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Journal articles on the topic 'Modules (Algebra)'

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Published: 4 June 2021

Last updated: 29 July 2024

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1

Vasilchenko, A. N. "PROPERTIES OF DUAL MODULES OVER STEENROD ALGEBRA." Vestnik of Samara University. Natural Science Series 20, no. 7 (May 30, 2017): 9–16. http://dx.doi.org/10.18287/2541-7525-2014-20-7-9-16.

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Properties of annulators and modules generated by annulators, including dual modules over Steenrod algebra are studied. Properties of Kroneker pairing are proved using general properties of Steenrod algebra and dual algebra as graded connected Hopf algebras. Isomorphisms between modules generated by annulators and dual modules over dual Stennrod algebra are proved. It is shown that these modules are Hopf comodules induced by coproduct in dual Steenrod algebra. All generators of these modules are found. The method of ﬁnding basis of module of indecomposable elements, viewed as vector space over cyclic ﬁeld for some of the studied modules

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2

Guo, Xiangqian, and Genqiang Liu. "Jet modules for the centerless Virasoro-like algebra." Journal of Algebra and Its Applications 18, no. 01 (January 2019): 1950002. http://dx.doi.org/10.1142/s0219498819500026.

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In this paper, we studied the jet modules for the centerless Virasoro-like algebra which is the Lie algebra of the Lie group of the area-preserving diffeomorphisms of a [Formula: see text]-torus. The jet modules are certain natural modules over the Lie algebra of semi-direct product of the centerless Virasoro-like algebra and the Laurent polynomial algebra in two variables. We reduce the irreducible jet modules to the finite-dimensional irreducible modules over some infinite-dimensional Lie algebra and then characterize the irreducible jet modules with irreducible finite dimensional modules over [Formula: see text]. To determine the indecomposable jet modules, we use the technique of polynomial modules in the sense of [Irreducible representations for toroidal Lie algebras, J. Algebras 221 (1999) 188–231; Weight modules over exp-polynomial Lie algebras, J. Pure Appl. Algebra 191 (2004) 23–42]. Consequently, indecomposable jet modules are described using modules over the algebra [Formula: see text], which is the “positive part” of a Block type algebra studied first by [Some infinite-dimensional simple Lie algebras in characteristic [Formula: see text] related to those of Block, J. Pure Appl. Algebra 127(2) (1998) 153–165] and recently by [A [Formula: see text]-graded generalization of the Witt-algebra, preprint; Classification of simple Lie algebras on a lattice, Proc. London Math. Soc. 106(3) (2013) 508–564]).

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3

FRISK, ANDERS, and VOLODYMYR MAZORCHUK. "PROPERLY STRATIFIED ALGEBRAS AND TILTING." Proceedings of the London Mathematical Society 92, no. 1 (December 19, 2005): 29–61. http://dx.doi.org/10.1017/s0024611505015431.

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We study the properties of tilting modules in the context of properly stratified algebras. In particular, we answer the question of when the Ringel dual of a properly stratified algebra is properly stratified itself, and show that the class of properly stratified algebras for which the characteristic tilting and cotilting modules coincide is closed under taking the Ringel dual. Studying stratified algebras whose Ringel dual is properly stratified, we discover a new Ringel-type duality for such algebras, which we call the two-step duality. This duality arises from the existence of a new (generalized) tilting module for stratified algebras with properly stratified Ringel dual. We show that this new tilting module has a lot of interesting properties; for instance, its projective dimension equals the projectively defined finitistic dimension of the original algebra, it guarantees that the category of modules of finite projective dimension is contravariantly finite, and, finally, it allows one to compute the finitistic dimension of the original algebra in terms of the projective dimension of the characteristic tilting module.

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4

Strade, Helmut. "Representations of the (р2 - 1)-Dimensional Lie Algebras of R. E. Block." Canadian Journal of Mathematics 43, no. 3 (June 1, 1991): 580–616. http://dx.doi.org/10.4153/cjm-1991-035-x.

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AbstractFor all algebras G, such that is an algebra mentioned in the title, the modules of dimension ≤ p2 are determined. The module homomorphisms from the tensor product of these modules into a third module of the same type are described. We also give the central extensions of the algebras .

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5

Adamović, Dražen, and Gordan Radobolja. "Self-dual and logarithmic representations of the twisted Heisenberg–Virasoro algebra at level zero." Communications in Contemporary Mathematics 21, no. 02 (February 27, 2019): 1850008. http://dx.doi.org/10.1142/s0219199718500086.

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This paper is a continuation of [D. Adamović and G. Radobolja, Free field realization of the twisted Heisenberg–Virasoro algebra at level zero and its applications, J. Pure Appl. Algebra 219(10) (2015) 4322–4342]. We present certain new applications and generalizations of the free field realization of the twisted Heisenberg–Virasoro algebra [Formula: see text] at level zero. We find explicit formulas for singular vectors in certain Verma modules. A free field realization of self-dual modules for [Formula: see text] is presented by combining a bosonic construction of Whittaker modules from [D. Adamović, R. Lu and K. Zhao, Whittaker modules for the affine Lie algebra [Formula: see text], Adv. Math. 289 (2016) 438–479; arXiv:1409.5354] with a construction of logarithmic modules for vertex algebras. As an application, we prove that there exists a non-split self-extension of irreducible self-dual module which is a logarithmic module of rank two. We construct a large family of logarithmic modules containing different types of highest weight modules as subquotients. We believe that these logarithmic modules are related with projective covers of irreducible modules in a suitable category of [Formula: see text]-modules.

