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2. Some remarks on Hilbertian fields (An appendix to the paper “Galois averages” by R. Massy)
- Author
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Jensen, C.U. and Massy, Richard
- Subjects
- *
ALGEBRA , *NUMBER theory , *MATHEMATICAL analysis , *ARITHMETIC functions - Abstract
Abstract: The paper gives proofs of some results just claimed in [R. Massy, Galois averages, J. Number Theory 113 (2005) 244–275]. For instance, it is proved that for a finite non-trivial separable extension , , of Hilbertian fields finitely generated over their prime field, the quotient group , for the corresponding multiplicative groups of non-zero elements, cannot be a torsion group of finite exponent. [Copyright &y& Elsevier]
- Published
- 2006
- Full Text
- View/download PDF
3. Tensor 2-product for [formula omitted]: Extensions to the negative half.
- Author
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McMillan, Matthew
- Subjects
- *
LIE algebras , *ALGEBRA - Abstract
In a recent paper, the author defined an operation of tensor product for a large class of 2-representations of U + , the positive half of the 2-category associated to sl 2. In this paper, we prove that the operation extends to give an operation of tensor product for 2-representations of the full 2-category U : when the inputs are 2-representations of the full U , the 2-product is also a 2-representation of the full U. As in the previous paper, the 2-product is given for a simple 2-representation L (1) and an abelian 2-representation V taken from the 2-category of algebras. This is the first construction of an operation of tensor product for higher representations of a full Lie algebra in the abelian setting. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
4. On the common slot property for symbol algebras.
- Author
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Sivatski, Alexander S.
- Subjects
- *
COMMONS , *ALGEBRA , *SIGNS & symbols , *LAURENT series - Abstract
Let k be a field, let n ≥ 2 be a nonsquarefree integer not divisible by the characteristic of k. Assume that all roots of unity of degree n are contained in k. In the first part of the paper we consider pairs of symbol algebras over k with common slots D 1 ≃ (e , x) n ≃ (r , u) n , D 2 ≃ (e , y) n ≃ (r , v) n , exp D 1 = exp D 2 = n , and show that in general (e , x , y) n ≠ (r , u , v) n. As a consequence we prove that in general it is impossible to connect the pair { (e , x) n ; (e , y) n } and the pair { (r , u) n ; (r , v) n } by a chain of pairs of symbol algebras with a common slot and isomorphic to (D 1 ; D 2) in such a way that any two neighboring pairs in the chain are obtained from one another by a "natural" transformation. In the second part of the paper we prove that in contrast to the case n = 2 for any n divisible by 4 there exist symbol algebras D 1 , D 2 with deg D 1 = deg D 2 = n and exp D 1 = exp D 2 = n without common slot such that i D 1 + j D 2 is a symbol algebra of degree n for any i , j ∈ Z. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
5. Transposed Poisson structures on Lie incidence algebras.
- Author
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Kaygorodov, Ivan and Khrypchenko, Mykola
- Subjects
- *
LIE algebras , *POISSON algebras , *COMMUTATION (Electricity) , *ALGEBRA - Abstract
Let X be a finite connected poset, K a field of characteristic zero and I (X , K) the incidence algebra of X over K seen as a Lie algebra under the commutator product. In the first part of the paper we show that any 1 2 -derivation of I (X , K) decomposes into the sum of a central-valued 1 2 -derivation, an inner 1 2 -derivation and a 1 2 -derivation associated with a map σ : X < 2 → K that is constant on chains and cycles in X. In the second part of the paper we use this result to prove that any transposed Poisson structure on I (X , K) is the sum of a structure of Poisson type, a mutational structure and a structure determined by λ : X e 2 → K , where X e 2 is the set of (x , y) ∈ X 2 such that x < y is a maximal chain not contained in a cycle. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
6. Derivations, extensions, and rigidity of subalgebras of the Witt algebra.
- Author
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Buzaglo, Lucas
- Subjects
- *
ABSTRACT algebra , *ALGEBRA , *C*-algebras , *FINITE differences , *LIE algebras - Abstract
Let k be an algebraically closed field of characteristic 0. We study some cohomological properties of Lie subalgebras of the Witt algebra W = Der (k [ t , t − 1 ]) and the one-sided Witt algebra W ≥ − 1 = Der (k [ t ]). In the first part of the paper, we consider finite codimension subalgebras of W ≥ − 1. We compute derivations and one-dimensional extensions of such subalgebras. These correspond to Ext U (L) 1 (M , L) , where L is a subalgebra of W ≥ − 1 and M is a one-dimensional representation of L. We find that these subalgebras exhibit a kind of rigidity: their derivations and extensions are controlled by the full one-sided Witt algebra. As an application of these computations, we prove that any isomorphism between finite codimension subalgebras of W ≥ − 1 extends to an automorphism of W ≥ − 1. The second part of the paper is devoted to explaining the observed rigidity. We define a notion of "completely non-split extension" and prove that W ≥ − 1 is the universal completely non-split extension of any of its subalgebras of finite codimension. In some sense, this means that even when studying subalgebras of W ≥ − 1 as abstract Lie algebras, they remember that they are contained in W ≥ − 1. We also consider subalgebras of infinite codimension, explaining the similarities and differences between the finite and infinite codimension situations. Almost all of the results above are also true for subalgebras of the Witt algebra. We summarise results for W at the end of the paper. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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7. Comments on a paper “A Hermitian Morita theorem for algebras with anti-structure”
- Author
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Dasgupta, Bhanumati
- Subjects
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ALGEBRA , *MATHEMATICS , *MATHEMATICAL analysis , *ALGORITHMS - Abstract
Abstract: In 1.9 of the paper [A. Hahn, A Hermitian Morita theorem for algebras with anti-structure, J. Algebra 93 (1985) 215–235], should be replaced by . This leads to minor changes in the rest of the paper where the ring should be replaced by its opposite and vice versa. [Copyright &y& Elsevier]
- Published
- 2007
- Full Text
- View/download PDF
8. Complete description of invariant, associative pseudo-Euclidean metrics on left Leibniz algebras via quadratic Lie algebras.
