A class of non-weight modules of 𝑈𝑝(𝖘𝖑2) and Clebsch–Gordan type formulas

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yan-an Cai ◽  
Hongjia Chen ◽  
Xiangqian Guo ◽  
Yao Ma ◽  
Mianmian Zhu
Keyword(s):  
Quantum Group ◽  
Tensor Products ◽  
Simple Modules ◽  
Isomorphism Classes ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Direct Sum Decomposition ◽  
Free Modules ◽  
Weight Modules

Abstract In this paper, we construct a class of new modules for the quantum group U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) which are free of rank 1 when restricted to C ⁢ [ K ± 1 ] \mathbb{C}[K^{\pm 1}] . The irreducibility of these modules and submodule structure for reducible ones are determined. It is proved that any C ⁢ [ K ± 1 ] \mathbb{C}[K^{\pm 1}] -free U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) -module of rank 1 is isomorphic to one of the modules we constructed, and their isomorphism classes are obtained. We also investigate the tensor products of the C ⁢ [ K ± 1 ] \mathbb{C}[K^{\pm 1}] -free modules with finite-dimensional simple modules over U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) , and for the generic cases, we obtain direct sum decomposition formulas for them, which are similar to the well-known Clebsch–Gordan formula for tensor products between finite-dimensional weight modules over U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) .

scholarly journals The h-vector of a Gorenstein codimension three domain

1995 ◽  
Vol 138 ◽  
pp. 113-140 ◽  
Author(s):  
E. De Negri ◽  
G. Valla
Keyword(s):  
Hilbert Function ◽  
Additive Group ◽  
Dimensional Vector ◽  
Infinite Field ◽  
Homogeneous Ideal ◽  
Dimensional Vector Space ◽  
Direct Sum ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Direct Sum Decomposition

Let k be an infinite field and A a standard G-algebra. This means that there exists a positive integer n such that A = R/I where R is the polynomial ring R := k[Xv …, Xn] and I is an homogeneous ideal of R. Thus the additive group of A has a direct sum decomposition A = ⊕ At where AiAj ⊆ Ai+j. Hence, for every t ≥ 0, At is a finite-dimensional vector space over k. The Hilbert Function of A is defined by

When is every module with essential socle a direct sum of automorphism-invariant modules?

2019 ◽  
Vol 19 (10) ◽  
pp. 2050185
Author(s):  
Shahabaddin Ebrahimi Atani ◽  
Saboura Dolati Pish Hesari ◽  
Mehdi Khoramdel
Keyword(s):  
Commutative Ring ◽  
Principal Ideal ◽  
Essential Extension ◽  
Principal Ideal Ring ◽  
Von Neumann ◽  
Simple Modules ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Ideal Ring ◽  
Von Neumann Regular

The purpose of this paper is to study the structure of rings over which every essential extension of a direct sum of a family of simple modules is a direct sum of automorphism-invariant modules. We show that if [Formula: see text] is a right quotient finite dimensional (q.f.d.) ring satisfying this property, then [Formula: see text] is right Noetherian. Also, we show a von Neumann regular (semiregular) ring [Formula: see text] with this property is Noetherian. Moreover, we prove that a commutative ring with this property is an Artinian principal ideal ring.

The Torsion Free Pieri Formula

1998 ◽  
Vol 50 (2) ◽  
pp. 266-289 ◽  
Author(s):  
D. J. Britten ◽  
F. W. Lemire
Keyword(s):  
Tensor Product ◽  
Arbitrary Degree ◽  
Simple Modules ◽  
Infinite Dimensional ◽  
Complete Reducibility ◽  
Finite Dimensional ◽  
Pieri Formula ◽  
Free Modules ◽  
Central Characters ◽  
Torsion Free

AbstractCentral to the study of simple infinite dimensional g𝓵(n, C)-modules having finite dimensional weight spaces are the torsion free modules. All degree 1 torsion free modules are known. Torsion free modules of arbitrary degree can be constructed by tensoring torsion free modules of degree 1 with finite dimensional simple modules. In this paper, the central characters of such a tensor product module are shown to be given by a Pieri-like formula, complete reducibility is established when these central characters are distinct and an example is presented illustrating the existence of a nonsimple indecomposable submodule when these characters are not distinct.