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6

ERDMANN, KARIN. "EXT-FINITE MODULES FOR WEAKLY SYMMETRIC ALGEBRAS WITH RADICAL CUBE ZERO." Journal of the Australian Mathematical Society 102, no. 1 (September 19, 2016): 108–21. http://dx.doi.org/10.1017/s1446788716000331.

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Assume that $A$ is a finite-dimensional algebra over some field, and assume that $A$ is weakly symmetric and indecomposable, with radical cube zero and radical square nonzero. We show that such an algebra of wild representation type does not have a nonprojective module $M$ whose ext-algebra is finite dimensional. This gives a complete classification of weakly symmetric indecomposable algebras which have a nonprojective module whose ext-algebra is finite dimensional. This shows in particular that existence of ext-finite nonprojective modules is not equivalent with the failure of the finite generation condition (Fg), which ensures that modules have support varieties.

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7

Khovanov, Mikhail. "Hopfological algebra and categorification at a root of unity: The first steps." Journal of Knot Theory and Its Ramifications 25, no. 03 (March 2016): 1640006. http://dx.doi.org/10.1142/s021821651640006x.

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Any finite-dimensional Hopf algebra [Formula: see text] is Frobenius and the stable category of [Formula: see text]-modules is triangulated monoidal. To [Formula: see text]-comodule algebras we assign triangulated module-categories over the stable category of [Formula: see text]-modules. These module-categories are generalizations of homotopy and derived categories of modules over a differential graded algebra. We expect that, for suitable [Formula: see text], our construction could be a starting point in the program of categorifying quantum invariants of 3-manifolds.

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8

Grewcoe, Clay James, Larisa Jonke, Toni Kodžoman, and George Manolakos. "From Hopf Algebra to Braided L∞-Algebra." Universe 8, no. 4 (April 1, 2022): 222. http://dx.doi.org/10.3390/universe8040222.

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We show that an L∞-algebra can be extended to a graded Hopf algebra with a codifferential. Then, we twist this extended L∞-algebra with a Drinfel’d twist, simultaneously twisting its modules. Taking the L∞-algebra as its own (Hopf) module, we obtain the recently proposed braided L∞-algebra. The Hopf algebra morphisms are identified with the strict L∞-morphisms, whereas the braided L∞-morphisms define a more general L∞-action of twisted L∞-algebras.

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9

Liu, Dong, Yufeng Pei, and Limeng Xia. "Representations for three-point Lie algebras of genus zero." International Journal of Mathematics 30, no. 14 (October 17, 2019): 1950070. http://dx.doi.org/10.1142/s0129167x19500708.

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In this paper, we study representations for three-point Lie algebras of genus zero based on the Cox–Jurisich’s presentations. We construct two functors which transform simple restricted modules with nonzero levels over the standard affine algebras into simple modules over the three-point affine algebras of genus zero. As a corollary, vertex representations are constructed for the three-point affine algebra of genus zero using vertex operators. Moreover, we construct a Fock module for certain quotient of three-point Virasoro algebra of genus zero.

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10

Thiel, U. "Champ: a Cherednik algebraMagmapackage." LMS Journal of Computation and Mathematics 18, no. 1 (2015): 266–307. http://dx.doi.org/10.1112/s1461157015000054.

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We present a computer algebra package based onMagmafor performing computations in rational Cherednik algebras with arbitrary parameters and in Verma modules for restricted rational Cherednik algebras. Part of this package is a new general Las Vegas algorithm for computing the head and the constituents of a module with simple head in characteristic zero, which we develop here theoretically. This algorithm is very successful when applied to Verma modules for restricted rational Cherednik algebras and it allows us to answer several questions posed by Gordon in some specific cases. We can determine the decomposition matrices of the Verma modules, the graded$G$-module structure of the simple modules, and the Calogero–Moser families of the generic restricted rational Cherednik algebra for around half of the exceptional complex reflection groups. In this way we can also confirm Martino’s conjecture for several exceptional complex reflection groups.Supplementary materials are available with this article.

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11

BAVULA, VOLODYMYR V., and TAO LU. "THE PRIME IDEALS AND SIMPLE MODULES OF THE UNIVERSAL ENVELOPING ALGEBRA U(𝔟⋉V2)." Glasgow Mathematical Journal 62, S1 (July 29, 2019): S77—S98. http://dx.doi.org/10.1017/s0017089519000302.

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AbstractLet 𝔟 be the Borel subalgebra of the Lie algebra 𝔰𝔩2 and V2 be the simple two-dimensional 𝔰𝔩2-module. For the universal enveloping algebra $\[{\cal A}: = U(\gb \ltimes {V_2})\]$ of the semi-direct product 𝔟⋉V2 of Lie algebras, the prime, primitive and maximal spectra are classified. Please approve edit to the sentence “The sets of completely prime…”.The sets of completely prime ideals of $\[{\cal A}\]$ are described. The simple unfaithful $\[{\cal A}\]$-modules are classified and an explicit description of all prime factor algebras of $\[{\cal A}\]$ is given. The following classes of simple U(𝔟⋉V2)-modules are classified: the Whittaker modules, the 𝕂[X]-torsion modules and the 𝕂[E]-torsion modules.

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12

Asefa, Dadi. "Gorenstein-Projective Modules over Upper Triangular Matrix Artin Algebras." Journal of Mathematics 2021 (November 19, 2021): 1–8. http://dx.doi.org/10.1155/2021/8127282.

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Gorenstein-projective module is an important research topic in relative homological algebra, representation theory of algebras, triangulated categories, and algebraic geometry (especially in singularity theory). For a given algebra A , how to construct all the Gorenstein-projective A -modules is a fundamental problem in Gorenstein homological algebra. In this paper, we describe all complete projective resolutions over an upper triangular Artin algebra Λ = A M B A 0 B . We also give a necessary and sufficient condition for all finitely generated Gorenstein-projective modules over Λ = A M B A 0 B .