- Author
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Abid, Fatima-Ezzahrae and Boucetta, Mohamed
- Subjects
- *
LIE algebras , *NONASSOCIATIVE algebras , *ASSOCIATIVE algebras , *COMMUTATIVE algebra , *ALGEBRA , *ASSOCIATIVE rings , *EUCLIDEAN algorithm , *BILINEAR forms - Abstract
A pseudo-Euclidean non-associative algebra (g , •) is a finite dimensional algebra over a field K that has a metric, i.e., a bilinear, symmetric, and non-degenerate form 〈 , 〉. The metric is considered L-invariant (resp. R-invariant) if all left multiplications (resp. right multiplications) are skew-symmetric. The metric is called associative if 〈 u • v , w 〉 = 〈 u , v • w 〉 for all u , v , w ∈ g. These three notions coincide when g is a Lie algebra and in this case g endowed with the metric is known as a quadratic Lie algebra. This paper provides a complete description of L-invariant, R-invariant, or associative pseudo-Euclidean metrics on left Leibniz algebras over a commutative field of characteristic zero. It shows that a left Leibniz algebra with an associative metric is also right Leibniz and can be obtained easily from its underlying Lie algebra, which is a quadratic Lie algebra. Additionally, it shows that at the core of a left Leibniz algebra endowed with a L-invariant or R-invariant metric, there are two Lie algebras with one quadratic and the left Leibniz algebra can be built from these Lie algebras. We derive many important results from this complete description. Finally, the paper provides a list of left Leibniz algebras with an associative metric up to dimension 6, as well as a list of left Leibniz algebras with an L-invariant metric, up to dimension 4, and R-invariant metric up to dimension 5. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
9. Determination of some almost split sequences in morphism categories.
- Author
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Hafezi, Rasool and Eshraghi, Hossein
- Subjects
- *
REPRESENTATION theory , *DYNKIN diagrams , *ARTIN algebras , *MORPHISMS (Mathematics) , *ALGEBRA - Abstract
Almost split sequences lie in the heart of Auslander-Reiten theory. This paper deals with the structure of almost split sequences with certain ending terms in the morphism category of an Artin algebra Λ. Firstly we try to interpret the Auslander-Reiten translates of particular objects in the morphism category in terms of the Auslander-Reiten translations within the category of Λ-modules, and then use them to calculate almost split sequences. In classical representation theory of algebras, it is quite important to recognize the middle term of almost split sequences. As such, another part of the paper is devoted to discuss the middle term of certain almost split sequences in the morphism category of Λ. As an application, we restrict in the last part of the paper to self-injective algebras and present a structural theorem that illuminates a link between representation-finite morphism categories and Dynkin diagrams. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
10. Gorensteinness in Rees algebras of powers of parameter ideals.
- Author
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Goto, Shiro and Iai, Shin-ichiro
- Subjects
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EXPONENTS , *NOETHERIAN rings , *ALGEBRA , *COHEN-Macaulay rings , *GORENSTEIN rings , *IDEALS (Algebra) , *LOCAL rings (Algebra) - Abstract
This paper gives a necessary and sufficient condition for Gorensteinness in Rees algebras of the d th power of parameter ideals in certain Noetherian local rings of dimension d ≥ 2. The main result of this paper produces many Gorenstein Rees algebras over non-Cohen-Macaulay local rings. For example, the Rees algebra R (q d) = ⊕ i ≥ 0 q d i is Gorenstein for every parameter ideal q that is a reduction of the maximal ideal in a d -dimensional Buchsbaum local ring of depth 1 and multiplicity 2. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
11. On the endomorphism algebra of Specht modules in even characteristic.
- Author
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Geranios, Haralampos and Higgins, Adam
- Subjects
- *
MODULES (Algebra) , *ENDOMORPHISM rings , *ENDOMORPHISMS , *ALGEBRA - Abstract
Over fields of characteristic 2, Specht modules may decompose and there is no upper bound for the dimension of their endomorphism algebra. A classification of the (in)decomposable Specht modules and a closed formula for the dimension of their endomorphism algebra remain two important open problems in the area. In this paper, we introduce a novel description of the endomorphism algebra of the Specht modules and provide infinite families of Specht modules with one-dimensional endomorphism algebra. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
12. Skew axial algebras of Monster type.
- Author
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Turner, Michael
- Subjects
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ALGEBRA , *CLASSIFICATION - Abstract
Skew axets were first defined by McInroy and Shpectorov where they used the term of axets to classify shapes of an algebra. When they first submitted their paper, it was not known if skew axial algebras exist and now we will present such examples with axet X ′ (1 + 2). Looking at 2-generated primitive axial algebras of Monster type, we will be able to state and prove the classification of such algebras with axet X ′ (1 + 2). We will conclude by looking at larger skew axets and give a suggestion on how one could extend the classification. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
13. Classification of simple Harish-Chandra modules over the generalized Witt algebras.
- Author
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Lü, Rencai and Xue, Yaohui
- Subjects
- *
ALGEBRA , *CLASSIFICATION - Abstract
In this paper, we classify simple Harish-Chandra modules over simple generalized Witt algebras. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
