Hochschild Cohomology of <Emphasis Type="Italic">q</Emphasis>-Schur Algebras

We compute the Hochschild cohomology of any block of q-Schur algebras. We focus on the even part of this Hochschild cohomology ring. To compute the Hochschild cohomology of q-Schur algebras, we prove the following two results: first, we construct two graded algebra surjections between the Hochschild cohomologies of quasi-hereditary algebras because all q-Schur algebras over a field are quasi-hereditary. Second, we give the graded algebra isomorphism of Hochschild cohomologies by using a certain derive equivalence.     

Hochschild cohomology ring modulo nilpotence of a one point extension of a quiver algebra defined by two cycles and a quantum-like relation

We consider a one point extension algebra B of a quiver algebra A _q over a field k defined by two cycles and a quantum-like relation depending on a nonzero element q in k. We determine the Hochschild cohomology ring of B modulo nilpotence and show that if q is a root of unity, then B is a counterexample to Snashall-Solberg’s conjecture.     

Off-Shell Supersymmetry and Filtered Clifford Supermodules

An off-shell representation of supersymmetry is a representation of the super Poincaré algebra on a dynamically unconstrained space of fields. We describe such representations formally, in terms of the fields and their spacetime derivatives, and we interpret the physical concept of engineering dimension as an integral grading. We prove that formal graded off-shell representations of one-dimensional N-extended supersymmetry, i.e., the super Poincaré algebra \(\mathfrak {p}^{1|N}\), correspond to filtered Clifford supermodules over Cl(N). We also prove that formal graded off-shell representations of two-dimensional (p,q)-supersymmetry, i.e., the super Poincaré algebra \(\mathfrak {p}^{1,1|p,q}\), correspond to bifiltered Clifford supermodules over Cl(p + q). Our primary tools are Rees superalgebras and Rees supermodules, the formal deformations of filtered superalgebras and supermodules, which give a one-to-one correspondence between filtered spaces and graded spaces with even degree-shifting injections. This generalizes the machinery used by Gerstenhaber to prove that every filtered algebra is a deformation of its associated graded algebra. Our treatment extends the notion of Rees algebras and modules to filtrations which are compatible with a supersymmetric structure. We also describe the analogous constructions for bifiltrations and bigradings.     

Stratified modules over an extension algebra

Let A be a standard Koszul standardly stratified algebra and X an A-module. The paper investigates conditions which imply that the module Ext* _A (X) over the Yoneda extension algebra A* is filtered by standard modules. In particular, we prove that the Yoneda extension algebra of A is also standardly stratified. This is a generalization of similar results on quasi-hereditary and on graded standardly stratified algebras.     

A new characterization for some extensions of PSL(2, <Emphasis Type="Italic">q)</Emphasis> for some <Emphasis Type="Italic">q</Emphasis> by some character degrees

In [22] (Tong-Viet H P, Simple classical groups of Lie type are determined by their character degrees, J. Algebra, 357 (2012) 61–68), the following question arose: Which groups can be uniquely determined by the structure of their complex group algebras? The authors in [12] (Khosravi B et al., Some extensions of PSL(2,p²) are uniquely determined by their complex group algebras, Comm. Algebra, 43(8) (2015) 3330–3341) proved that each extension of PSL(2,p²) of order 2|PSL(2,p²)| is uniquely determined by its complex group algebra. In this paper we continue this work. Let p be an odd prime number and q = p or q = p³. Let M be a finite group such that |M| = h|PSL(2,q), where h is a divisor of |Out(PSL(2,q))|. Also suppose that M has an irreducible character of degree q and 2p does not divide the degree of any irreducible character of M. As the main result of this paper we prove that M has a unique nonabelian composition factor which is isomorphic to PSL(2,q). As a consequence of our result we prove that M is uniquely determined by its order and some information on its character degrees which implies that M is uniquely determined by the structure of its complex group algebra.     

3<Emphasis Type="Italic">nj</Emphasis>-symbols and identities for <Emphasis Type="Italic">q</Emphasis>-Bessel functions

The 6j-symbols for representations of the q-deformed algebra of polynomials on \(\mathrm {SU}(2)\) are given by Jackson’s third q-Bessel functions. This interpretation leads to several summation identities for the q-Bessel functions. Multivariate q-Bessel functions are defined, which are shown to be limit cases of multivariate Askey–Wilson polynomials. The multivariate q-Bessel functions occur as 3nj-symbols.     

