Authors:John Hutchens; Nathaniel Schwartz Pages: 487 - 510 Abstract: We continue a study of automorphisms of order 2 of algebraic groups. In particular we look at groups of type G2 over fields k of characteristic two. Let C be an octonion algebra over k; then Aut(C) is a group of type G2 over k. We characterize automorphisms of order 2 and their corresponding fixed point groups for Aut(C) by establishing a connection between the structure of certain four dimensional subalgebras of C and the elements in Aut(C) that induce inner automorphisms of order 2. These automorphisms relate to certain quadratic forms which, in turn, determine the Galois cohomology of the fixed point groups of the involutions. The characteristic two case is unique because of the existence of four dimensional totally singular subalgebras. Over finite fields we show how our results coincide with known results, and we establish a classification of automorphisms of order 2 over infinite fields of characteristic two. PubDate: 2018-06-01 DOI: 10.1007/s10468-017-9723-y Issue No:Vol. 21, No. 3 (2018)

Authors:P. Petrullo; D. Senato Pages: 511 - 527 Abstract: Starting from the invariant theory of binary forms, we extend the classical notion of covariants and introduce the ring of \(\mathcal {T}\) -covariants. This ring consists of maps defined on a ring of polynomials in one variable which commute with all the translation operators. We study this ring and we show some of its meaningful features. We state an analogue of the classical Hermite reciprocity law, and recover the Hilbert series associated with a suitable double grading via the elementary theory of partitions. Together with classical covariants of binary forms other remarkable mathematical notions, such as orthogonal polynomials and cumulants, turn out to have a natural and simple interpretation in this algebraic framework. As a consequence, a Heine integral representation for the cumulants of a random variable is obtained. PubDate: 2018-06-01 DOI: 10.1007/s10468-017-9724-x Issue No:Vol. 21, No. 3 (2018)

Authors:Frederik Caenepeel; Fred Van Oystaeyen Pages: 529 - 550 Abstract: We continue the study of glider representations of finite groups G with given structure chain of subgroups e ⊂ G 1 ⊂… ⊂ G d = G. We give a characterization of irreducible gliders of essential length e ≤ d which in the case of p-groups allows to prove some results about classical representation theory. The paper also contains an introduction to generalized character theory for glider representations and an extension of the decomposition groups in the Clifford theory. Furthermore, we study irreducible glider representations for products of groups and nilpotent groups. PubDate: 2018-06-01 DOI: 10.1007/s10468-017-9725-9 Issue No:Vol. 21, No. 3 (2018)

Authors:Hiroki Matsui Pages: 551 - 563 Abstract: Classification problems of subcategories have been deeply considered so far. In this paper, we discuss classifying dense resolving and dense coresolving subcategories of exact categories via their Grothendieck groups. This study is motivated by the classification of dense triangulated subcategories of triangulated categories due to Thomason. PubDate: 2018-06-01 DOI: 10.1007/s10468-017-9726-8 Issue No:Vol. 21, No. 3 (2018)

Authors:Weideng Cui Pages: 565 - 578 Abstract: In this paper, we prove that the pairwise orthogonal primitive idempotents of generic cyclotomic Birman-Murakami-Wenzl algebras can be constructed by consecutive evaluations of a certain rational function. In the Appendix, we prove a similar result for generic cyclotomic Nazarov-Wenzl algebras. A consequence of the constructions is a one-parameter family of fusion procedures for the cyclotomic Hecke algebra and its degenerate analogue. PubDate: 2018-06-01 DOI: 10.1007/s10468-017-9727-7 Issue No:Vol. 21, No. 3 (2018)

Authors:Ashish Gupta; Arnab Dey Sarkar Pages: 579 - 587 Abstract: The Gelfand–Kirillov dimension has gained importance since its introduction as a tool in the study of non-commutative infinite dimensional algebras and their modules. In this paper we show a dichotomy for the Gelfand–Kirillov dimension of simple modules over certain simple rings of differential operators. We thus answer a question of J. C. McConnell in Representations of solvable Lie algebras V. On the Gelfand-Kirillov dimension of simple modules. McConnell (J. Algebra 76(2), 489–493, 1982) concerning this dimension for a class of algebras that arise as simple homomorphic images of solvable lie algebras. We also determine the Gelfand–Kirillov dimension of an induced module. PubDate: 2018-06-01 DOI: 10.1007/s10468-017-9728-6 Issue No:Vol. 21, No. 3 (2018)

