A Note on the Dimension Theory of ℤ n -Graded Rings

In this note we want to generalize some of the results in [1 Brewer , J. , Montgomery , P. , Rutter E. , Heinzer , W. ( 1973 ). Krull dimension of polynomial rings in “Conference on Commutative Algebra, Lawrence 1972.” . Springer Lecture Notes in Mathematics 311 : 26 – 45 .[Crossref] , [Google Scholar]] from polynomial rings in several indeterminates to arbitrary ? ⁿ -graded commutative rings. We will prove an analogue of Jaffard's Special Chain Theorem and a similar result for the height of a prime ideal 𝔭 over its graded core 𝔭*.     

The infinitesimal and the little q-Schur superalgebras

In this paper, based on the results in [8 Du, J., Gu, H.-X. (2014). A realization of the quantum supergroup U(𝔤𝔩_m|n). J. Algebra 404:60–99.[Web of Science ®] , [Google Scholar]] we give a monomial basis for q-Schur superalgebra and then a presentation for it. The presentation is different from that in [12 El Turkey, H., Kujawa, J. (2012). Presenting Schur superalgebras. Pacific J. Math., 262(2):285–316.[Crossref], [Web of Science ®] , [Google Scholar]]. Imitating [3 Cox, A. G. (1997). On some applications of infinitesimal methods to quantum groups and related algebras. Ph.D. Thesis. University of London. [Google Scholar]] and [7 Du, J., Fu, Q., Wang, J.-P. (2005). Infinitesimal quantum 𝔤𝔩_n and little q-Schur algebras. J. Algebra 287:199–233.[Crossref], [Web of Science ®] , [Google Scholar]], we define the infinitesimal and the little q-Schur superalgebras. We give a “weight idempotent presentation” for infinitesimal q-Schur superalgebras. The BLM bases and monomial bases of little q-Schur superalgebras are obtained, and dimension formulas of infinitesimal and little q-Schur superalgebras are deduced.     

On the Cohomology of the Lie Superalgebra of Contact Vector Fields on S 1|m

The main result of this article is the explicit calculation of the first cohomology space H ¹(𝒦(3), 𝒮Ψ𝒟𝒪(S ^1|3)) of the Lie superalgebra 𝒦(3) of contact vector fields on the supercircle S ^1|3 with coefficients in the module of superpseudodifferential operators 𝒮Ψ𝒟𝒪(S ^1|3). For the supercicles of dimensional 1 | 0, 1 | 1, and 1 | 2, the first cohomology space is computed, respectively, in the following articles: [2 Agrebaoui , B. , Ben Fraj , N. ( 2004 ). On the cohomology of the Lie superalgebra of contact vector fields on S ^1|1 . Belletin de la Société Royale des Sciences de Liège 72 ( 6 ): 365 – 375 . [Google Scholar], 3 Agrebaoui , B. , Ben Fraj , N. , Omri , S. ( 2006 ). On the cohomology of the Lie superalgebra of contact vector fields on S ^1|2 . J. Nonlinear Math. Phys. 13 ( 4 ): 523 – 534 .[Taylor &; Francis Online] , [Google Scholar], 14 Ovsienko , V. , Roger , C. ( 1999 ). Deforming the Lie algebra of vector fields on S ¹ inside the Lie algebra of pseudodifferential operators on S ¹ . AMS Transl. Ser. 2, (Adv. Math. Sci.) 194 : 211 – 227 . [Google Scholar]]. The case m ≥ 4 is still out of reach, but we give a lower bound for the dimension of the cohomology space and exhibit three nontrivial, 1-cocycles.     

