Rectifications of A∞-Algebras

In this article, in the setting of connected DG-modules, we prove that, for any A _∞-algebra (M, {m _i } _i≥1), there is a chain contraction from a DG-algebra A _M onto the DG-module M such that the A _∞-algebra structure induced by perturbation theory on M is precisely the original one. In fact, the mentioned DG-algebra can be considered a rectification of the A _∞-algebra in the sense of (Boardman and Vogt, 1973 Boardman , J. M. , Vogt , R. M. ( 1973 ). Homotopy Invariant Algebraic Structures on Topological Spaces . Lecture Notes in Mathematics . Springer-Verlag , pp. 347 . [Google Scholar]). Appropiate dual results are given for A _∞-coalgebras.     

Application of Full Quivers of Representations of Algebras,to Polynomial Identities

In [7 Belov , A. , Rowen , L. H. , Vishne , U. Full quivers of representations of algebras. To appear in Trans. Amer. Math. Soc.  [Google Scholar]] we introduced the notion of full quivers of representations of algebras, which are more explicit than quivers of algebras, and better suited for algebras over finite fields. Here, we consider full quivers as a combinatorial tool in order to describe PI-varieties of algebras. We apply the theory to clarify the proofs of diverse topics in the literature: Determining which relatively free algebras are weakly Noetherian, determining when relatively free algebras are finitely presented, presenting a quick proof for the rationality of the Hilbert series of a relatively free PI-algebra, and explaining counterexamples to Specht's conjecture for varieties of Lie algebras.     

Modular Irreducible Representations of Weyl Groups of Type B n

In this article, we build all the irreducible representations of a Weyl group of type B _n over a field of characteristic p > 2. This construction is based on [2 Aguado , J. , Araujo , J. ( 1998 ). Representations of the symmetric group 𝔖 _n on K[x ₁,…, x _n ] . Revista de la Unión Matemática Argentina 41 . [Google Scholar]], and in fact it extends the results of that article for the modular irreducible representations of the symmetric group.     

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.     

Projective Brauer Theory

In this article, we define and study a new concept of projective Brauer characters. Following the direct approach as taken in Navarro (1998 Navarro , G. ( 1998 ). Characters and Blocks of Finite Groups. Cambridge , UK : Cambridge University Press . [Google Scholar]) and Isaacs (1975 Isaacs , I. M. ( 1975 ). Character Theory of Finite Groups . New York : Academic Press . [Google Scholar]), we study the resulting projective blocks of projective Frobenius characters and projective Brauer characters of a given group determined by certain linkage condition as an introduction to a projective extension of classic Brauer theory.     

On the Attached Prime Ideals of Local Cohomology Modules

In this article, we shall prove some new properties about attached prime ideals over local cohomology modules. Also we generalize some of the results of [2 Brodmann , M. P. , Sharp , R. Y. ( 1998 ). Local Cohomology; An Algebraic Introduction with Geometric Applications . Cambridge : Cambridge University Press .[Crossref] , [Google Scholar]].     

Categorical Realizations of Quivers

We introduce and study categorical realizations of quivers. This construction generalizes comma categories and includes representations of quivers on categories, twisted representations of quivers (in the sense of [9 Gothen, P. B., King, Alastair D. (2005). Homological algebra of twisted quiver bundles. J. London Math. Soc. (2) 71(1):85–99.[Crossref], [Web of Science ®] , [Google Scholar]]), and bilinear pairings as special cases. In this general context, we prove the cancelation from direct sums, and show the existence and uniqueness of decomposition into a sum of indecomposable objects, provided certain assumptions hold. This yields similar results for the special cases just mentioned. Using similar ideas, we also prove a version of Fitting's Lemma for natural transformations between functors.     

Generic Representation Theory of the Heisenberg Group

In this article we extend a result for representations of the additive group G_a given in [4 Suslin , A. , Friedlander , E. M. , Bendel , C. P. ( 1997 ). Infinitesimal 1-parameter subgroups and cohomology . Journal of the AMS 10 ( 3 ): 693 – 728 .[Web of Science ®] , [Google Scholar]] to the Heisenberg group H₁. Namely, if p is greater than 2d, then all d-dimensional characteristic p representations for H₁ can be factored into commuting products of representations, with each factor arising from a representation of the Lie algebra of H₁, and conversely any commuting collection of Lie algebra representations gives rise to a representation of H₁ in this fashion. In this sense, for a fixed dimension and large enough p, all representations for H₁ look generically like representations for direct powers of it over a field of characteristic zero.

The following originally appeared as Chapter 13 of the author's dissertation [1 Crumley , M. ( 2010 ). Ultraproducts of Tannakian Categories and Generic Representation Theory of Unipotent Algebraic Groups. Ph.D. dissertation, The University of Toledo, Department of Mathematics . [Google Scholar]].     