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13

LI, HAISHENG. "ON ABELIAN COSET GENERALIZED VERTEX ALGEBRAS." Communications in Contemporary Mathematics 03, no. 02 (May 2001): 287–340. http://dx.doi.org/10.1142/s0219199701000366.

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This paper studies the algebraic aspect of a general abelian coset theory with a work of Dong and Lepowsky as our main motivation. It is proved that the vacuum space ΩV (or the space of highest weight vectors) of a Heisenberg algebra in a general vertex operator algebra V has a natural generalized vertex algebra structure in the sense of Dong and Lepowsky and that the vacuum space ΩW of a V-module W is a natural ΩV-module. The automorphism group Aut ΩVΩV of the adjoint ΩV-module is studied and it is proved to be a central extension of a certain torsion free abelian group by C×. For certain subgroups A of Aut ΩVΩV, certain quotient algebras [Formula: see text] of ΩV are constructed. Furthermore, certain functors among the category of V-modules, the category of ΩV-modules and the category of [Formula: see text]-modules are constructed and irreducible ΩV-modules and [Formula: see text]-modules are classified in terms of irreducible V-modules. If the category of V-modules is semisimple, then it is proved that the category of [Formula: see text]-modules is semisimple.

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14

NIEPER-WISSKIRCHEN, MARC A. "OPERADS AND JET MODULES." International Journal of Geometric Methods in Modern Physics 02, no. 06 (December 2005): 1133–86. http://dx.doi.org/10.1142/s0219887805000995.

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Let A be an algebra over an operad in a cocomplete closed symmetric monoidal category. We study the category of A-modules. We define certain symmetric product functors of such modules generalizing the tensor product of modules over commutative algebras, which we use to define the notion of a jet module. This in turn generalizes the notion of a jet module over a module over a classical commutative algebra. We are able to define Atiyah classes (i.e., obstructions to the existence of connections) in this generalized context. We use certain model structures on the category of A-modules to study the properties of these Atiyah classes. The purpose of the paper is not to present any really deep theorem. It is more about the right concepts when dealing with modules over an algebra that is defined over an arbitrary operad, i.e., the aim is to show how to generalize various classical constructions, including modules of jets, the Atiyah class and the curvature, to the operadic context. For convenience of the reader and for the purpose of defining the notations, the basic definitions of the theory of operads and model categories are included.

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15

Chen, Qiufan, and Yan-an Cai. "Modules over algebras related to the Virasoro algebra." International Journal of Mathematics 26, no. 09 (August 2015): 1550070. http://dx.doi.org/10.1142/s0129167x15500706.

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In this paper, we consider a class of non-weight modules for some algebras related to the Virasoro algebra: The algebra Vir (a, b), the twisted deformative Schrödinger–Virasoro Lie algebras and the Schrödinger algebra. We study the modules whose restriction to the Cartan subalgebra (modulo center) are free of rank 1 for these algebras. Moreover, the simplicities of these modules are determined.

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16

ONDRUS, MATTHEW, and EMILIE WIESNER. "WHITTAKER MODULES FOR THE VIRASORO ALGEBRA." Journal of Algebra and Its Applications 08, no. 03 (June 2009): 363–77. http://dx.doi.org/10.1142/s0219498809003370.

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Whittaker modules have been well studied in the setting of complex semisimple Lie algebras. Their definition can easily be generalized to certain other Lie algebras with triangular decomposition, including the Virasoro algebra. We define Whittaker modules for the Virasoro algebra and obtain analogues to several results from the classical setting, including a classification of simple Whittaker modules by central characters and composition series for general Whittaker modules.

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17

Mazorchuk, Volodymyr, and Kaiming Zhao. "Characterization of Simple Highest Weight Modules." Canadian Mathematical Bulletin 56, no. 3 (September 1, 2013): 606–14. http://dx.doi.org/10.4153/cmb-2011-199-5.

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Abstract.We prove that for simple complex finite dimensional Lie algebras, affine Kac–Moody Lie algebras, the Virasoro algebra, and the Heisenberg–Virasoro algebra, simple highest weight modules are characterized by the property that all positive root elements act on these modules locally nilpotently. We also show that this is not the case for higher rank Virasoro algebras and for Heisenberg algebras.

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18

Zhao, Xiangui, and Yang Zhang. "Gelfand-Kirillov Dimensions of Modules over Differential Difference Algebras." Algebra Colloquium 23, no. 04 (September 26, 2016): 701–20. http://dx.doi.org/10.1142/s1005386716000596.

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Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension of a finitely generated module over a differential difference algebra through a computational method: Gröbner-Shirshov basis method. We develop the Gröbner-Shirshov basis theory of differential difference algebras, and of finitely generated modules over differential difference algebras, respectively. Then, via Gröbner-Shirshov bases, we give algorithms for computing the Gelfand-Kirillov dimensions of cyclic modules and finitely generated modules over differential difference algebras.

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19

Sun, Jiancai. "Twisted tensor products of φ-coordinated modules for nonlocal vertex algebras." Forum Mathematicum 32, no. 2 (March 1, 2020): 433–46. http://dx.doi.org/10.1515/forum-2019-0165.

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AbstractWe study twisted tensor products of ϕ-coordinated modules of nonlocal vertex algebras. Furthermore, we introduce the notion of ϕ-coordinated module twistor for a ϕ-coordinated module of a nonlocal vertex algebra.

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20

Eklof, Paul C., and Hans-Christian Mez. "Modules of existentially closed algebras." Journal of Symbolic Logic 52, no. 1 (March 1987): 54–63. http://dx.doi.org/10.2307/2273861.