14. From quantum loop superalgebras to super Yangians.
- Author
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Lin, Hongda, Wang, Yongjie, and Zhang, Honglian
- Subjects
- *
ALGEBRA , *SUPERALGEBRAS , *LIE superalgebras , *ARGUMENT - Abstract
The goal of this paper is to generalize a statement by Drinfeld, asserting that Yangians can be constructed as limit forms of the quantum loop algebras, to the super case. We establish a connection between quantum loop superalgebra and super Yangian of the general linear Lie superalgebra gl M | N in RTT type presentation. In particular, we derive the Poincaré-Birkhoff-Witt(PBW) theorem for the quantum loop superalgebra U q (Lgl M | N). Additionally, we investigate the application of the same argument to twisted super Yangian of the ortho-symplectic Lie superalgebra. For this purpose, we introduce the twisted quantum loop superalgebra as a one-sided coideal of U q (Lgl M | 2 n) with respect to the comultiplication. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
15. Evaluation maps for affine quantum Schur algebras.
- Author
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Fu, Qiang and Liu, Mingqiang
- Subjects
- *
AFFINE algebraic groups , *HECKE algebras , *MODULES (Algebra) , *ALGEBRA , *POLYNOMIALS - Abstract
For a ∈ C ⁎ there are two natural evaluation maps ev a and ev a from the affine Hecke algebra H ▵ (r) C to the Hecke algebra H (r) C. The maps ev a and ev a induce evaluation maps ev ˜ a and ev ˜ a from the affine quantum Schur algebra S ▵ (n , r) C to the quantum Schur algebra S (n , r) C , respectively. In this paper we prove that the evaluation map ev ˜ a (resp. ev ˜ a) is compatible with the evaluation map Ev a (resp. Ev (− 1) n a q n ) for quantum affine sl n. Furthermore we compute the Drinfeld polynomials associated with the simple S ▵ (n , r) C -modules which come from the simple S (n , r) C -modules via the evaluation maps ev ˜ a. Then we characterize finite-dimensional irreducible S ▵ (n , r) C -modules which are irreducible as S (n , r) C -modules for n > r. As an application, we characterize finite-dimensional irreducible modules for the affine Hecke algebra H ▵ (r) C which are irreducible as modules for the Hecke algebra H (r) C. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
16. Symplectic structures, product structures and complex structures on Leibniz algebras.
- Author
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Tang, Rong, Xu, Nanyan, and Sheng, Yunhe
- Subjects
- *
ALGEBRA , *BILINEAR forms , *VECTOR spaces , *PHASE space , *JORDAN algebras - Abstract
In this paper, a symplectic structure on a Leibniz algebra is defined to be a symmetric nondegenerate bilinear form satisfying certain compatibility condition, and a phase space of a Leibniz algebra is defined to be a symplectic Leibniz algebra satisfying certain conditions. We show that a Leibniz algebra has a phase space if and only if there is a compatible Leibniz-dendriform algebra, and phase spaces of Leibniz algebras are one-to-one corresponds to Manin triples of Leibniz-dendriform algebras. Product (paracomplex) structures and complex structures on Leibniz algebras are studied in terms of decompositions of Leibniz algebras. A para-Kähler structure on a Leibniz algebra is defined to be a symplectic structure and a paracomplex structure satisfying a compatibility condition. We show that a symplectic Leibniz algebra admits a para-Kähler structure if and only if the Leibniz algebra is the direct sum of two Lagrangian subalgebras as vector spaces. A complex product structure on a Leibniz algebra consists of a complex structure and a product structure satisfying a compatibility condition. A pseudo-Kähler structure on a Leibniz algebra is defined to be a symplectic structure and a complex structure satisfying a compatibility condition. Various properties and relations of complex product structures and pseudo-Kähler structures are studied. In particular, Leibniz-dendriform algebras give rise to complex product structures and pseudo-Kähler structures naturally. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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17. An algebraic framework for the Drinfeld double based on infinite groupoids.
- Author
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Zhou, Nan and Wang, Shuanhong
- Subjects
- *
GROUPOIDS , *DRINFELD modules , *ALGEBRA , *HOPF algebras - Abstract
In this paper we mainly consider the notion of Drinfeld double for two weak multiplier Hopf (⁎-)algebras which are paired with each other. Then we show that the Drinfeld double is again a weak multiplier Hopf (⁎-)algebra. Furthermore, we study integrals on the Drinfeld double. Finally, we establish the correspondence between modules over a Drinfeld double D (A) and Yetter-Drinfeld modules over a weak algebraic quantum group A. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
18. Plenary train algebras of rank m and backcrossing identity.
- Author
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Bayara, Joseph and Coulibaly, Siaka
- Subjects
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IDEMPOTENTS , *ALGEBRA - Abstract
This paper concerns commutative plenary train algebras of rank m and their idempotents. We obtain the Peirce decomposition of these algebras having an idempotent element and the multiplication table of Peirce components when the plenary train roots are mutually different. We show that a backcrossing algebra is a plenary train algebra of rank m if and only if, it is a principal train one of rank m. For the backcrossing train algebras, we confirm a first conjecture of Juan Carlos Gutiérrez Fernández on the relation between plenary train roots and principal train roots. A second conjecture on the existence of idempotent in train algebras also obtains a positive answer in the class of backcrossing algebras. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
19. Perazzo hypersurfaces and the weak Lefschetz property.
- Author
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Miró-Roig, Rosa M. and Pérez, Josep
- Subjects
- *
HILBERT functions , *HOMOGENEOUS polynomials , *ALGEBRA , *HYPERSURFACES - Abstract
We deal with Perazzo hypersurfaces X = V (f) in P n + 2 defined by a homogeneous polynomial f (x 0 , x 1 , ... , x n , u , v) = p 0 (u , v) x 0 + p 1 (u , v) x 1 + ⋯ + p n (u , v) x n + g (u , v) , where p 0 , p 1 , ... , p n are algebraically dependent but linearly independent forms of degree d − 1 in K [ u , v ] and g is a form in K [ u , v ] of degree d. Perazzo hypersurfaces have vanishing hessian and, hence, the associated graded artinian Gorenstein algebra A f fails the strong Lefschetz property. In this paper, we first determine the maximum and minimum Hilbert function of A f , we prove that the Hilbert function of A f is always unimodal and we determine when A f satisfies the weak Lefschetz property. We illustrate our results with many examples and we show that our results do not generalize to Perazzo hypersurfaces X = V (f) in P n + 3 defined by a homogeneous polynomial f (x 0 , x 1 , ... , x n , u , v , w) = p 0 (u , v , w) x 0 + p 1 (u , v , w) x 1 + ⋯ + p n (u , v , w) x n + g (u , v , w) , where p 0 , p 1 , ... , p n are algebraically dependent but linearly independent forms of degree d − 1 in K [ u , v , w ] and g is a form in K [ u , v , w ] of degree d. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
