Search

This paper presents an E -derivation analogue of a result on derivations due to van den Essen, Wright and Zhao. We prove that the image of a locally finite K - E -derivation of polynomial algebras in two variables over a field K of characteristic zero is a Mathieu subspace. This result together with that of van den Essen, Wright and Zhao confirms the LFED conjecture in the case of polynomial algebras in two variables. [ABSTRACT FROM AUTHOR]

In this paper we show that the Poisson analogue of the Noether's Problem has a positive solution for essentially all symplectic reflection groups — the analogue of complex reflection groups in the symplectic world. Our proofs are constructive, and generalize and refine previously known results. The results of this paper can be thought as analogues of the Noncommutative Noether Problem and the Gelfand-Kirillov Conjecture for rational Cherednik algebras in the quasi-classical limit. An abstract framework to understand these results is introduced. As a consequence for complex reflection groups, we obtain the Poisson rationality of the Calogero-Moser spaces associated to any of them, and we verify the Gelfand-Kirillov Conjecture for trigonometric Cherednik algebras and the Poisson rationality of their corresponding Calogero-Moser spaces. [ABSTRACT FROM AUTHOR]

In this paper, we consider the homogeneous coordinate rings A (Y n , d) ≅ K [ Ω n , d ] of monomial projections Y n , d of Veronese varieties parameterized by subsets Ω n , d of monomials of degree d in n + 1 variables where: (1) Ω n , d contains all monomials supported in at most s variables and, (2) Ω n , d is a set of monomial invariants of a finite diagonal abelian group G ⊂ GL (n + 1 , K) of order d. Our goal is to study when K [ Ω n , d ] is a quadratic algebra and, if so, when K [ Ω n , d ] is Koszul or G-quadratic. For the family (1), we prove that K [ Ω n , d ] is quadratic when s ≥ ⌈ n + 2 2 ⌉. For the family (2), we completely characterize when K [ Ω 2 , d ] is quadratic in terms of the group G ⊂ GL (3 , K) , and we prove that K [ Ω 2 , d ] is quadratic if and only if it is Koszul. We also provide large families of examples where K [ Ω n , d ] is G-quadratic. [ABSTRACT FROM AUTHOR]

In this paper, we introduce new invariants to a singularity (V , 0) , i.e., the derivation Lie algebras L k (V) of the higher Nash blow-up local algebra M k (V). A new conjecture about the non-existence of negative weighted derivations of L k (V) for weighted homogeneous isolated hypersurface singularities is proposed. We verify this conjecture partially. Moreover, we compute the Lie algebra L 2 (V) for binomial isolated singularities. We also formulate a sharp upper estimate conjecture for the dimension of L k (V) for weighted homogeneous isolated hypersurface singularities and verify this conjecture for a large class of singularities. [ABSTRACT FROM AUTHOR]

For a number field F and an odd prime number p , let F ˜ be the compositum of all Z p -extensions of F and Λ ˜ the associated Iwasawa algebra. Let G S (F ˜) be the Galois group over F ˜ of the maximal extension which is unramified outside p -adic and infinite places. In this paper we study the Λ ˜ -module X S (− i) (F ˜) : = H 1 (G S (F ˜) , Z p (− i)) and its relationship with X (F ˜ (μ p)) (i − 1) Δ , the Δ : = Gal (F ˜ (μ p) / F ˜) -invariant of the Galois group over F ˜ (μ p) of the maximal abelian unramified pro- p -extension of F ˜ (μ p). More precisely, we show that under a decomposition condition, the pseudo-nullity of the Λ ˜ -module X (F ˜ (μ p)) (i − 1) Δ is implied by the existence of a Z p d -extension L with X S (− i) (L) : = H 1 (G S (L) , Z p (− i)) being without torsion over the Iwasawa algebra associated to L , and which contains a Z p -extension F ∞ satisfying H 2 (G S (F ∞) , Q p / Z p (i)) = 0. As a consequence we obtain a sufficient condition for the validity of Greenberg's generalized conjecture when the integer i ≡ 1 mod [ F (μ p) : F ]. This existence is fulfilled for (p , i) -regular fields. [ABSTRACT FROM AUTHOR]