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AbstractThe underlying modules of existentially closed ⊿-algebras are studied. Among other things, it is proved that they are all elementarily equivalent, and that all of them are existentially closed as modules if and only if ⊿ is regular. It is also proved that every saturated module in the appropriate elementary equivalence class underlies an ex. ⊿-algebra. Applications to some problems in module theory are given. A number of open questions are mentioned.

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21

Kleshchev, Alexander S., and David J. Steinberg. "Homomorphisms between standard modules over finite-type KLR algebras." Compositio Mathematica 153, no. 3 (March 2017): 621–46. http://dx.doi.org/10.1112/s0010437x16008204.

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Khovanov–Lauda–Rouquier (KLR) algebras of finite Lie type come with families of standard modules, which under the Khovanov–Lauda–Rouquier categorification correspond to PBW bases of the positive part of the corresponding quantized enveloping algebra. We show that there are no non-zero homomorphisms between distinct standard modules and that all non-zero endomorphisms of a standard module are injective. We present applications to the extensions between standard modules and modular representation theory of KLR algebras.

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22

ÁGOSTON, ISTVÁN, and ERZSÉBET LUKÁCS. "STRATIFYING PAIRS OF SUBCATEGORIES FOR CPS-STRATIFIED ALGEBRAS." Journal of Algebra and Its Applications 12, no. 04 (March 10, 2013): 1250201. http://dx.doi.org/10.1142/s0219498812502015.

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Two special types of module subcategories are defined over stratified algebras of Cline, Parshall and Scott. We show that for every stratified algebra there exists a (not necessarily unique) cotorsion pair of subcategories which describe to a large extent the stratification structure of the algebra. These subcategories generalize the notion of modules with standard and costandard filtration for standardly stratified and quasi-hereditary algebras.

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23

Bavula, V. V., and T. Lu. "Classification of Simple Weight Modules over the Schrödinger Algebra." Canadian Mathematical Bulletin 61, no. 1 (March 1, 2018): 16–39. http://dx.doi.org/10.4153/cmb-2017-017-7.

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AbstractA classification of simple weight modules over the Schrödinger algebra is given. The Krull and the global dimensions are found for the centralizer (H) (and some of its prime factor algebras) of the Cartan element H in the universal enveloping algebra of the Schrödinger (Lie) algebra. The simple (H)-modules are classified. The Krull and the global dimensions are found for some (prime) factor algebras of the algebra (over the centre). It is proved that some (prime) factor algebras of and (H) are tensor homological/Krull minimal.

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24

Marczinzik, René. "Simple reflexive modules over Artin algebras." Journal of Algebra and Its Applications 18, no. 10 (August 6, 2019): 1950193. http://dx.doi.org/10.1142/s0219498819501937.

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Let [Formula: see text] be an Artin algebra. It is well known that [Formula: see text] is selfinjective if and only if every finitely generated [Formula: see text]-module is reflexive. In this paper, we pose and motivate the question whether an algebra [Formula: see text] is selfinjective if and only if every simple module is reflexive. We give a positive answer to this question for large classes of algebras which include for example all Gorenstein algebras and all QF-3 algebras.

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Chen, Hongjia, and Xiangqian Guo. "Non-weight modules over the Heisenberg–Virasoro algebra and the W algebra W(2,2)." Journal of Algebra and Its Applications 16, no. 05 (April 12, 2017): 1750097. http://dx.doi.org/10.1142/s0219498817500979.

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In this paper, we construct and study some non-weight modules for the Heisenberg–Virasoro algebra and the [Formula: see text] algebra [Formula: see text]. We determine the modules, whose restriction to the universal enveloping algebra of the degree-[Formula: see text] part (modulo center) are free of rank [Formula: see text] for these two algebras. In the most interesting case, this degree-[Formula: see text] part is not the Cartan subalgebra. We also determine the simplicity of these modules, which provide new simple modules for the [Formula: see text] algebra [Formula: see text].

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26

SUN, JIANCAI, and HENGYUN YANG. "TWISTED TENSOR PRODUCT MODULES OVER MÖBIUS TWISTED TENSOR PRODUCT NONLOCAL VERTEX ALGEBRAS." International Journal of Mathematics 24, no. 05 (May 2013): 1350033. http://dx.doi.org/10.1142/s0129167x1350033x.

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This is the third part in a series of papers developing a twisted tensor product theory for nonlocal vertex algebras and its modules. In this paper we introduce and study twisted tensor product modules over Möbius twisted tensor product nonlocal vertex algebras. Among the main results, we find the isomorphic relation between the opposite Möbius twisted tensor product nonlocal vertex algebra and twisted tensor product of opposite Möbius nonlocal vertex algebras. And we also establish the isomorphism between two twisted tensor product contragredient modules. Furthermore, we study iterated twisted tensor product modules over iterated twisted tensor product nonlocal vertex algebras and find conditions for constructing an iterated twisted tensor product module of three factors.

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27

WALKER, G., and R. M. W. WOOD. "Flag modules and the hit problem for the Steenrod algebra." Mathematical Proceedings of the Cambridge Philosophical Society 147, no. 1 (July 2009): 143–71. http://dx.doi.org/10.1017/s0305004109002382.