20. Symplectic orbits of unimodular rows.
- Author
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Syed, Tariq
- Subjects
- *
ORBITS (Astronomy) , *SYMPLECTIC groups , *ALGEBRA , *ORBIT method , *MATRICES (Mathematics) - Abstract
For a smooth affine algebra R of dimension d ≥ 3 over a field k and an invertible alternating matrix χ of rank 2 n , the group S p (χ) of invertible matrices of rank 2 n over R which are symplectic with respect to χ acts on the right on the set U m 2 n (R) of unimodular rows of length 2 n over R. In this paper, we prove that S p (χ) acts transitively on U m 2 n (R) if k is algebraically closed, d ! ∈ k × and 2 n ≥ d. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
21. Extension dimensions of derived and stable equivalent algebras.
- Author
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Zhang, Jinbi and Zheng, Junling
- Subjects
- *
ARTIN algebras , *ALGEBRA - Abstract
The extension dimensions of an Artin algebra give a reasonable way of measuring how far an algebra is from being representation-finite. In this paper we mainly study the behavior of the extensions dimensions of algebras under different equivalences. We show that the difference of the extension dimensions of two derived equivalent algebras is bounded above by the length of the tilting complex associated with the derived equivalence, and that the extension dimension is an invariant under the stable equivalence. In addition, we provide two sufficient conditions such that the extension dimension is an invariant under particular derived equivalences. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
22. Class numbers of multinorm-one tori.
- Author
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Hung, Fan-Yun and Yu, Chia-Fu
- Subjects
- *
ALGEBRA , *GENERALIZATION - Abstract
We present a formula for the class number of a multinorm one torus T L / k associated to any étale algebra L over a global field k. This is deduced from a formula for analogues of invariants introduced by T. Ono, which are interpreted as a generalization of Gauss genus theory. This paper includes the variants of Ono's invariant for arbitrary S -ideal class numbers and the narrow version, generalizing results of Katayama, Morishita, Ono and Sasaki. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
23. The completion of d-abelian categories.
- Author
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Ebrahimi, Ramin and Nasr-Isfahani, Alireza
- Subjects
- *
HOMOLOGICAL algebra , *CLUSTER algebras , *ABELIAN categories , *ALGEBRA - Abstract
Let A be a finite-dimensional algebra, and M be a d -cluster tilting subcategory of mod A. From the viewpoint of higher homological algebra, a natural question to ask is when M induces a d -cluster tilting subcategory in Mod A. In this paper, we investigate this question in a more general form. We consider M as an essentially small d -abelian category, known to be equivalent to a d -cluster tilting subcategory of an abelian category A. The completion of M , denoted by Ind (M) , is defined as the universal completion of M with respect to filtered colimits. We explore Ind (M) and demonstrate its equivalence to the full subcategory L d (M) of Mod M , comprising left d -exact functors. Notably, Ind (M) as a subcategory of Mod M Eff (M) falls short of being a d -cluster tilting subcategory since it satisfies all properties of a d -cluster tilting subcategory except d -rigidity. For a d -cluster tilting subcategory M of mod A , M → consists of all filtered colimits of objects from M , is a generating-cogenerating, functorially finite subcategory of Mod A. The question of whether M → is a d -rigid subcategory remains unanswered. However, if it is indeed d -rigid, it qualifies as a d -cluster tilting subcategory. In the case d = 2 , employing cotorsion theory, we establish that M → is a 2-cluster tilting subcategory if and only if M is of finite type. Thus, the question regarding whether M → is a d -cluster tilting subcategory of Mod A appears to be equivalent to Iyama's question about the finiteness of M. Furthermore, for general d , we address the problem and present several equivalent conditions for Iyama's question. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
24. (Co)homology and crossed module for BiHom-associative algebras.
- Author
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Huang, Danli and Liu, Ling
- Subjects
- *
MODULES (Algebra) , *ASSOCIATIVE algebras , *LINEAR operators , *ASSOCIATIVE rings , *ALGEBRA - Abstract
BiHom-associative algebras are generalized associative algebras with two multiplicative linear maps. In this paper, we give the Hochschild homology and cyclic homology structure of BiHom-associative algebras. Then, generalize the dual bimodule action to define the cyclic cohomology. Finally, we introduce the crossed modules of BiHom-associative algebras and show that the Hochschild cohomology of BiHom-associative algebra classifies crossed modules. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
25. Descriptions of strongly multiplicity free representations for simple Lie algebras.
- Author
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Sun, Bin-Ni and Zhao, Yufeng
- Subjects
- *
LIE algebras , *MULTIPLICITY (Mathematics) , *UNIVERSAL algebra , *ALGEBRA , *ENDOMORPHISMS , *ENDOMORPHISM rings - Abstract
Let g be a complex simple Lie algebra and Z (g) be the center of the universal enveloping algebra U (g). Denote by V λ the finite-dimensional irreducible g -module with highest weight λ. Lehrer and Zhang defined the notion of strongly multiplicity free representations for simple Lie algebras motivated by studying the structure of the endomorphism algebra End U (g) (V λ ⊗ r) in terms of the quotients of the Kohno's infinitesimal braid algebra. Kostant introduced the g -invariant endomorphism algebras R λ (g) = (End V λ ⊗ U (g)) g and R λ , π (g) = (End V λ ⊗ π (U (g))) g. In this paper, we give some other criteria for a multiplicity free representation to be strongly multiplicity free by classifying the pairs (g , V λ) , which are multiplicity free and for such pairs, R λ (g) and R λ , π (g) are generated by generalizations of the quadratic Casimir elements of Z (g). [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