On <Emphasis Type="Italic">G</Emphasis>-rigid surfaces

Rigid algebraic varieties form an important class of complex varieties that exhibit interesting geometric phenomena. In this paper we propose a natural extension of rigidity to complex projective varieties with a finite group action (G-varieties) and focus on the first nontrivial case, namely, on G-rigid surfaces that can be represented as desingularizations of Galois coverings of the projective plane with Galois group G. We obtain local and global G-rigidity criteria for these G-surfaces and present several series of such surfaces that are rigid with respect to the action of the deck transformation group.     

On the centers of quantum groups of <Emphasis Type="Italic">A</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-type

Let g be the finite dimensional simple Lie algebra of type An, and let U? = U _q (g,Λ) and U = U _q (g,Q) be the quantum groups defined over the weight lattice and over the root lattice, respectively. In this paper, we find two algebraically independent central elements in U? for all n ≥ 2 and give an explicit formula of the Casimir elements for the quantum group U? = U _q (g,Λ), which corresponds to the Casimir element of the enveloping algebra U(g). Moreover, for n = 2 we give explicitly generators of the center subalgebras of the quantum groups U? = U _q (g,Λ) and U = U _q (g,Q).     

Galois subfields of tame division algebras

We show that a finite-dimensional tame division algebra D over a Henselian field F has a maximal subfield Galois over F if and only if its residue division algebra \(\overline D \) has a maximal subfield Galois over the residue field \(\overline F \).     

Cleft extensions of Koszul twisted Calabi–Yau algebras

Let H be a twisted Calabi–Yau (CY) Hopf algebra and σ a 2-cocycle on H. Let A be an N-Koszul twisted CY algebra such that A is a graded H^σ- module algebra. We show that the cleft extension A#_σH is also a twisted CY algebra. This result has two consequences. Firstly, the smash product of an N-Koszul twisted CY algebra with a twisted CY Hopf algebra is still a twisted CY algebra. Secondly, the cleft objects of a twisted CY Hopf algebra are all twisted CY algebras. As an application of this property, we determine which cleft objects of U(D, λ), a class of pointed Hopf algebras introduced by Andruskiewitsch and Schneider, are Calabi–Yau algebras.     

Small Representations for Affine <Emphasis Type="Italic">q</Emphasis>-Schur Algebras

When the parameter \(q\in \mathbb {C}^{*}\) is not a root of unity, simple modules of affine q-Schur algebras have been classified in terms of Frenkel–Mukhin’s dominant Drinfeld polynomials (Deng et al. 2012). We compute these Drinfeld polynomials associated with the simple modules of an affine q-Schur algebra which come from the simple modules of the corresponding q-Schur algebra via the evaluation maps.     

Cube root Ramanujan formulas and elementary Galois theory

The cube root Ramanujan formulas are explained from the point of view of Galois theory. Let F be a cyclic cubic extension of a field K. It is proved that the normal closure over K of a pure cubic extension of F contains a certain pure cubic extension of K. The proposed proof can be generalized to radicals of any prime degree q. In the case where the base field K is the field of rational numbers and the field F is embedded in the cyclotomic extension obtained by adding the pth roots of unity, the corresponding simple radical extension of the field of rational numbers is explicitly constructed. The proof of the main result illustrates Hilbert’s Theorem 90. An example of a particular formula generalizing Ramanujan’s formulas for degree 5 is given. A necessary condition for nested radical expressions of depth 2 to be contained in the normal closure of a pure cubic extension of the field F is given.     

ON THE COADJOINT ORBITS OF MAXIMAL UNIPOTENT SUBGROUPS OF REDUCTIVE GROUPS

Let G be a simple algebraic group defined over an algebraically closed field of characteristic 0 or a good prime for G. Let U be a maximal unipotent subgroup of G and \( \mathfrak{u} \) its Lie algebra. We prove the separability of orbit maps and the connectedness of centralizers for the coadjoint action of U on (certain quotients of) the dual \( \mathfrak{u} \)* of \( \mathfrak{u} \). This leads to a method to give a parametrization of the coadjoint orbits in terms of so-called minimal representatives which form a disjoint union of quasi-affine varieties. Moreover, we obtain an algorithm to explicitly calculate this parametrization which has been used for G of rank at most 8, except E₈.When G is defined and split over the field of q elements, for q the power of a good prime for G, this algorithmic parametrization is used to calculate the number k(U(q); \( \mathfrak{u} \)*(q)) of coadjoint orbits of U(q) on \( \mathfrak{u} \)*(q). Since k(U(q), \( \mathfrak{u} \)*(q)) coincides with the number k(U(q)) of conjugacy classes in U(q), these calculations can be viewed as an extension of the results obtained in [11]. In each case considered here there is a polynomial h(t) with integer coefficients such that for every such q we have k(U(q)) = h(q). We also explain implications of our results for a parametrization of the irreducible complex characters of U(q).     