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AbstractThe ‘hit problem’ of F. P. Peterson in algebraic topology asks for a minimal generating set for the polynomial algebraP(n) =2[x1,. . .,xn] as a module over the Steenrod algebra2. An equivalent problem is to find an2-basis for the subringK(n) of elementsfin the dual Hopf algebraD(n), a divided power algebra, such thatSqk(f)=0 for allk> 0. The Steenrod kernelK(n) is a2GL(n,2)-module dual to the quotientQ(n) ofP(n) by the hit elements+2P(n). A submoduleS(n) ofK(n) is obtained as the image of a family of maps from the permutation moduleFl(n) ofGL(n,2) on complete flags in ann-dimensional vector spaceVover2. We use the Schubert cell decomposition of the flags to calculateS(n) in degrees$d =\sum_{i=1}^n (2^{\lambda_i}-1)$, where λ1> λ2> ⋅⋅⋅ > λn≥ 0. When λn= 0, we define a2GL(n,2)-module map δ:Qd(n) →Q2d+n−1(n) analogous to the well-known isomorphismQd(n) →Q2d+n(n) of M. Kameko. When λn−1≥ 2, we show that δ is surjective and δ*:S2d+n−1(n)→Sd(n) is an isomorphism.

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Liu, Dong, Yufeng Pei, and Limeng Xia. "Whittaker modules for the super-Virasoro algebras." Journal of Algebra and Its Applications 18, no. 11 (August 19, 2019): 1950211. http://dx.doi.org/10.1142/s0219498819502116.

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In this paper, we define and study Whittaker modules for the super-Viraoro algebras, including the Neveu-Schwarz algebra and the Ramond algebra. We classify the simple Whittaker modules and obtain necessary and sufficient conditions for irreducibility of these modules.

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29

Mazorchuk, Volodymyr, and Kaiming Zhao. "Supports of weight modules over Witt algebras." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 141, no. 1 (February 2011): 155–70. http://dx.doi.org/10.1017/s0308210509000912.

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As the first step towards a classification of simple weight modules with finite dimensional weight spaces over Witt algebras Wn, we explicitly describe the supports of such modules. We also obtain some descriptions of the support of an arbitrary simple weight module over a ℤn-graded Lie algebra $\mathfrak{g}$ having a root space decomposition $\smash{\bigoplus_{\alpha\in\mathbb{Z}^n}\mathfrak{g}_\alpha}$ with respect to the abelian subalgebra $\mathfrak{g}_0$, with the property $\smash{[\mathfrak{g}_\alpha,\mathfrak{g}_\beta] = \mathfrak{g}_{\alpha+\beta}}$ for all α, β ∈ ℤn, α ≠ β (this class contains the algebra Wn).

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Bavula, V. V., and T. H. Lenagan. "A Bernstein-Gabber-Joseph theorem for affine algebras." Proceedings of the Edinburgh Mathematical Society 42, no. 2 (June 1999): 311–32. http://dx.doi.org/10.1017/s0013091500020277.

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Bernstein's famous result, that any non-zero module M over the n-th Weyl algebra An satisfies GKdim(M)≥GKdim(An)/2, does not carry over to arbitrary simple affine algebras, as is shown by an example of McConnell. Bavula introduced the notion of filter dimension of simple algebra to explain this failure. Here, we introduce the faithful dimension of a module, a variant of the filter dimension, to investigate this phenomenon further and to study a revised definition of holonomic modules. We compute the faithful dimension for certain modules over a variant of the McConnell example to illustrate the utility of this new dimension.

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31

Böhm, Gabriella. "Comodules over weak multiplier bialgebras." International Journal of Mathematics 25, no. 05 (May 2014): 1450037. http://dx.doi.org/10.1142/s0129167x14500372.

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This is a sequel paper of [Weak multiplier bialgebras, Trans. Amer. Math. Soc., in press] in which we study the comodules over a regular weak multiplier bialgebra over a field, with a full comultiplication. Replacing the usual notion of coassociative coaction over a (weak) bialgebra, a comodule is defined via a pair of compatible linear maps. Both the total algebra and the base (co)algebra of a regular weak multiplier bialgebra with a full comultiplication are shown to carry comodule structures. Kahng and Van Daele's integrals [The Larson–Sweedler theorem for weak multiplier Hopf algebras, in preparation] are interpreted as comodule maps from the total to the base algebra. Generalizing the counitality of a comodule to the multiplier setting, we consider the particular class of so-called full comodules. They are shown to carry bi(co)module structures over the base (co)algebra and constitute a monoidal category via the (co)module tensor product over the base (co)algebra. If a regular weak multiplier bialgebra with a full comultiplication possesses an antipode, then finite-dimensional full comodules are shown to possess duals in the monoidal category of full comodules. Hopf modules are introduced over regular weak multiplier bialgebras with a full comultiplication. Whenever there is an antipode, the Fundamental Theorem of Hopf Modules is proven. It asserts that the category of Hopf modules is equivalent to the category of firm modules over the base algebra.

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32

Asefa, Dadi. "Gorenstein-Projective Modules over Morita Rings." Algebra Colloquium 28, no. 03 (July 26, 2021): 521–32. http://dx.doi.org/10.1142/s1005386721000407.

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Let [Formula: see text] be a Morita ring which is an Artin algebra. In this paper we investigate the relations between the Gorenstein-projective modules over a Morita ring [Formula: see text] and the algebras [Formula: see text] and [Formula: see text]. We prove that if [Formula: see text] is a Gorenstein algebra and both [Formula: see text] and [Formula: see text] (resp., both [Formula: see text] and [Formula: see text]) have finite projective dimension, then [Formula: see text] (resp., [Formula: see text]) is a Gorenstein algebra. We also discuss when the CM-freeness and the CM-finiteness of a Morita ring [Formula: see text] is inherited by the algebras [Formula: see text] and [Formula: see text].

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33

Ondrus, Matthew, and Emilie Wiesner. "The Restriction of Polynomial Modules for the Virasoro Algebra to sl2(C)." Algebra Colloquium 29, no. 03 (July 26, 2022): 491–508. http://dx.doi.org/10.1142/s1005386722000372.