26. Slack Hopf monads.
- Author
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Bruguières, Alain, Haim, Mariana, and López Franco, Ignacio
- Subjects
- *
HOPF algebras , *ALGEBROIDS , *GENERALIZATION , *MAGMAS , *ALGEBRA - Abstract
Hopf monads generalise Hopf algebras. They clarify several aspects of the theory of Hopf algebras and capture several related structures such as weak Hopf algebras and Hopf algebroids. However, important parts of Hopf algebra theory are not reached by Hopf monads, most noticeably Drinfeld's quasi-Hopf algebras. In this paper we introduce a generalisation of Hopf monads, that we call slack Hopf monads. This generalisation retains a clean theory and is flexible enough to encompass quasi-Hopf algebras as examples. A slack Hopf monad is a colax magma monad T on a magma category C such that the forgetful functor U T : C T → C 'slackly' preserves internal Homs. We give a number of different descriptions of slack Hopf monads, and study special cases such as slack Hopf monads on cartesian categories and k -linear exact slack Hopf monads on Vect k , that is comagma algebras such that a modified fusion operator is invertible. In particular, we characterise quasi-Hopf algebras in terms of slackness. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
27. Grothendieck rings of towers of generalized Weyl algebras in the finite orbit case.
- Author
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Hartwig, Jonas T. and Rosso, Daniele
- Subjects
- *
ORBITS (Astronomy) , *ALGEBRA , *TENSOR products , *INDECOMPOSABLE modules , *ORBIT method - Abstract
Previously we showed that the tensor product of a weight module over a generalized Weyl algebra (GWA) with a weight module over another GWA is a weight module over a third GWA. In this paper we compute tensor products of simple and indecomposable weight modules over generalized Weyl algebras supported on a finite orbit. This allows us to give a complete presentation by generators and relations of the Grothendieck ring of the categories of weight modules over a tower of generalized Weyl algebras in this setting. We also obtain partial results about the split Grothendieck ring. We described the case of infinite orbits in previous work. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
28. Denominator vectors and dimension vectors from triangulated surfaces.
- Author
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Yurikusa, Toshiya
- Subjects
- *
CLUSTER algebras , *INDECOMPOSABLE modules , *INTERSECTION numbers , *ALGEBRA - Abstract
In a categorification of skew-symmetric cluster algebras, each cluster variable corresponds with an indecomposable module over the associated Jacobian algebra. Buan, Marsh and Reiten studied when the denominator vector of each cluster variable in an acyclic cluster algebra coincides with the dimension vector of the corresponding module. In this paper, we give analogues of their results for cluster algebras from triangulated surfaces by comparing two kinds of intersection numbers of tagged arcs. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
29. Mixed standardization and Ringel duality.
- Author
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Adachi, Takahide and Tsukamoto, Mayu
- Subjects
- *
ALGEBRA , *GENERALIZATION , *STANDARDIZATION , *DUALITY theory (Mathematics) - Abstract
Dlab–Ringel's standardization method gives a realization of a standardly stratified algebra. In this paper, we construct mixed stratified algebras, which are a generalization of standardly stratified algebras, following Dlab–Ringel's standardization method. Moreover, we study a Ringel duality of mixed stratified algebras from the viewpoint of stratifying systems. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
30. Automorphisms of extensions of Lie-Yamaguti algebras and inducibility problem.
- Author
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Goswami, Saikat, Mishra, Satyendra Kumar, and Mukherjee, Goutam
- Subjects
- *
AUTOMORPHISM groups , *ALGEBRA , *GROUP algebras , *LIE algebras , *NONASSOCIATIVE algebras , *AUTOMORPHISMS - Abstract
Lie-Yamaguti algebras generalize both the notions of Lie algebras and Lie triple systems. In this paper, we consider the inducibility problem for automorphisms of Lie-Yamaguti algebra extensions. More precisely, given an abelian extension [Display omitted] of a Lie-Yamaguti algebra L , we are interested in finding the pairs (ϕ , ψ) ∈ Aut (V) × Aut (L) , which are inducible by an automorphism in Aut (L ˜). We connect the inducibility problem to the (2 , 3) -cohomology of Lie-Yamaguti algebra. In particular, we show that the obstruction for a pair of automorphisms in Aut (V) × Aut (L) to be inducible lies in the (2 , 3) -cohomology group H (2 , 3) (L , V). We develop the Wells exact sequence for Lie-Yamaguti algebra extensions, which relates the space of derivations, automorphism groups, and (2 , 3) -cohomology groups of Lie-Yamaguti algebras. As an application, we describe certain automorphism groups of semi-direct product Lie-Yamaguti algebras. In a sequel, we apply our results to discuss inducibility problem for nilpotent Lie-Yamaguti algebras of index 2. We give examples of infinite families of such nilpotent Lie-Yamaguti algebras and characterize the inducible pairs of automorphisms for extensions arising from these examples. Finally, we write an algorithm to find out all the inducible pairs of automorphisms for extensions arising from nilpotent Lie-Yamaguti algebras of index 2. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