The Larson–Sweedler theorem for weak multiplier Hopf algebras

The Larson–Sweedler theorem says that a finite-dimensional bialgebra with a faithful integral is a Hopf algebra [15 Larson, R. G., Sweedler, M. E. (1969). An associative orthogonal bilinear form for Hopf algebras. Amer. J. Math. 91:75–93.[Crossref], [Web of Science ®] , [Google Scholar]]. The result has been generalized to finite-dimensional weak Hopf algebras by Vecsernyés [44 Vecsernyés, P. (2003). Larson–Sweedler theorem and the role of grouplike elements in weak Hopf algebras. J. Algebra 270:471–520. See also arXiv: 0111045v3 [math.QA] for an extended version.[Crossref], [Web of Science ®] , [Google Scholar]]. In this paper, we show that the result is still true for weak multiplier Hopf algebras. The notion of a weak multiplier bialgebra was introduced by Böhm et al. in [4 Böhm, G., Gómez-Torecillas, J., López-Centella, E. (2015). Weak multiplier bialgebras. Weak multiplier bialgebras. 367(12):8681–8872. See also arXiv: 1306.1466 [math.QA]. [Google Scholar]]. In this note it is shown that a weak multiplier bialgebra with a regular and full coproduct is a regular weak multiplier Hopf algebra if there is a faithful set of integrals. Weak multiplier Hopf algebras are introduced and studied in [40 Van Daele, A., Wang, S. (2015). Weak multiplier Hopf algebras I. The main theory. J. Ange. Math. (Crelles J.) 705:155–209, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/crelle-2013-0053, July 2013. See also arXiv:1210.4395v1 [math.RA].[Web of Science ®] , [Google Scholar]]. Integrals on (regular) weak multiplier Hopf algebras are treated in [43 Van Daele, A., Wang, S. (2016). Weak multiplier Hopf algebras III. Integrals and duality. Preprint University of Leuven (Belgium) and Southeast University of Nanjing (China), See arXiv: 1701.04951.v3 [math.RA]. [Google Scholar]]. This result is important for the development of the theory of locally compact quantum groupoids in the operator algebra setting, see [13 Kahng, B.-J., Van Daele, A. A class of C*-algebraic locally compact quantum groupoids I. Preprint Canisius College Buffalo (USA) and University of Leuven (Belgium). [Google Scholar]] and [14 Kahng, B.-J., Van Daele, A. A class of C*-algebraic locally compact quantum groupoids II. Preprint Canisius College Buffalo (USA) and University of Leuven (Belgium). [Google Scholar]]. Our treatment of this material is motivated by the prospect of such a theory.     

Extremal loop weight modules and tensor products for quantum toroidal algebras

We define integrable representations of quantum toroidal algebras of type A by tensor product, using the Drinfeld “coproduct.” This allows us to recover the vector representations recently introduced by Feigin–Jimbo–Miwa–Mukhin [7 Feigin, B., Jimbo, M., Miwa, T., Mukhin, E. (2013). Representations of quantum toroidal 𝔤𝔩_n. J. Algebra 380:78–108.[Crossref], [Web of Science ®] , [Google Scholar]] and constructed by the author [21 Macdonald, I. G. (1995). Symmetric Functions and Hall Polynomials. 2nd ed. Oxford: Oxford Math. Monographs, 1979. [Google Scholar]] as a subfamily of extremal loop weight modules. In addition we get new extremal loop weight modules as subquotients of tensor powers of vector representations. As an application we obtain finite-dimensional representations of quantum toroidal algebras by specializing the quantum parameter at roots of unity.     

Conjugacy Classes of P-Cycles of Type D in Alternating Groups

We classify the conjugacy classes of p-cycles of type D in alternating groups. This finishes the open cases in [3 Andruskiewitsch , N. , Fantino , F. , Graña , M. , Vendramin , L. ( 2011 ). Finite-dimensional pointed Hopf algebras with alternating groups are trivial . Ann. Mat. Pura Appl 190 : 225 – 245 .[Web of Science ®] , [Google Scholar]]. Also we determine all the subracks of those conjugacy classes which are not of type D.     

Poisson Hopf algebra related to a twisted quantum group

A Poisson algebra ?[G] considered as a Poisson version of the twisted quantized coordinate ring ?_q,p[G], constructed by Hodges et al. [11 Hodges, T. J., Levasseur, T., Toro, M. (1997). Algebraic structure of multi-parameter quantum groups. Adv. Math. 126:52–92.[Crossref], [Web of Science ®] , [Google Scholar]], is obtained and its Poisson structure is investigated. This establishes that all Poisson prime and primitive ideals of ?[G] are characterized. Further it is shown that ?[G] satisfies the Poisson Dixmier-Moeglin equivalence and that Zariski topology on the space of Poisson primitive ideals of ?[G] agrees with the quotient topology induced by the natural surjection from the maximal ideal space of ?[G] onto the Poisson primitive ideal space.     