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The Lie algebra [Formula: see text] may be regarded in a natural way as a subalgebra of the infinite-dimensional Virasoro Lie algebra, so it is natural to consider connections between the representation theory of the two algebras. In this paper, we explore the restriction to [Formula: see text] of certain induced modules for the Virasoro algebra. Specifically, we consider Virasoro modules induced from so-called polynomial subalgebras, and we show that the restriction of these modules results in twisted versions of familiar modules such as Verma modules and Whittaker modules.

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34

HE, JUNHUA, and YOUJUN TAN. "DECOMPOSABLY-GENERATED MODULES OF SIMPLE LIE ALGEBRAS." Journal of Algebra and Its Applications 11, no. 02 (April 2012): 1250023. http://dx.doi.org/10.1142/s0219498811005415.

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It is shown that there are finitely many irreducible finite-dimensional orthogonal modules V (up to isomorphism) over any complex simple Lie algebras such that Spin0(V) is decomposably-generated in the sense of Panyushev [The exterior algebra and "Spin" of an orthogonal 𝔤-module, Trans. Groups6 (2001) 371–396]. The case of simple Lie algebras of type A is discussed.

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35

Pei, Yufeng, and Jinwei Yang. "Strongly graded vertex algebras generated by vertex Lie algebras." Communications in Contemporary Mathematics 21, no. 08 (October 20, 2019): 1850069. http://dx.doi.org/10.1142/s0219199718500694.

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We construct three families of vertex algebras along with their modules from appropriate vertex Lie algebras, using the constructions in [Vertex Lie algebra, vertex Poisson algebras and vertex algebras, in Recent Developments in Infinite-Dimensional Lie Algebras and Conformal Field Theory[Formula: see text] Proceedings of an International Conference at University of Virginia[Formula: see text] May 2000, in Contemporary Mathematics, Vol. 297 (American Mathematical Society, 2002), pp. 69–96] by Dong, Li and Mason. These vertex algebras are strongly graded vertex algebras introduced in [Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, I: Introduction and strongly graded algebras and their generalized modules, in Conformal Field Theories and Tensor Categories[Formula: see text] Proceedings of a Workshop Held at Beijing International Center for Mathematics Research, eds. C. Bai, J. Fuchs, Y.-Z. Huang, L. Kong, I. Runkel and C. Schweigert, Mathematical Lectures from Beijing University, Vol. 2 (Springer, New York, 2014), pp. 169–248] by Huang, Lepowsky and Zhang in their logarithmic tensor category theory and can also be realized as vertex algebras associated to certain well-known infinite dimensional Lie algebras. We classify irreducible [Formula: see text]-gradable weak modules for these vertex algebras by determining their Zhu’s algebras. We find examples of strongly graded generalized modules for these vertex algebras that satisfy the [Formula: see text]-cofiniteness condition introduced in [Differential equations and logarithmic intertwining operators for strongly graded vertex algebra, Comm. Contemp. Math. 19(2) (2017) 1650009] by the second author. In particular, by a result of the second author [Differential equations and logarithmic intertwining operators for strongly graded vertex algebra, Comm. Contemp. Math. 19(2) (2017) 1650009, 26 pp.], the convergence and extension property for products and iterates of logarithmic intertwining operators in [Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, VII: Convergence and extension properties and applications to expansion for intertwining maps, preprint (2011); arXiv:1110.1929 ] among such strongly graded generalized modules is verified.

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36

Chen, Haibo, Jianzhi Han, Yucai Su, and Ying Xu. "Loop Schrödinger–Virasoro Lie conformal algebra." International Journal of Mathematics 27, no. 06 (June 2016): 1650057. http://dx.doi.org/10.1142/s0129167x16500579.

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In this paper, we introduce two kinds of Lie conformal algebras, associated with the loop Schrödinger–Virasoro Lie algebra and the extended loop Schrödinger–Virasoro Lie algebra, respectively. The conformal derivations, the second cohomology groups of these two conformal algebras are completely determined. And nontrivial free conformal modules of rank one and [Formula: see text]-graded free intermediate series modules over these two conformal algebras are also classified in the present paper.

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37

Das, Paramita, Shamindra Kumar Ghosh, and Ved Prakash Gupta. "Affine Modules and the Drinfeld Center." MATHEMATICA SCANDINAVICA 118, no. 1 (March 7, 2016): 119. http://dx.doi.org/10.7146/math.scand.a-23301.

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Given a finite index subfactor, we show that the affine morphisms at zero level in the affine category over the planar algebra associated to the subfactor is isomorphic to the fusion algebra of the subfactor as a $*$-algebra. This identification paves the way to analyze the structure of affine $P$-modules with weight zero for any subfactor planar algebra $P$ (possibly having infinite depth). Further, for irreducible depth two subfactor planar algebras, we establish an additive equivalence between the category of affine $P$-modules and the center of the category of $N$-$N$-bimodules generated by $L^2(M)$; this partially verifies a conjecture of Jones and Walker.

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38

CRISP, TYRONE. "FREDHOLM MODULES OVER GRAPH -ALGEBRAS." Bulletin of the Australian Mathematical Society 92, no. 2 (June 19, 2015): 302–15. http://dx.doi.org/10.1017/s0004972715000556.

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We present two applications of explicit formulas, due to Cuntz and Krieger, for computations in $K$-homology of graph $C^{\ast }$-algebras. We prove that every $K$-homology class for such an algebra is represented by a Fredholm module having finite-rank commutators, and we exhibit generating Fredholm modules for the $K$-homology of quantum lens spaces.