31. The parity of Lusztig's restriction functor and Green's formula.
- Author
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Fang, Jiepeng, Lan, Yixin, and Xiao, Jie
- Subjects
- *
QUANTUM groups , *REPRESENTATIONS of algebras , *FINITE fields , *SEMISIMPLE Lie groups , *SHEAF theory , *ALGEBRA , *HOMOMORPHISMS , *HOMOLOGICAL algebra - Abstract
Our investigation in the present paper is based on three important results. (1) In [14] , Ringel introduced Hall algebra for representations of a quiver over finite fields and proved the elements corresponding to simple representations satisfy the quantum Serre relation. This gives a realization of the nilpotent part of quantum group if the quiver is of finite type. (2) In [6] , Green found a homological formula for the representation category of the quiver and equipped Ringel's Hall algebra with a comultiplication. The generic form of the composition subalgebra of Hall algebra generated by simple representations realizes the nilpotent part of quantum group of any type. (3) In [11] , Lusztig defined induction and restriction functors for the perverse sheaves on the variety of representations of the quiver which occur in the direct images of constant sheaves on flag varieties, and he found a formula between his induction and restriction functors which gives the comultiplication as algebra homomorphism for quantum group. In the present paper, we prove the formula holds for all semisimple complexes with Weil structure. This establishes the categorification of Green's formula. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
32. Diagram automorphisms and canonical bases for quantum affine algebras, II.
- Author
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Ma, Ying, Shoji, Toshiaki, and Zhou, Zhiping
- Subjects
- *
ALGEBRA , *BIJECTIONS , *FINITE, The - Abstract
Let U q − be the negative part of the quantum enveloping algebra, and σ the algebra automorphism on U q − induced from a diagram automorphism. Let U _ q − be the quantum algebra obtained from σ , and B ˜ (resp. B _ ˜) the canonical signed basis of U q − (resp. U _ q −). Assume that U q − is simply-laced of finite or affine type. In our previous papers [10] , [11] , we have proved by an elementary method, that there exists a natural bijection B ˜ σ ≃ B _ ˜ in the case where σ is admissible. In this paper, we show that such a bijection exists even if σ is not admissible, possibly except some small rank cases. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
33. Quotients of the Highwater algebra and its cover.
- Author
-
Franchi, C., Mainardis, M., and M c Inroy, J.
- Subjects
- *
ALGEBRA , *AUTOMORPHISM groups , *FINITE simple groups - Abstract
Primitive axial algebras of Monster type are a class of non-associative algebras with a strong link to finite (especially simple) groups. The motivating example is the Griess algebra, with the Monster as its automorphism group. A crucial step towards the understanding of such algebras is the explicit description of the 2-generated symmetric objects. Recent work of Yabe, and Franchi and Mainardis shows that any such algebra is either explicitly known, or is a quotient of the infinite-dimensional Highwater algebra H , or its characteristic 5 cover H ˆ. In this paper, we complete the classification of symmetric axial algebras of Monster type by determining the quotients of H and H ˆ. We proceed in a unified way, by defining a cover of H in all characteristics. This cover has a previously unseen fusion law and provides an insight into why the Highwater algebra has a cover which is of Monster type only in characteristic 5. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
34. Vertex algebras and TKK algebras.
- Author
-
Chen, Fulin, Ding, Lingen, and Wang, Qing
- Subjects
- *
ALGEBRA , *VERTEX operator algebras , *LIE algebras , *COMPLEX numbers , *C*-algebras , *ISOMORPHISM (Mathematics) - Abstract
In this paper, we associate the TKK algebra G ˆ (J) with vertex algebras through twisted modules. Firstly, we prove that for any complex number ℓ , the category of restricted G ˆ (J) -modules of level ℓ is canonically isomorphic to the category of σ -twisted V C g ˆ (ℓ , 0) -modules, where V C g ˆ (ℓ , 0) is a vertex algebra arising from the toroidal Lie algebra of type C 2 and σ is an isomorphism of V C g ˆ (ℓ , 0) induced from the involution of this toroidal Lie algebra. Secondly, we prove that for any nonnegative integer ℓ , the integrable restricted G ˆ (J) -modules of level ℓ are exactly the σ -twisted modules for the quotient vertex algebra L C g ˆ (ℓ , 0) of V C g ˆ (ℓ , 0). Finally, we classify the irreducible 1 2 N -graded σ -twisted L C g ˆ (ℓ , 0) -modules. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
35. Morphisms and extensions between bricks over preprojective algebras of type A.
- Author
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Hanson, Eric J. and You, Xinrui
- Subjects
- *
ALGEBRA , *BRICKS - Abstract
The bricks over preprojective algebras of type A are known to be in bijection with certain combinatorial objects called "arcs". In this paper, we show how one can use arcs to compute bases for the Hom-spaces and first extension spaces between bricks. We then use this description to classify the "weak exceptional sequences" over these algebras. Finally, we explain how our result relates to a similar combinatorial model for the exceptional sequences over hereditary algebras of type A. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
36. Central polynomials of the second-order matrix algebra with graded involution.
- Author
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Cruz, J.P. and Vieira, A.C.
- Subjects
- *
MATRICES (Mathematics) , *POLYNOMIALS , *ALGEBRA - Abstract
Let F be an infinite field and M 1 , 1 (F) be the algebra of 2 × 2 matrices over F endowed with non-trivial Z 2 -grading. We consider the involutions ⁎ defined on M 1 , 1 (F) which preserve the homogeneous components of the grading. In this paper, we deal with the ⁎-superalgebra (M 1 , 1 (F) , ⁎) and determine the generators of its ideal of (Z 2 , ⁎) -identities, considering that F has characteristic zero and also, we explicitly construct the generators of its space of central (Z 2 , ⁎) -polynomials, when the characteristic of F is different from 2. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
37. Representations of map extended Witt algebras.
- Author
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Sharma, Sachin S., Chakraborty, Priyanshu, Pandey, Ritesh Kumar, and Eswara Rao, S.