We prove a version of the Deligne conjecture for n -fold monoidal abelian categories A over a field k of characteristic 0, assuming some compatibility and non-degeneracy conditions for A . The output of our construction is a weak Leinster ( n , 1 ) -algebra over k , a relaxed version of the concept of Leinster n -algebra in A lg ( k ) . The difference between the Leinster original definition and our relaxed one is apparent when n > 1 , for n = 1 both concepts coincide. We believe that there exists a functor from weak Leinster ( n , 1 ) -algebras over k to C • ( E n + 1 , k ) -algebras, well-defined when k = Q , and preserving weak equivalences. For the case n = 1 such a functor is constructed in [31] by elementary simplicial methods, providing (together with this paper) a complete solution for 1-monoidal abelian categories. Our approach to Deligne conjecture is divided into two parts. The first part, completed in the present paper, provides a construction of a weak Leinster ( n , 1 ) -algebra over k , out of an n -fold monoidal k -linear abelian category (provided the compatibility and non-degeneracy condition are fulfilled). The second part (still open for n > 1 ) is a passage from weak Leinster ( n , 1 ) -algebras to C • ( E n + 1 , k ) -algebras. As an application, we prove in Theorem 8.1 that the Gerstenhaber–Schack complex of a Hopf algebra over a field k of characteristic 0 admits a structure of a weak Leinster ( 2 , 1 ) -algebra over k extending the Yoneda structure. It relies on our earlier construction [30] of a 2-fold monoidal structure on the abelian category of tetramodules over a bialgebra. [ABSTRACT FROM AUTHOR]

In this paper, we study equivalences between the categories of quasi–coherent sheaves on non–commutative noetherian schemes. In particular, we give a new proof of Căldăraru's conjecture about Morita equivalences of Azumaya algebras on noetherian schemes. Moreover, we derive necessary and sufficient condition for two reduced non–commutative curves to be Morita equivalent. [ABSTRACT FROM AUTHOR]

Let G be connected reductive algebraic group defined over an algebraically closed field of characteristic p > 0 and suppose that p is a good prime for the root system of G , the derived subgroup of G is simply connected and the Lie algebra g = Lie (G) admits a non-degenerate (Ad G) -invariant symmetric bilinear form. Given a linear function χ on g we denote by U χ (g) the reduced enveloping algebra of g associated with χ. By the Kac–Weisfeiler conjecture (now a theorem), any irreducible U χ (g) -module has dimension divisible by p d (χ) where 2 d (χ) is the dimension of the coadjoint G -orbit containing χ. In this paper we give a positive answer to the natural question raised in the 1990s by Kac, Humphreys and the first-named author and show that any algebra U χ (g) admits a module of dimension p d (χ). [ABSTRACT FROM AUTHOR]

Abstract: This paper consists of three parts. In the first part we prove that Zhu''s and -algebras in type A have the same dimensions. In the second part we compute the graded decomposition of the -algebras in type A, thus proving the Gaberdiel–Gannon conjecture. Our main tool is the theory of abelian radicals, which we develop in the third part. [Copyright &y& Elsevier]

Abstract: In this paper, we compute the congruences of Catalan and Motzkin numbers modulo 4 and 8. In particular, we prove the conjecture proposed by Deutsch and Sagan that no Motzkin number is a multiple of 8. [Copyright &y& Elsevier]

Abstract: Let be the set of all primes. The following result is proved: For any nonzero integer a, the set contains arbitrarily long sequences which have the same largest prime factor. We give an application to the dynamics of the w function which extends the “seven” in Theorem 2.14 of [Wushi Goldring, Dynamics of the w function and primes, J. Number Theory 119 (2006) 86–98] to any positive integer. Beyond this we also establish a relation between a result of congruent covering systems and a question on the dynamics of the w function. This implies that the answer to Conjecture 2.16 of Goldring''s paper is negative. Two conjectures are posed. [Copyright &y& Elsevier]

Abstract: In 1982–1983, E. Nochka proved a conjecture of Cartan on defects of holomorphic curves in relative to a possibly degenerate set of hyperplanes. This was further explained by W. Chen in his 1987 thesis, and subsequently simplified by M. Ru and P.-M. Wong in 1991. The proof involved assigning weights to the hyperplanes. This paper provides further simplification of the proof of the construction of the weights, by bringing back the use of the convex hull in working with the “Nochka diagram.” [Copyright &y& Elsevier]

In this paper, we connect the rigidity problem and the coarse Baum-Connes conjecture for Roe algebras. In particular, we show that if X and Y are two uniformly locally finite metric spaces such that their Roe algebras are ⁎-isomorphic, then X and Y are coarsely equivalent provided either X or Y satisfies the coarse Baum-Connes conjecture with coefficients. It is well-known that coarse embeddability into a Hilbert space implies the coarse Baum-Connes conjecture with coefficients. On the other hand, we provide a new example of a finitely generated group satisfying the coarse Baum-Connes conjecture with coefficients but which does not coarsely embed into a Hilbert space. [ABSTRACT FROM AUTHOR]