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Sun, Jiancai. "General twisting of ϕ1-coordinated modules of nonlocal vertex algebras." International Journal of Mathematics 29, no. 14 (December 2018): 1850098. http://dx.doi.org/10.1142/s0129167x18500982.

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We introduce a notion of [Formula: see text]-coordinated module twistor for a [Formula: see text]-coordinated module of a nonlocal vertex algebra. Furthermore, we study twisted tensor products and L-R-twisted tensor products of [Formula: see text]-coordinated modules of nonlocal vertex algebras and we unify these two constructions by [Formula: see text]-coordinated module twistors.

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40

Adachi, Takahide, Osamu Iyama, and Idun Reiten. "-tilting theory." Compositio Mathematica 150, no. 3 (December 3, 2013): 415–52. http://dx.doi.org/10.1112/s0010437x13007422.

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AbstractThe aim of this paper is to introduce $\tau $-tilting theory, which ‘completes’ (classical) tilting theory from the viewpoint of mutation. It is well known in tilting theory that an almost complete tilting module for any finite-dimensional algebra over a field $k$ is a direct summand of exactly one or two tilting modules. An important property in cluster-tilting theory is that an almost complete cluster-tilting object in a 2-CY triangulated category is a direct summand of exactly two cluster-tilting objects. Reformulated for path algebras $kQ$, this says that an almost complete support tilting module has exactly two complements. We generalize (support) tilting modules to what we call (support) $\tau $-tilting modules, and show that an almost complete support $\tau $-tilting module has exactly two complements for any finite-dimensional algebra. For a finite-dimensional $k$-algebra $\Lambda $, we establish bijections between functorially finite torsion classes in $ \mathsf{mod} \hspace{0.167em} \Lambda $, support $\tau $-tilting modules and two-term silting complexes in ${ \mathsf{K} }^{\mathrm{b} } ( \mathsf{proj} \hspace{0.167em} \Lambda )$. Moreover, these objects correspond bijectively to cluster-tilting objects in $ \mathcal{C} $ if $\Lambda $ is a 2-CY tilted algebra associated with a 2-CY triangulated category $ \mathcal{C} $. As an application, we show that the property of having two complements holds also for two-term silting complexes in ${ \mathsf{K} }^{\mathrm{b} } ( \mathsf{proj} \hspace{0.167em} \Lambda )$.

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ADAMOVIĆ, DRAŽEN, and OZREN PERŠE. "FUSION RULES AND COMPLETE REDUCIBILITY OF CERTAIN MODULES FOR AFFINE LIE ALGEBRAS." Journal of Algebra and Its Applications 13, no. 01 (August 20, 2013): 1350062. http://dx.doi.org/10.1142/s021949881350062x.

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We develop a new method for obtaining branching rules for affine Kac–Moody Lie algebras at negative integer levels. This method uses fusion rules for vertex operator algebras of affine type. We prove that an infinite family of ordinary modules for affine vertex algebra of type A investigated in our previous paper J. Algebra319 (2008) 2434–2450, is closed under fusion. Then, we apply these fusion rules on explicit bosonic realization of level -1 modules for the affine Lie algebra of type [Formula: see text], obtain a new proof of complete reducibility for these representations, and the corresponding decomposition for ℓ ≥ 3. We also obtain the complete reducibility of the associated level -1 modules for affine Lie algebra of type [Formula: see text]. Next, we notice that the category of [Formula: see text] modules at level -2ℓ + 3 has the isomorphic fusion algebra. This enables us to decompose certain [Formula: see text] and [Formula: see text]-modules at negative levels.

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CARVALHO, PAULA A. A. B., and IAN M. MUSSON. "MONOLITHIC MODULES OVER NOETHERIAN RINGS." Glasgow Mathematical Journal 53, no. 3 (August 1, 2011): 683–92. http://dx.doi.org/10.1017/s0017089511000267.

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AbstractWe study finiteness conditions on essential extensions of simple modules over the quantum plane, the quantised Weyl algebra and Noetherian down-up algebras. The results achieved improve the ones obtained by Carvalho et al. (Carvalho et al., Injective modules over down-up algebras, Glasgow Math. J. 52A (2010), 53–59) for down-up algebras.

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43

Popescu, Dorin. "Polynomial rings and their projective modules." Nagoya Mathematical Journal 113 (March 1989): 121–28. http://dx.doi.org/10.1017/s0027763000001288.

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Let R be a regular noetherian ring. A central question concerning projective modules over polynomial R-algebras is the following.(1.1) BASS-QUILLEN CONJECTURE ([2] Problem IX, [10]). Every finitely generated projective module P over a polynomial R-algebra R[T], T = (T1,…, Tn) is extended from R, i.e.P≊R[T]⊗R P/(T)P.

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Ariki, Susumu, Andrew Mathas, and Hebing Rui. "Cyclotomic Nazarov-Wenzl Algebras." Nagoya Mathematical Journal 182 (June 2006): 47–134. http://dx.doi.org/10.1017/s0027763000026842.

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AbstractNazarov [Naz96] introduced an infinite dimensional algebra, which he called the affine Wenzl algebra, in his study of the Brauer algebras. In this paper we study certain “cyclotomic quotients” of these algebras. We construct the irreducible representations of these algebras in the generic case and use this to show that these algebras are free of rank rn(2n−1)!! (when Ω is u-admissible). We next show that these algebras are cellular and give a labelling for the simple modules of the cyclotomic Nazarov-Wenzl algebras over an arbitrary field. In particular, this gives a construction of all of the finite dimensional irreducible modules of the affine Wenzl algebra.

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Matsumoto, Kengo. "C∗-Algebras Associated with HilbertC∗-Quad Modules of Finite Type." International Journal of Mathematics and Mathematical Sciences 2014 (2014): 1–21. http://dx.doi.org/10.1155/2014/952068.