Singular Solutions to the Quantum Yang–Baxter Equations

Let X and A be weak Hopf algebras in the sense of Li (1998 Li , F. ( 1998 ). Weak Hopf algebras and some new solutions of the quantum Yang–Baxter equation . J. Algebra 208 ( 1 ): 72 – 100 .[Crossref], [Web of Science ®] , [Google Scholar]). As in the case of Hopf algebras (Majid, 1990 Majid , S. ( 1990 ). Quasitriangular Hopf algebras and Yang–Baxter equations . Internat. J. Modern Phys. A 5 : 1 – 91 . [Google Scholar]), a weak bicrossed coproduct X∞ _R A is constructed by means of good regular R-matrices of the weak Hopf algebras X and A. Using this, we provide a new framework of obtaining singular solutions of the quantum Yang–Baxter equation by constructing weak quasitriangular structures over X∞ _R A when both X and A admit a weak quasitriangular structure. Finally, two explicit examples are given.     

Irreducible W-Graphs for Type D 4 and D 5

ABSTRACT

The “W-graph” concept was introduced by Kazhdan and Lusztig in their influential article Kazhdan and Lusztig (1979 Kazhdan , D. , Lusztig , G. ( 1979 ). Representations of Coxeter groups and Hecke algebras . Invent. Math. 53 : 165 – 184 . [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]). If W is a Coxeter group, then a W-graph provides a method for constructing a matrix representation of the Hecke algebra ? associated with W (the degree of the representation being the number of vertices of the W-graph). The aim of this note is to explicitly construct all the irreducible representations of ? when W is of type D ₄ and D ₅.     

About the Cohomology of the Lie Superalgebra of Vector Fields on ℝ n|n

In this article, we compute the first space of cohomology of Vect (? ^n|n ), the Lie superalgebra of vector fields on the supermanifold ? ^n|n with coefficients in 𝒻 (? ^n|n ), the space of smooth functions on ? ^n|n . We give a super analog of the cohomologies of vector fields that where studied for instance by Fuchs [2 Fuchs , D. B. ( 1986 ). Cohomology of Infinite Dimensional Lie Algebras . New York : Consultants Bureau . [Google Scholar]]. This work allows us to classify the deformations of the action of Vect(? ^n|n ) on 𝒻 (? ^n|n ).     

Unitary Modules for EALAs Co-ordinatized by a Quantum Torus

Abstract

A class of unitary modules is constructed for an extended affine Lie algebra (EALA) co-ordinatized by a quantum torus for |q| = 1 by extending the method developed by Kac and Raina (Kac, V. G., Raina, A. K. (1987 Kac, V. G. and Raina, A. K. 1987. “Highest weight representations of infinite dimensional Lie algebras”. Singapore: World Scientific.  [Google Scholar]). Highest Weight Representations of Infinite Dimensional Lie Algebras. Singapore: World Scientific.) which uses infinite wedge products. By twisting the module by an automorphism (defined by the author Eswara Rao (Eswara Rao, S. (2001 Eswara Rao, S. 2001. “A class of integrable modules for the core of EALA co-ordinated by Quantum Tori”. TIFR preprint [Google Scholar]). A class of integrable modules for the core of EALA co-ordinated by Quantum Tori. TIFR preprint.) which commutes with anti-linear anti-automorphism, we construct a large class of unitary modules for the EALA.     

Double Ore Extensions Versus Iterated Ore Extensions

Motivated by the construction of new examples of Artin–Schelter regular algebras of global dimension four, Zhang and Zhang [6 Zhang , J. J. , Zhang , J. ( 2008 ). Double Ore extensions . J. Pure Appl. Algebra 212 ( 12 ): 2668 – 2690 .[Crossref], [Web of Science ®] , [Google Scholar]] introduced an algebra extension A _P [y ₁, y ₂; σ, δ, τ] of A, which they called a double Ore extension. This construction seems to be similar to that of a two-step iterated Ore extension over A. The aim of this article is to describe those double Ore extensions which can be presented as iterated Ore extensions of the form A[y ₁; σ₁, δ₁][y ₂; σ₂, δ₂]. We also give partial answers to some questions posed in Zhang and Zhang [6 Zhang , J. J. , Zhang , J. ( 2008 ). Double Ore extensions . J. Pure Appl. Algebra 212 ( 12 ): 2668 – 2690 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Cone Length for DG Modules and Global Dimension of DG Algebras