The Generalized C.M.Z.-Theorem and a Drinfel'd Double Construction for WT-Coalgebras and Graded Quantum Groupoids

Let π be a group. In this article, we introduce the notions of a weak Doi–Hopf π-module and a weak π-twisted smash product. We show that the Yetter–Drinfel'd π-modules over a weak crossed Hopf π-coalgebra (WT-coalgebra) are special cases as these new weak Doi–Hopf π-modules, generalizing the main result by Caenepeel et al. (1997 Caenepeel , S. , Militaru , G. , Zhu , S. ( 1997 ). Crossed modules and Doi–Hopf modules . Israel J. Math. 100 : 221 – 248 .[Crossref] , [Google Scholar]) and that the Drinfel'd double for WT-coalgebras (Van Daele and Wang, 2008 Van Daele , A. , Wang , S. H. ( 2008 ). New braided crossed categories and Drinfel'd quantum double for weak Hopf group-coalgebras . Comm. Algebra 36 : 2341 – 2386 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) appears as, a type of such a weak π -twisted smash product, respectively. Finally, starting from a weak Hopf algebra endowed with an action of a group π by weak Hopf automorphisms, we construct a quasitriangular weak Hopf π -coalgebra by a twisted double method, generalizing the main result in Virelizier (2005 Virelizier , A. ( 2005 ). Graded quantum groups and quasitriangular Hopf group-coalgebras . Comm. Algebra 33 ( 9 ): 3029 – 3050 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). This method allows us to obtain nontrivial examples of quasitriangular weak Hopf π-coalgebras.     

Weyl Modules Associated to Kac–Moody Lie Algebras

Local Weyl modules were originally defined for affine Lie algebras by Chari and Pressley in [5 Chari, V., Pressley, A. (2001). Weyl modules for classical and quantum affine algebras. Represent. Theory 5:191–223 (electronic).[Crossref] , [Google Scholar]]. In this paper we extend the notion of local Weyl modules for a Lie algebra 𝔤 ?A, where 𝔤 is any Kac–Moody algebra and A is any finitely generated commutative associative algebra with unit over ?, and prove a tensor product decomposition theorem which generalizes result in [2 Chari, V., Fourier, G., Khandai, T. (2010). A categorical approach to Weyl modules. Transform. Groups 15(3):517–549.[Crossref], [Web of Science ®] , [Google Scholar], 5 Chari, V., Pressley, A. (2001). Weyl modules for classical and quantum affine algebras. Represent. Theory 5:191–223 (electronic).[Crossref] , [Google Scholar]].     

Bialgebra Cohomology of the Duals of a Class of Generalized Taft Algebras

We compute explicitly the bialgebra cohomology of the duals of the generalized Taft algebras, which are noncommutative, noncocommutative finite-dimensional Hopf algebras. In order to do this, we use an identification of this cohomology with an Ext algebra (Taillefer, 2004a Taillefer , R. ( 2004a ). Cohomology theories of Hopf bimodules and cup-product . Alg. and Representation Theory 7 : 471 – 490 . [Google Scholar]) and a result describing the Drinfeld double of the dual of a generalized Taft algebra up to Morita equivalence (Erdmann et al., 2006 Erdmann , K. , Green , E. L. , Snashall , N. , Taillefer , R. ( 2006 ). Representation theory of the Drinfeld doubles of a family of Hopf algebras . J. Pure and Applied Algebra 204 ( 2 ): 413 – 454 .[Crossref], [Web of Science ®] , [Google Scholar]).     

On the Construction of Finite Oscillator Dictionary

A finite oscillator dictionary which has important applications in sequences designs and the compressive sensing was introduced by Gurevich, Hadani, and Sochen in [3 Gurevich , S. , Hadani , R. , Sochen , N. (2008). The finite harmonic oscillator and its applications to sequences, communication and radar. IEEE Trans. Inform. Theory 54(9):4239–4253.[Crossref], [Web of Science ®] , [Google Scholar]]. In this paper, closed formulae of the finite split oscillator dictionary 𝔖 ^s are revisited by a simple proof, the structure of nonsplit tori of the group SL(2, 𝔽 _p ) is studied, and an explicit algorithm for computing the finite nonsplit oscillator dictionary 𝔖 ^ns is described.     