Authors:Yoshiyuki Kimura; Hironori Oya Pages: 589 - 604 Abstract: In this paper, we show that quantum twist maps, introduced by Lenagan-Yakimov, induce bijections between dual canonical bases of quantum nilpotent subalgebras. As a corollary, we show the unitriangular property between dual canonical bases and Poincaré-Birkhoff-Witt type bases under the “reverse” lexicographic order. We also show that quantum twist maps induce bijections between certain unipotent quantum minors. PubDate: 2018-06-01 DOI: 10.1007/s10468-017-9729-5 Issue No:Vol. 21, No. 3 (2018)

Authors:Teresa Conde Pages: 605 - 625 Abstract: The ADR algebra R A of an Artin algebra A is a right ultra strongly quasihereditary algebra (RUSQ algebra). In this paper we study the Δ-filtrations of modules over RUSQ algebras and determine the projective covers of a certain class of R A -modules. As an application, we give a counterexample to a claim by Auslander–Platzeck–Todorov, concerning projective resolutions over the ADR algebra. PubDate: 2018-06-01 DOI: 10.1007/s10468-017-9730-z Issue No:Vol. 21, No. 3 (2018)

Authors:Ernie Stitzinger; Ashley White Pages: 627 - 633 Abstract: We extend conjugacy results from Lie algebras to their Leibniz algebra generalizations. The proofs in the Lie case depend on anti-commutativity. Thus it is necessary to find other paths in the Leibniz case. Some of these results involve Cartan subalgebras. Our results can be used to extend other results on Cartan subalgebras. We show an example here and others will be shown in future work. PubDate: 2018-06-01 DOI: 10.1007/s10468-017-9731-y Issue No:Vol. 21, No. 3 (2018)

Authors:Sota Asai Pages: 635 - 681 Abstract: We deal with the finite-dimensional mesh algebras given by stable translation quivers. These algebras are self-injective, and thus the stable module categories have a structure of triangulated categories. Our main result determines the Grothendieck groups of these stable module categories. As an application, we give a complete classification of the mesh algebras up to stable equivalences. PubDate: 2018-06-01 DOI: 10.1007/s10468-017-9732-x Issue No:Vol. 21, No. 3 (2018)

Authors:Victor Ginzburg; Nick Rozenblyum Abstract: Let Bun G be the moduli space of G-bundles on a smooth complex projective curve. Motivated by a study of boundary conditions in mirror symmetry, Gaiotto (2016) associated to any symplectic representation of G a Lagrangian subvariety of T∗Bun G . We give a simple interpretation of (a generalization of) Gaiotto’s construction in terms of derived symplectic geometry. This allows to consider a more general setting where symplectic G-representations are replaced by arbitrary symplectic manifolds equipped with a Hamiltonian G-action and with an action of the multiplicative group that rescales the symplectic form with positive weight. PubDate: 2018-05-31 DOI: 10.1007/s10468-018-9801-9

Authors:Lauren Grimley; Christine Uhl Abstract: We consider deformations of quantum exterior algebras extended by finite groups. Among these deformations are a class of algebras which we call truncated quantum Drinfeld Hecke algebras in view of their relation to classical Drinfeld Hecke algebras. We give the necessary and sufficient conditions for which these algebras occur, using Bergman’s Diamond Lemma. We compute the relevant Hochschild cohomology to make explicit the connection between Hochschild cohomology and truncated quantum Drinfeld Hecke algebras. To demonstrate the variance of the allowed algebras, we compute both classical type examples and demonstrate an example that does not arise as a factor algebra of a quantum Drinfeld Hecke algebra. PubDate: 2018-05-31 DOI: 10.1007/s10468-018-9787-3

Abstract: Let R be a complete local Gorenstein ring of dimension one, with maximal ideal \(\mathfrak {m}\) . We show that if M is a Cohen-Macaulay R-module which begins an AR-sequence, then this sequence is produced by a particular endomorphism of \(\mathfrak {m}\) corresponding to a minimal prime ideal of R. We apply this result to determining the shape of some components of stable Auslander-Reiten quivers, which in the considered examples are shown to be tubes. PubDate: 2018-05-30 DOI: 10.1007/s10468-018-9805-5

Abstract: We introduce a symmetric monoidal category of modules over the direct limit queer superalgebra \({\mathfrak{q}}(\infty )\) . This category can be defined in two equivalent ways with the aid of the large annihilator condition. Tensor products of copies of the natural and the conatural representations are injective objects in this category. We obtain the socle filtrations and formulas for the tensor products of the indecomposable injectives. In addition, it is proven that the category is Koszul self-dual. PubDate: 2018-05-30 DOI: 10.1007/s10468-018-9803-7

