'$. In this talk, we provide some basic properties of ${\\rm TK}_1(E)$. In particular, give formulas to calculate ${\\rm TK}_1(E)$ in some special cases such as when $E$ is unramified, totally ramified or semiramified. The obtained results permit us to relate the group ${\\rm TK}_1(D)$ of a valued division ring $D$, which is finite dimensional over a Henselian center, to the group ${\\rm TK}_1({\\rm gr}(D))$ of the graded division ring ${\\rm gr}(D)$ associated to $D$.

Trần Ngọc Hội, Nguyễn Hữu Trí Nhật, Trần Trung Kiệt, Liêu Long Hồ
NHÓM CON CỦA NHÓM TUYẾN TÍNH TỔNG QUÁT CHỨA NHÓM CON SƠ CẤP TRÊN MỞ RỘNG VÀNH CÓ HẠNG HỮU HẠN

Cho R là một vành có đơn vị và S là một mở rộng vành của R sao cho S là một R-môđun hạng hữu hạn m. Khi đó, thông qua biểu diễn chính qui, S được xem như là một vành con của vành ma trận M( , ) m R và do đó GL( , ) n S là một nhóm con của nhóm tuyến tính tổng quát GL( , ). mn R Bài toán đặt ra là mô tả các nhóm con trung gian giữa E( , ) n S và GL( , ). mn R Khi R S = , bài toán đã được giải ứng với nhiều lớp vành khác nhau như vành giao hoán, vành chính qui, vành với điều kiện về hạng ổn định,…. Khi m = 2, bài toán đã được giải cho trường hơp R là tích trực tiếp các trường trong [2], [3], [4]. Khi m >1, trong [1], ShangZhi Li đã giải bài toán cho các vành chia R và S. Trong báo cáo này, chúng tôi chỉ ra rằng trong trường hợp tổng quát, với một số giả thiết về hạng ổn định ideal của R, ứng với mỗi nhóm con H như thế, tồn tại duy nhất một vành con A của M( , ) m R có chứa S sao cho E(n,A) <= H <= N_GL(mn,R)_(E(n,A)).

Nam Cao
ON THE DIAMETERS OF COMMUTING GRAPHS OF MATRIX ALGEBRAS OVER DIVISION RINGS

Let $F$ be a real-closed field and $D$ a $F$-division algebra. In this report, we prove that if $D$ is algebraic over $F$ then the graph $\Gamma(M_n (D))$ is connected and its diameter is at most four for any $n≥3$. If in addition, the division ring $D$ is noncommutative, we also have the same results for $n=2$. As a corollary, we show the diameter of the commuting graph of the matrix algebra of degree $n≥2$ over a generalized quaternion algebra $H_F (a,b)$, where $F$ is a real-closed field, two elements $a, b \in F^*$, is also at most 4. This fact is a strong improvement of the previous result by Akbari et al. asserting that this diameter is at most six in the case F is the field R of real numbers and $a=b=-1$.

Le Qui Danh
INTERSECTION GRAPH OF QUASI-NORMAL SKEW LINEAR GROUPS

The intersection graph of quasi-normal subgroups of a group $G$, denoted by $\Gamma_{\rm q}(G)$, is a graph defined as follows: the vertex set is the set of all nontrivial, proper quasi-normal subgroups of $G$, and two distinct vertices $H$ and $K$ are adjacent if $H\cap K\neq\langle 1\rangle$. In this report, we present that if $G$ is an arbitrary non-simple group, then the diameter of $\Gamma_{\rm q}(G)$ is in $\{0,1,2,\infty\}$. Besides, all general skew linear groups $\mathrm{GL}_n(D)$ over a division ring $D$ can be classified depending on the diameter of $\Gamma_{\rm q}(\mathrm{GL}_n(D))$.