First-order decidability and definability of integers in infinite algebraic extensions of the rational numbers

We extend results of Videla and Fukuzaki to define algebraic integers in large classes of infinite algebraic extensions of Q and use these definitions for some of the fields to show the first-order undecidabilitv. We also obtain a structural sufficient condition for definability of the ring of integers over its field of fractions. In particular, we show that the following propositions hold: (1) For any rational prime q and any positive rational integer m. algebraic integers are definable in any Galois extension of Q where the degree of any finite subextension is not divisible by q^m. (2) Given a prime q, and an integer m > 0, algebraic integers are definable in a cyclotomic extension (and any of its subfields) generated by any set \(\{ {\zeta _{{p^l}}}|l \in {Z_{ > 0,}}P \ne q\) is any prime such that q^{m +1} ∧(p — 1)}. (3) The first-order theory of Any Abelina Extension of Q With Finitely Many Rational Primes is undecidable and rational integers are definable in these extensions.We also show that under a condition on the splitting of one rational Q generated elliptic curve over the field in question is enough to have a definition of Z and to show that the field is undecidable.     

Koszulity and Koszul modules of dual extension algebras

Let A and B be algebras, and let T be the dual extension algebra of A and B. We provide a different method to prove that T is Koszul if and only if both A and B are Koszul. Furthermore, we prove that an algebra is Koszul if and only if one of its iterated dual extension algebras is Koszul, if and only if all its iterated dual extension algebras are Koszul. Finally, we give a necessary and sufficient condition for a dual extension algebra to have the property that all linearly presented modules are Koszul modules, which provides an effective way to construct algebras with such a property.     

The Injective Leavitt Complex

For a finite quiver Q without sinks, we consider the corresponding finite dimensional algebra A with radical square zero. We construct an explicit compact generator for the homotopy category of acyclic complexes of injective A-modules. We call such a generator the injective Leavitt complex of Q. This terminology is justified by the following result: the differential graded endomorphism algebra of the injective Leavitt complex of Q is quasi-isomorphic to the Leavitt path algebra of Q. Here, the Leavitt path algebra is naturally \(\mathbb {Z}\)-graded and viewed as a differential graded algebra with trivial differential.     

On a Class of Graded Ideals of Semigroup <Emphasis Type="Italic">C</Emphasis>*-Algebras

We present general results about graded C*-algebras and continue the previously initiated research of the C*-algebra generated by the left regular representation of an abelian semigroup. We study the invariant ideals of this C*-algebra invariant with respect to the representation of a compact group G in the automorphism group of this algebra. We prove that the invariance of the ideal is equivalent to the fact that this ideal is graded C*-algebra, that there is a maximum of all invariant ideals, and it is the commutator ideal. Separately we study a class of graded primitive ideals generated by a single projector.     

On the Malle conjecture and the self-twisted cover

For a large class of finite groups G, the number of Galois extensions E/Q of group G and discriminant |d _E | ≤ y is shown to grow at least like a power of y, for some specified positive exponent. The groups G are the regular Galois groups over Q and the counted extensions E/Q are obtained by specializing a given regular Galois extension F/Q(T). The extensions E/Q can further be prescribed any unramified local behavior at each suitably large prime p ≤ log(y)/δ for some δ ≥ 1. This result is a step toward the Malle conjecture on the number of Galois extensions of given group and bounded discriminant. The local conditions further make it a notable constraint on regular Galois groups over Q. The method uses a new version of Hilbert’s irreducibility theorem that counts the specialized extensions and not just the specialization points. A main tool for it is the self-twisted cover that we introduce.     

Specialization results and ramification conditions

Given a hilbertian field k of characteristic zero and a finite Galois extension E/k(T) with group G such that E/k is regular, we produce some specializations of E/k(T) at points t0 ∈ P¹(k) which have the same Galois group but also specified inertia groups at finitely many given primes. This result has two main applications. Firstly we conjoin it with previous works to obtain Galois extensions of Q of various finite groups with specified local behavior — ramified or unramified — at finitely many given primes. Secondly, in the case k is a number field, we provide criteria for the extension E/k(T) to satisfy this property: at least one Galois extension F/k of group G is not a specialization of E/k(T).     

New characterization of <Emphasis Type="Italic">S</Emphasis><Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>) by its noncommuting graph

Let G be a nonabelian group, and associate the noncommuting graph ?(G) with G as follows: the vertex set of ?(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. Let S ₄(q) be the projective symplectic simple group, where q is a prime power. We prove that if G is a group with ?(G) ? ?(S ₄(q)) then G ? S ₄(q).