Tilting modules over Auslander algebras of Nakayama algebras with radical cube zero

2021 ◽  
pp. 1-22
Author(s):  
Zongzhen Xie ◽  
Hanpeng Gao ◽  
Zhaoyong Huang
Keyword(s):  
Tilting Modules ◽  
Simple Modules ◽  
Isomorphism Classes ◽  
Finite Dimensional ◽  
Nakayama Algebras ◽  
Nakayama Algebra ◽  
Auslander Algebra

Let [Formula: see text] be the Auslander algebra of a finite-dimensional basic connected Nakayama algebra [Formula: see text] with radical cube zero and [Formula: see text] simple modules. Then the cardinality [Formula: see text] of the set consisting of isomorphism classes of basic tilting [Formula: see text]-modules is [Formula: see text]

scholarly journals CLASSIFICATION OF IRREDUCIBLE WEIGHT MODULES OVER THE TWISTED HEISENBERG–VIRASORO ALGEBRA

2010 ◽  
Vol 12 (02) ◽  
pp. 183-205 ◽  
Author(s):  
RENCAI LÜ ◽  
KAIMING ZHAO
Keyword(s):  
Virasoro Algebra ◽  
The Other ◽  
Simple Modules ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Highest Weight Modules ◽  
Intermediate Series ◽  
Weight Modules

In this paper, all irreducible weight modules with finite dimensional weight spaces over the twisted Heisenberg–Virasoro algebra are determined. There are two different classes of them. One class is formed by simple modules of intermediate series, whose nonzero weight spaces are all 1-dimensional; the other class consists of the irreducible highest weight modules and lowest weight modules.

scholarly journals Tropical ideals do not realise all Bergman fans

2021 ◽  
Vol 8 (3) ◽  
Author(s):  
Jan Draisma ◽  
Felipe Rincón
Keyword(s):  
Tropical Geometry ◽  
Tensor Products ◽  
Tropical Variety ◽  
Polyhedral Complex ◽  
Direct Sum ◽  
Weight One ◽  
Polyhedral Complexes

AbstractEvery tropical ideal in the sense of Maclagan–Rincón has an associated tropical variety, a finite polyhedral complex equipped with positive integral weights on its maximal cells. This leads to the realisability question, ubiquitous in tropical geometry, of which weighted polyhedral complexes arise in this manner. Using work of Las Vergnas on the non-existence of tensor products of matroids, we prove that there is no tropical ideal whose variety is the Bergman fan of the direct sum of the Vámos matroid and the uniform matroid of rank two on three elements and in which all maximal cones have weight one.

scholarly journals CHARACTER CLUSTERS FOR LIE ALGEBRA MODULES OVER A FIELD OF NONZERO CHARACTERISTIC

2013 ◽  
Vol 89 (2) ◽  
pp. 234-242 ◽  
Author(s):  
DONALD W. BARNES
Keyword(s):  
Lie Algebra ◽  
Saturated Formation ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Nonzero Characteristic ◽  
Composition Factors

AbstractFor a Lie algebra $L$ over an algebraically closed field $F$ of nonzero characteristic, every finite dimensional $L$-module can be decomposed into a direct sum of submodules such that all composition factors of a summand have the same character. Using the concept of a character cluster, this result is generalised to fields which are not algebraically closed. Also, it is shown that if the soluble Lie algebra $L$ is in the saturated formation $\mathfrak{F}$ and if $V, W$ are irreducible $L$-modules with the same cluster and the $p$-operation vanishes on the centre of the $p$-envelope used, then $V, W$ are either both $\mathfrak{F}$-central or both $\mathfrak{F}$-eccentric. Clusters are used to generalise the construction of induced modules.

Invariants of 3-manifolds derived from universal invariants of framed links

1995 ◽  
Vol 117 (2) ◽  
pp. 259-273 ◽  
Author(s):  
Tomotada Ohtsuki
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Independent Method ◽  
Topological Invariant ◽  
Finite Dimensional

Reshetikhin and Turaev [10] gave a method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra (e.g. a quantum group Uq(sl2)) using finite-dimensional representations of it. In this paper we give another independent method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra via universal invariants of framed links without using representations of the algebra. For Uq(sl2) these two methods give different invariants of 3-manifolds.

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