Quasi-Deformations of 𝔰𝔩2(𝔽) Using Twisted Derivations

In this article we apply a method devised in Hartwig, Larsson, and Silvestrov (2006 Hartwig , J. T. , Larsson , D. , Silvestrov , S. D. ( 2006 ). Deformations of Lie algebras using σ-derivations . J. Algebra 295 : 314 – 361 .[Crossref], [Web of Science ®] , [Google Scholar]) and Larsson and Silvestrov (2005a Larsson , D. , Silvestrov , S. D. (2005a). Quasi-hom-Lie algebras, Central extensions and 2-cocycle-like identities. J. Algebra 288:321–344.[Crossref], [Web of Science ®] , [Google Scholar]) to the simple 3-dimensional Lie algebra 𝔰𝔩₂(𝔽). One of the main points of this deformation method is that the deformed algebra comes endowed with a canonical twisted Jacobi identity. We show in the present article that when our deformation scheme is applied to 𝔰𝔩₂(𝔽) we can, by choosing parameters suitably, deform 𝔰𝔩₂(𝔽) into the Heisenberg Lie algebra and some other 3-dimensional Lie algebras in addition to more exotic types of algebras, this being in stark contrast to the classical deformation schemes where 𝔰𝔩₂(𝔽) is rigid.     

Larson–Sweedler Theorem and Some Properties of Discrete Type in (G-Cograded) Multiplier Hopf Algebras

We extend the Larson–Sweedler theorem to group-cograded multiplier Hopf algebras introduced in Abd El-hafez et al. (2004 Abd El-hafez , A. T. , Delvaux , L. , Van Daele , A. ( 2004 ). Group-cograded multiplier Hopf (?-)algebra. Math. QA/0404026 . To appear in Algebras and Representation Theory . [CSA]  [Google Scholar]), by showing that a group-cograded multiplier bialgebra with finite-dimensional unital components is a group-cograded multiplier Hopf algebra if and only if it possesses a nondegenerate left cointegral. We also generalize the theory of multiplier Hopf algebras of discrete type in Van Daele and Zhang (1999 Van Daele , A. , Zhang , Y. ( 1999 ). Multiplier Hopf algebras of discrete type . J. Algebra 214 : 400 – 417 . [CSA] [CROSSREF]  [Google Scholar]) to group-cograded multiplier Hopf algebras. Our results are applicable to Hopf group-coalgebras in the sense of Turaev (2000 Turaev , V. G. ( 2000 ). Homotopy field theory in dimension 3 and crossed group-categories . Preprint GT/0005291. [CSA]  [Google Scholar]). Finally, we study regular multiplier Hopf algebras of η -discrete type.     

Universal Central Extensions of the Matrix Leibniz Superalgebras 𝔰𝔩(m,n, 𝒜)

The universal central extensions and their extension kernels of the smatrix Lie superalgebra 𝔰𝔩(m, n, 𝒜), the Steinberg Lie superalgebra 𝔰𝔩(m, n, 𝒜) in category SLeib of Leibniz superalgebras are determined under a weak assumption (compared with Mikhalev and Pinchuk, 2000 Mikhalev , A. V. , Pinchuk , I. A. ( 2000 ). Universal central extension of the matrix Lie superalgebras sl(m, n, A) . Contemp. Math. 264 : 111 – 126 . [Google Scholar]) using the first Hochschild homology and the first cyclic homology group.     

The q-difference Noether problem for complex reflection groups and quantum OGZ algebras

For any complex reflection group G = G(m,p,n), we prove that the G-invariants of the division ring of fractions of the n:th tensor power of the quantum plane is a quantum Weyl field and give explicit parameters for this quantum Weyl field. This shows that the q-difference Noether problem has a positive solution for such groups, generalizing previous work by Futorny and the author [10 Futorny, V., Hartwig, J. T. (2014). Solution to a q-difference Noether problem and the quantum Gelfand–Kirillov conjecture for 𝔤𝔩_N. Math. Z. 276(1–2):1–37. [Google Scholar]]. Moreover, the new result is simultaneously a q-deformation of the classical commutative case and of the Weyl algebra case recently obtained by Eshmatov et al. [8 Eshmatov, F., Futorny, V., Ovsienko, S., Fernando Schwarz, J. (2015). Noncommutative Noether’s Problem for Complex Reflection Groups. Available at: http://arxiv.org/abs/1505.05626 [Google Scholar]].