Authors:Manuel Saorín; Alexander Zimmermann Abstract: Module structures of an algebra on a fixed finite dimensional vector space form an algebraic variety. Isomorphism classes correspond to orbits of the action of an algebraic group on this variety and a module is a degeneration of another if it belongs to the Zariski closure of the orbit. Riedtmann and Zwara gave an algebraic characterisation of this concept in terms of the existence of short exact sequences. Jensen, Su and Zimmermann, as well as independently Yoshino, studied the natural generalisation of the Riedtmann-Zwara degeneration to triangulated categories. The definition has an intrinsic non-symmetry. Suppose that we have a triangulated category in which idempotents split and either for which the endomorphism rings of all objects are artinian, or which is the category of compact objects in an algebraic compactly generated triangulated K-category. Then we show that the non-symmetry in the algebraic definition of the degeneration is inessential in the sense that the two possible choices which can be made in the definition lead to the same concept. PubDate: 2018-05-28 DOI: 10.1007/s10468-018-9799-z

Authors:V. F. Molchanov Abstract: We construct polynomial quantization (a variant of quantization in the spirit of Berezin) on para-Hermitian symmetric spaces. For that we use two approaches: (a) using a reproducing function, (b) using an “overgroup”. Also we show that the multiplication of symbols is an action of an overalgebra. PubDate: 2018-05-26 DOI: 10.1007/s10468-018-9784-6

Authors:Van C. Nguyen; Linhong Wang; Xingting Wang Abstract: In this paper, working over an algebraically closed field k of prime characteristic p, we introduce a concept, called Primitive Deformation, to provide a structured technique to classify certain finite-dimensional Hopf algebras which are Hopf deformations of restricted universal enveloping algebras. We illustrate this technique for the case when the restricted Lie algebra has dimension 3. Together with our previous classification results, we provide a complete classification of p3-dimensional connected Hopf algebras over k of characteristic p > 2. PubDate: 2018-05-25 DOI: 10.1007/s10468-018-9800-x

Authors:Elisabete Barreiro; Ivan Kaygorodov; José M. Sánchez Abstract: We study the structure of certain k-modules ð• over linear spaces ð•Ž with restrictions neither on the dimensions of ð• and ð•Ž nor on the base field ð”½. A basis \(\mathfrak {B} = \{v_{i}\}_{i\in I}\) of ð• is called multiplicative with respect to the basis \(\mathfrak {B}^{\prime } = \{w_{j}\}_{j \in J}\) of ð•Ž if for any \(\sigma \in S_{n}, i_{1},\dots ,i_{k} \in I\) and \(j_{k + 1},\dots , j_{n} \in J\) we have \([v_{i_{1}},\dots , v_{i_{k}}, w_{j_{k + 1}}, \dots , w_{j_{n}}]_{\sigma } \in \mathbb {F}v_{r_{\sigma }}\) for some r σ ∈ I. We show that if ð• admits a multiplicative basis then it decomposes as the direct sum \(\mathbb {V} = \bigoplus _{\alpha } V_{\alpha }\) of well described k-submodules V α each one admitting a multiplicative basis. Also the minimality of ð• is characterized in terms of the multiplicative basis and it is shown that the above direct sum is by means of the family of its minimal k-submodules, admitting each one a multiplicative basis. Finally we study an application of k-modules with a multiplicative basis over an arbitrary n-ary algebra with multiplicative basis. PubDate: 2018-05-23 DOI: 10.1007/s10468-018-9790-8

Authors:Jason Gaddis; Robert Won; Daniel Yee Abstract: A general criterion is given for when the center of a Taft algebra smash product is the fixed ring. This is applied to the study of the noncommutative discriminant. Our method relies on the Poisson methods of Nguyen, Trampel, and Yakimov, but also makes use of Poisson Ore extensions. Specifically, we fully determine the inner faithful actions of Taft algebras on quantum planes and quantum Weyl algebras. We compute the discriminant of the corresponding smash product and apply it to compute the Azumaya locus and restricted automorphism group. PubDate: 2018-05-18 DOI: 10.1007/s10468-018-9798-0

Authors:Karin Erdmann; Edward L. Green; Nicole Snashall; Rachel Taillefer Abstract: We return to the fusion rules for the Drinfeld double of the duals of the generalised Taft algebras that we studied in Erdmann et al. (J. Pure Appl. Algebra 204, 413-454 2006). We first correct some proofs and statements in Erdmann et al. (J. Pure Appl. Algebra 204, 413-454 2006) that were incorrect, using stable homomorphisms. We then complete this with new results on fusion rules for the modules we had not studied in Erdmann et al. (J. Pure Appl. Algebra 204, 413-454 2006) and a classification of endotrivial and algebraic modules. PubDate: 2018-05-18 DOI: 10.1007/s10468-018-9797-1