As the definition of free class of differential modules over a commutative ring in [1 Avramov , L. L. , Buchweitz , R.-O. , Iyengar , S. ( 2007 ). Class and rank of differential modules . Invent. Math. 169 : 1 – 35 .[Crossref], [Web of Science ®] , [Google Scholar]], we define DG free class for semifree DG modules over an Adams connected DG algebra A. For any DG A-modules M, we define its cone length as the least DG free classes of all semifree resolutions of M. The cone length of a DG A-module plays a similar role as projective dimension of a module over a ring does in homological ring theory. The left (resp., right) global dimension of an Adams connected DG algebra A is defined as the supremum of the set of cone lengths of all DG A-modules (resp., A ^op -modules). It is proved that the definition is a generalization of that of graded algebras. Some relations between the global dimension of H(A) and the left (resp. right) global dimension of A are discovered. When A is homologically smooth, we prove that the left (right) global dimension of A is finite and the dimension of D(A) and D ^c (A) are not bigger than the DG free class of a minimal semifree resolution X of the DG A ^e -module A.     

CONGRUENCE SUBGROUPS AND TWISTED COHOMOLOGY OF SLn (F[t]). II. FINITE FIELDS AND NUMBER FIELDS

Let K be the subgroup of SL _n (F[t]) consisting of matrices congruent to the identity modulo t. In [7] Knudson, K. 1998. Congruence Subgroups and Twisted Cohomology of SL_n(F[t]). J. Algebra, 207: 695–721.  [Google Scholar], the author conjectured that if F is a finite field, then H ₁(K) is the adjoint representation s l _n (F). A proof of this conjecture is provided in this note. The argument also works in case F is a number field. Applications to the cohomology of SL _n (F[t]) are included as is the study of the analogous question for SL _n (F[t, t ^?1]).     

Bounds for Arithmetic Degrees

ABSTRACT

Let I be a monomial ideal with minimal monomial generators m₁,…, m_s, and assume that deg(m₁) ≥deg(m₂) ≥ … ≥deg(m_s). Among other things, we prove that the arithmetic degree of I is bounded above by deg(m₁)…deg(m_mht(I)), where mht(I) is the maximal height of associated primes of I. This bound is shaper than the one given in [12 Sturmfels, B., Trung, N. V., Vogel, W. (1995). Bounds on degrees of projective schemes. Math. Ann. 302:417–432.[Crossref], [Web of Science ®] , [Google Scholar]] and more natural than the one given in [9 Hoa, L. T., Trung, N. V. (1998). On the Castelnuovo-Mumford regularity and the arithmetic degree of monomial ideals. Math. Z. 229:519–537.[Crossref], [Web of Science ®] , [Google Scholar]]. In addition, we point out that adeg(I) ≠ adeg(Gin(I)) in general and conjecture that adeg(I) = adeg(Gin(I)) if and only if R/I is sequentially Cohen–Macaulay.     

Continuous Cohomology of Permutation Groups on Profinite Modules

ABSTRACT

Model theorists have made use of low-dimensional continuous cohomology of infinite permutation groups on profinite modules, see Ahlbrandt and Ziegler (1991 Ahlbrandt , G. , Ziegler , M. ( 1991 ). What's so special about (?/4?)^ω? Archive for Math. Logic 31 : 115 – 132 . [CSA] [Crossref] , [Google Scholar]), Evans (1997b Evans , D. M. ( 1997b ). Computation of first cohomology groups of finite covers . J. Algebra 193 : 214 – 238 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]), Evans et al. (1997 Evans , D. M. , Ivanov , A. A. , Macpherson , H. D. ( 1997 ). Finite covers . In: Evans , D. M. , ed. Model Theory of Groups and Automorphism Groups . London Mathematical Society Lecture Notes 244 . Cambridge : Cambridge Univ Press , pp. 1 – 72 .[Crossref] , [Google Scholar]), and Hodges and Pillay (1994 Hodges , W. , Pillay , A. ( 1994 ). Cohomology of structures and some problems of Ahlbrandt and Ziegler . J. London Math. Soc. 50 ( 2 ): 1 – 16 . [CSA] [Crossref] , [Google Scholar]), for example. We expand the module category in order to widen the cohomological toolkit. For an important class of groups we use these tools to establish criteria for finiteness of cohomology.     

On Generalized CB-Degenerations of Algebras

We introduce a concept generalizing classical degenerations of algebras (defined by structure constants) and Crawley-Boevey degenerations introduced in [3 Crawley-Boevey , W. W. ( 1995 ). Tameness of biserial algebras . Arch. Math. 65 : 399 – 407 .[Crossref], [Web of Science ®] , [Google Scholar]]. We prove that if A ₀ is such a generalized degeneration of A ₁ and the algebras have equal dimensions, then A ₀ is a degeneration of A ₁ in the classical sense.     

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]).