- Subjects
- *
ALGEBRA , *VERTEX operator algebras - Abstract
In this paper, we classify irreducible modules for map extended Witt algebras with finite dimensional weight spaces. They turn out to be either modules with uniformly bounded weight spaces or highest weight modules. We further prove that all these modules are single point evaluation modules (n ≥ 2). So they are actually irreducible modules for extended Witt algebras. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
38. Foldings of KLR algebras.
- Author
-
Ma, Ying, Shoji, Toshiaki, and Zhou, Zhiping
- Subjects
- *
KAC-Moody algebras , *ALGEBRA - Abstract
Let U q − be the negative half of the quantum group associated to the Kac-Moody algebra g , and U _ q − the quantum group obtained by a folding of g. Let A = Z [ q , q − 1 ]. McNamara showed that U _ q − is categorified over a certain extension ring A ˜ of A , by using the folding theory of KLR algebras. He posed a question whether A ˜ coincides with A or not. In this paper, we give an affirmative answer for this problem. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
39. Simple Heisenberg-Virasoro modules from Weyl algebra modules.
- Author
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Li, Shujuan, Zhu, Junyi, and Guo, Xiangqian
- Subjects
- *
ISOMORPHISM (Mathematics) , *ALGEBRA - Abstract
In this paper, we constructed and studied two classes of modules over the Heisenberg-Virasoro algebra HVir from Weyl algebra modules, parallel to the similar construction for Virasoro algebra in [13,9]. More precisely, given any module P over the degree-1 Weyl algebra, we define the HVir-modules M (P , V) , where V is a restricted module over the nonnegative part a of HVir, and L (P , W , ξ , b , c) , where W is a restricted module over another subalgebra b ⊆ HVir and ξ , b , c ∈ C with ξ ≠ 0. We determined the irreducibility of these modules as well as the isomorphisms within each class respectively. Moreover, any two such modules from different classes are never isomorphic although they might be closely related, as we showed in the last section. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
40. Commuting maps and identities with inverses on alternative division rings.
- Author
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Ferreira, Bruno Leonardo Macedo, Julius, Hayden, and Smigly, Douglas
- Subjects
- *
DIVISION rings , *ASSOCIATIVE rings , *IDENTITIES (Mathematics) , *CAYLEY numbers (Algebra) , *ALGEBRA - Abstract
Let D be an alternative division ring with characteristic different from two. The purpose of this paper is to characterize additive mappings f , g : D → D satisfying certain identities studied by Vukman, Brešar, and Catalano previously on an associative division ring. We also present some new identities concerning Lie and Jordan products, and provide a complete description of commuting maps on octonion algebras. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
41. Gonosomal algebras and associated discrete-time dynamical systems.
- Author
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Rozikov, U.A., Shoyimardonov, S.K., and Varro, R.
- Subjects
- *
DISCRETE-time systems , *DYNAMICAL systems , *NONASSOCIATIVE algebras , *ALGEBRA , *LETHAL mutations - Abstract
In this paper we study the discrete-time dynamical systems associated with gonosomal algebras used as algebraic model in the sex-linked genes inheritance. We show that the class of gonosomal algebras is disjoint from the other non-associative algebras usually studied (Lie, alternative, Jordan, associative power). To each gonosomal algebra, with the mapping x ↦ 1 2 x 2 , an evolution operator W is associated that gives the state of the offspring population at the birth stage, then from W we define the operator V which gives the frequency distribution of genetic types. We study discrete-time dynamical systems generated by these two operators, in particular we show that the various stability notions of the equilibrium points are preserved by passing from W to V. Moreover, for the evolution operators associated with genetic disorders in the case of a diallelic gonosomal lethal gene we give complete analysis of fixed and limit points of the dynamical systems. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
42. Jordan algebras of a degenerate bilinear form: Specht property and their identities.
- Author
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Fideles, Claudemir and Martino, Fabrizio
- Subjects
- *
BILINEAR forms , *IDENTITIES (Mathematics) , *JORDAN algebras , *GROBNER bases , *ALGEBRA , *POLYNOMIALS - Abstract
Let K be a field and let J n , k be the Jordan algebra of a degenerate symmetric bilinear form b of rank n − k over K. Then one can consider the decomposition J n , k = B n − k ⊕ D k , where B n − k represents the corresponding Jordan algebra, denoted as B n − k = K ⊕ V. In this algebra, the restriction of b on the (n − k) -dimensional subspace V is non-degenerate, while D k accounts for the degenerate part of J n , k. This paper aims to provide necessary and sufficient conditions to check if a given multilinear polynomial is an identity for J n , k. As a consequence of this result and under certain hypothesis on the base field, we exhibit a finite basis for the T -ideal of polynomial identities of J n , k. Over a field of characteristic zero, we also prove that the ideal of identities of J n , k satisfies the Specht property. Moreover, similar results are obtained for weak identities, trace identities and graded identities with a suitable Z 2 -grading as well. In all of these cases, we employ methods and results from Invariant Theory. Finally, as a consequence from the trace case, we provide a counterexample to the embedding problem given in [8] in case of infinite dimensional Jordan algebras with trace. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
43. On Rees algebras of ideals and modules over hypersurface rings.
- Author
-
Weaver, Matthew
- Subjects
- *
MODULES (Algebra) , *PROJECTIVE modules (Algebra) , *IDEALS (Algebra) , *ALGEBRA - Abstract
The acquisition of the defining equations of Rees algebras is a natural way to study these algebras and allows certain invariants and properties to be deduced. In this paper, we consider Rees algebras of codimension 2 perfect ideals of hypersurface rings and produce a minimal generating set for their defining ideals. Then, using generic Bourbaki ideals, we study Rees algebras of modules with projective dimension one over hypersurface rings. We describe the defining ideal of such algebras and determine Cohen-Macaulayness and other invariants. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
44. 2-generated axial algebras of Monster type (2β,β).