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A HilbertC∗-quad module of finite type has a multistructure of HilbertC∗-bimodules with two finite bases. We will construct aC∗-algebra from a HilbertC∗-quad module of finite type and prove its universality subject to certain relations among generators. Some examples of theC∗-algebras from HilbertC∗-quad modules of finite type will be presented.

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46

Guo, Shuangjian, Xiaohui Zhang, and Shengxiang Wang. "Total integrals of Doi Hom-Hopf modules." Journal of Algebra and Its Applications 15, no. 04 (February 19, 2016): 1650069. http://dx.doi.org/10.1142/s0219498816500699.

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Let [Formula: see text] be a monoidal Hom-Hopf algebra, [Formula: see text] a right [Formula: see text]-Hom-comodule algebra and [Formula: see text] a right [Formula: see text]-Hom-module coalgebra. We first investigate the criterion for the existence of a total integral of [Formula: see text] in the setting of monoidal Hom-Hopf algebras. Also, we prove that there exists a total integral [Formula: see text] if and only if any representation of the pair [Formula: see text] is injective in a functorial way, which generalizes Menini and Militaru’s result. Finally, we extend to the category of [Formula: see text]-Doi Hom-Hopf modules a result of Doi on projectivity of every relative [Formula: see text]-Hopf module as an [Formula: see text]-module.

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47

Blecher, David P. "On Morita's fundamental theorem for $C^*$-algebras." MATHEMATICA SCANDINAVICA 88, no. 1 (March 1, 2001): 137. http://dx.doi.org/10.7146/math.scand.a-14319.

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We give a solution, via operator spaces, of an old problem in the Morita equivalence of $C^*$-algebras. Namely, we show that $C^*$-algebras are strongly Morita equivalent in the sense of Rieffel if and only if their categories of left operator modules are isomorphic via completely contractive functors. Moreover, any such functor is completely isometrically isomorphic to the Haagerup tensor product (= interior tensor product) with a strong Morita equivalence bimodule. An operator module over a $C^*$-algebra $\mathcal A$ is a closed subspace of some B(H) which is left invariant under multiplication by $\pi(\mathcal\ A)$, where $\pi$ is a*-representation of $\mathcal A$ on $H$. The category $_{\mathcal{AHMOD}}$ of *-representations of $\mathcal A$ on Hilbert space is a full subcategory of the category $_{\mathcal{AOMOD}}$ of operator modules. Our main result remains true with respect to subcategories of $OMOD$ which contain $HMOD$ and the $C^*$-algebra itself. It does not seem possible to remove the operator space framework; in the very simplest cases there may exist no bounded equivalence functors on categories with bounded module maps as morphisms (as opposed to completely bounded ones). Our proof involves operator space techniques, together with a $C^*$-algebra argument using compactness of the quasistate space of a $C^*$-algebra, and lowersemicontinuity in the enveloping von Neumann algebra.

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Hassanzadeh, Mohammad. "On Cyclic Cohomology of ×-Hopf algebras." Journal of K-Theory 13, no. 1 (January 2, 2014): 147–70. http://dx.doi.org/10.1017/is013011021jkt246.

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AbstractIn this paper we study the cyclic cohomology of certain ×-Hopf algebras: universal enveloping algebras, quantum algebraic tori, the Connes-Moscovici ×-Hopf algebroids and the Kadison bialgebroids. Introducing their stable anti Yetter-Drinfeld modules and cocyclic modules, we compute their cyclic cohomology. Furthermore, we provide a pairing for the cyclic cohomology of ×-Hopf algebras which generalizes the Connes-Moscovici characteristic map to ×-Hopf algebras. This enables us to transfer the ×-Hopf algebra cyclic cocycles to algebra cyclic cocycles.

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Guljaš, Boris. "Hilbert \(C^{*}\)-modules in which all relatively strictly closed submodules are complemented." Glasnik Matematicki 56, no. 2 (December 23, 2021): 343–74. http://dx.doi.org/10.3336/gm.56.2.08.

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We give the characterization and description of all full Hilbert modules and associated algebras having the property that each relatively strictly closed submodule is orthogonally complemented. A strict topology is determined by an essential closed two-sided ideal in the associated algebra and a related ideal submodule. It is shown that these are some modules over hereditary algebras containing the essential ideal isomorphic to the algebra of (not necessarily all) compact operators on a Hilbert space. The characterization and description of that broader class of Hilbert modules and their associated algebras is given. As auxiliary results we give properties of strict and relatively strict submodule closures, the characterization of orthogonal closedness and orthogonal complementing property for single submodules, relation of relative strict topology and projections, properties of outer direct sums with respect to the ideals in \(\ell_\infty\) and isomorphisms of Hilbert modules, and we prove some properties of hereditary algebras and associated hereditary modules with respect to the multiplier algebras, multiplier Hilbert modules, corona algebras and corona modules.

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Soltanmoradi, Shabani, Davood Ebrahimi Bagha, and Pourbahri Rahpeyma. "Weak module amenability for the second dual of a Banach algebra." Acta et Commentationes Universitatis Tartuensis de Mathematica 25, no. 2 (November 17, 2021): 297–306. http://dx.doi.org/10.12697/acutm.2021.25.19.

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In this paper we study the weak module amenability of Banach algebras which are Banach modules over another Banach algebra with compatible actions. We show that for every module derivation D : A ↦ ( A/J_A )∗ if D∗∗(A∗∗) ⊆ WAP (A/J_A ), then weak module amenability of A∗∗ implies that of A. Also we prove that under certain conditions for the module derivation D, if A∗∗ is weak module amenable then A is also weak module amenable.

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