Second, we introduce a new family of algebras called quantum OGZ algebras. They are natural quantizations of the OGZ algebras introduced by Mazorchuk [18 Mazorchuk, V. (1999). Orthogonal Gelfand-Zetlin algebras, I. Beiträge Algebra Geom. 40(2):399–415. [Google Scholar]] originating in the classical Gelfand–Tsetlin formulas. Special cases of quantum OGZ algebras include the quantized enveloping algebra of 𝔤𝔩_n and quantized Heisenberg algebras. We show that any quantum OGZ algebra can be naturally realized as a Galois ring in the sense of Futorny-Ovsienko [11 Futorny, V., Ovsienko, S. (2010). Galois orders in skew monoid rings. J. Algebra 324:598–630.[Crossref], [Web of Science ®] , [Google Scholar]], with symmetry group being a direct product of complex reflection groups G(m,p,r_k).

Finally, using these results, we prove that the quantum OGZ algebras satisfy the quantum Gelfand–Kirillov conjecture by explicitly computing their division ring of fractions.     

On the Twisting and Drinfel'd Double for Multiplier Hopf Algebras

We define the concept of “semiprime” for preradicals and for submodules, and we prove some properties that relate both of them. Related concepts are defined in article by Bican et al. [2 Bican , L. , Jambor , P. , Kepka , T. , Nemec , P. ( 1980 ). Prime and coprime modules . Fundamenta Mathematicae CVII , 33 – 45 . [Google Scholar]] and by Van den Berg and Wisbauer [9 Van den Berg , J. , Wisbauer , R. ( 2001 ). Duprime and dusemiprime modules . Journal of Pure and Applied Algebra 165 : 337 – 356 .[Crossref], [Web of Science ®] , [Google Scholar]]. For any ring, we compare the least semiprime preradical, the Jacobson radical and the join of all nilpotent preradicals, and we characterize V-rings in terms of these three preradicals. We study the least semiprime preradical above any preradical and we prove some of its properties. Using “Amitsur constructions” we define another related operators and prove some of their properties.     

A Regularized Hybrid Steepest Descent Method for Variational Inclusions

This article is concerned with a generalization of the hybrid steepest descent method from variational inequalities to the multivalued case. This will be reached by replacing the multivalued operator by its Yosida approximate, which is always Lipschitz continuous. It is worth mentioning that the hybrid steepest descent method is an algorithmic solution to variational inequality problems over the fixed point set of certain nonexpansive mappings and has remarkable applicability to the constrained nonlinear inverse problems like image recovery and MIMO communication systems (see, e.g., [9 I. Yamada , M. Yukawa , and M. Yamagishi ( 2011 ). Minimizing the moreau envelope of nonsmooth convex functions over the fixed point set of certain quasi-nonexpansive mappings . In Fixed Point Algorithms for Inverse Problems in Science and Engineering ( H.H. Bauschke , R. Burachik , P.L. Combettes , V. Elser , D.R. Luke , and H. Wolkowicz , eds.), Springer-Verlag , New York , Chapter 17 , pp. 343 – 388 . [Google Scholar], 10 I. Yamada , Ogura , and N. Shirakawa ( 2002 ). A numerically robust hybrid steepest descent method for the convexly constrained generalized inverse problems . In Inverse Problems, Image Analysis and Medical Imaging. Contemporary Mathematics ( Z. Nashed and O. Scherzer , eds.), American Mathematical Society , Providence , RI , Vol. 313 , pp. 269 – 305 . [Google Scholar]]).     

Tensor Products of Modules Over a Certain Class of Quantum Doubles

The quantum doubles of a certain class of rank two pointed Hopf algebras are considered. The socle of the tensor product of two such modules is computed, and formulas similar to the ones in [4 Chen , H. X. ( 2000 ). Irreducible representations of a class of quantum doubles . J. Alg. 225 : 391 – 409 . [Google Scholar]] are obtained. Cases when such a tensor product is completely irreducible are also given in the last section.