- Author
-
Franchi, Clara, Mainardis, Mario, and Shpectorov, Sergey
- Subjects
- *
NONASSOCIATIVE algebras , *ALGEBRA , *ASSOCIATIVE algebras , *FINITE simple groups , *JORDAN algebras - Abstract
Axial algebras of Monster type (α , β) are a class of non-associative algebras that includes, besides associative algebras, other important examples such as the Jordan algebras and the Griess algebra. 2-generated primitive axial algebras of Monster type (α , β) naturally split into three cases: the case when α ∉ { 2 β , 4 β } , the case α = 4 β and α = 2 β. In this paper we give a complete classification all 2-generated primitive axial algebras of Monster type (2 β , β). [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
45. Characterization of simple smooth modules.
- Author
-
Ma, Yao, Nguyen, Khoa, Tantubay, Santanu, and Zhao, Kaiming
- Subjects
- *
KAC-Moody algebras , *LIE algebras , *ALGEBRA - Abstract
In this paper, we characterize simple smooth modules over some infinite-dimensional Z -graded Lie algebras. More precisely, we prove that if one specific positive root element of a Z -graded Lie algebra g locally finitely acts on a simple g -module V , then V is a smooth g -module. These infinite-dimensional Z -graded Lie algebras include the Virasoro algebra, affine-Virasoro algebras, the (twisted, mirror) Heisenberg-Virasoro algebras, the planar Galilean conformal algebra, and many others. This result for untwisted affine Kac-Moody algebras holds unless we change the condition from "locally finitely" to "locally nilpotently". We also show that these are not the case for the Heisenberg algebra. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
46. Graded dimensions and monomial bases for the cyclotomic quiver Hecke superalgebras.
- Author
-
Hu, Jun and Shi, Lei
- Subjects
- *
SUPERALGEBRAS , *CYCLOTOMIC fields , *ALGEBRA , *POLYNOMIALS , *GROBNER bases - Abstract
In this paper we derive a closed formula for the (Z × Z 2) -graded dimension of the cyclotomic quiver Hecke superalgebra R Λ (β) associated to an arbitrary Cartan superdatum (A , P , Π , Π ∨) , polynomials (Q i , j (x 1 , x 2)) i , j ∈ I , β ∈ Q n + and Λ ∈ P +. As applications, we obtain a necessary and sufficient condition for which e (ν) ≠ 0 in R Λ (β). We construct an explicit monomial basis for the bi-weight space e (ν ˜) R Λ (β) e (ν ˜) , where ν ˜ is a certain specific n -tuple defined in (1.4). In particular, this gives rise to a monomial basis for the cyclotomic odd nilHecke algebra. Finally, we consider the case when β = α 1 + α 2 + ⋯ + α n with α 1 , ⋯ , α n distinct. We construct an explicit monomial basis of R Λ (β) and show that it is indecomposable in this case. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
47. Restricted modules and associated vertex algebras of extended Heisenberg-Virasoro algebra.
- Author
-
Guo, Hongyan and Li, Huaimin
- Subjects
- *
INFINITE dimensional Lie algebras , *ALGEBRA , *LIE algebras - Abstract
In this paper, a family of infinite dimensional Lie algebras L ˜ is introduced and investigated, called the extended Heisenberg-Virasoro algebra, denoted by L ˜. These Lie algebras are related to the N = 2 superconformal algebra and the Bershadsky-Polyakov algebra. We study restricted modules and associated vertex algebras of the Lie algebra L ˜. More precisely, we construct its associated vertex (operator) algebras V L ˜ (ℓ 123 , 0) , and show that the category of vertex algebra V L ˜ (ℓ 123 , 0) -modules is equivalent to the category of restricted L ˜ -modules of level ℓ 123. Then we give uniform constructions of simple restricted L ˜ -modules. Also, we present several equivalent characterizations of simple restricted modules over L ˜. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
48. On some classes of solvable Leibniz algebras and their completeness.
- Author
-
Abdurasulov, K.K., Omirov, B.A., and Rakhimov, I.S.
- Subjects
- *
LIE algebras , *ALGEBRA , *ISOMORPHISM (Mathematics) , *COMPLETENESS theorem - Abstract
The paper is devoted studying solvable Leibniz algebras with a nilradical possessing the codimension equals the number of its generators. We describe this class in non-split nilradical case up to isomorphism. Then the case of split nilradical is worked out. We show that the results obtained earlier on this class of Leibniz algebras come as particular cases of the results of this paper. It is shown that such a solvable extension is unique. Finally, we prove that the solvable Leibniz algebras considered are complete. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
49. t-Adic symmetrization map on the harmonic algebra.
- Author
-
Ono, Masataka
- Subjects
- *
ALGEBRA , *HARMONIC maps - Abstract
Bachmann, Takeyama and Tasaka introduced the Q -linear map ϕ on the harmonic algebra H 1 , which we call the symmetrization map in this paper. They calculated ϕ (w) explicitly for an element w in H 1 related to the multiple zeta values of Mordell–Tornheim type. In this paper, we introduce its t -adic generalization ϕ ˆ and calculate ϕ ˆ (w) for elements w in H 1 〚 t 〛 constructed from the theory of 2-colored rooted trees. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
50. Poisson Noether's Problem and Poisson rationality.
- Author
-
Schwarz, João
- Subjects
- *
SYMPLECTIC groups , *POISSON algebras , *ALGEBRA , *LOGICAL prediction , *TRIGONOMETRIC functions - Abstract
In this paper we show that the Poisson analogue of the Noether's Problem has a positive solution for essentially all symplectic reflection groups — the analogue of complex reflection groups in the symplectic world. Our proofs are constructive, and generalize and refine previously known results. The results of this paper can be thought as analogues of the Noncommutative Noether Problem and the Gelfand-Kirillov Conjecture for rational Cherednik algebras in the quasi-classical limit. An abstract framework to understand these results is introduced. As a consequence for complex reflection groups, we obtain the Poisson rationality of the Calogero-Moser spaces associated to any of them, and we verify the Gelfand-Kirillov Conjecture for trigonometric Cherednik algebras and the Poisson rationality of their corresponding Calogero-Moser spaces. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
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