Results 1  10
of
165
Lifting of quantum linear spaces and pointed Hopf algebras of order p3
 J Algebra
, 1998
"... Abstract. We propose the following principle to study pointed Hopf algebras, or more generally, Hopf algebras whose coradical is a Hopf subalgebra. Given such a Hopf algebra A, consider its coradical filtration and the associated graded coalgebra gr A. Then gr A is a graded Hopf algebra, since the c ..."
Abstract

Cited by 81 (16 self)
 Add to MetaCart
Abstract. We propose the following principle to study pointed Hopf algebras, or more generally, Hopf algebras whose coradical is a Hopf subalgebra. Given such a Hopf algebra A, consider its coradical filtration and the associated graded coalgebra gr A. Then gr A is a graded Hopf algebra, since the coradical A0 of A is a Hopf subalgebra. In addition, there is a projection π: gr A → A0; let R be the algebra of coinvariants of π. Then, by a result of Radford and Majid, R is a braided Hopf algebra and gr A is the bosonization (or biproduct) of R and A0: gr A ≃ R#A0. The principle we propose to study A is first to study R, then to transfer the information to gr A via bosonization, and finally to lift to A. In this article, we apply this principle to the situation when R is the simplest braided Hopf algebra: a quantum linear space. As consequences of our technique, we obtain the classification of pointed Hopf algebras of order p 3 (p an odd prime) over an algebraically closed field of characteristic zero; with the same hypothesis, the characterization of the pointed Hopf algebras whose coradical is abelian and has index p or p 2; and an infinite family of pointed, nonisomorphic, Hopf algebras of the same dimension. This last result gives a negative
An introduction to commutative and noncommutative Gröbner bases
 Theoretical Computer Science
, 1994
"... In 1965, Buchberger introduced the notion of Gröbner bases for a polynomial ideal and an algorithm (Buchberger Algorithm) for their computation ([B1],[B2]). Since the end of the Seventies, Gröbner bases have been an essential tool in the development of computational ..."
Abstract

Cited by 71 (3 self)
 Add to MetaCart
In 1965, Buchberger introduced the notion of Gröbner bases for a polynomial ideal and an algorithm (Buchberger Algorithm) for their computation ([B1],[B2]). Since the end of the Seventies, Gröbner bases have been an essential tool in the development of computational
KazhdanLusztig polynomials and character formulae for the Lie superalgebra gl(mn
 J. AMS
"... The problem of computing the characters of the finite dimensional irreducible representations of the Lie superalgebra gl(mn) over C was solved a few years ago by V. Serganova [19]. In this article, we present an entirely different approach. We also formulate a precise ..."
Abstract

Cited by 43 (5 self)
 Add to MetaCart
The problem of computing the characters of the finite dimensional irreducible representations of the Lie superalgebra gl(mn) over C was solved a few years ago by V. Serganova [19]. In this article, we present an entirely different approach. We also formulate a precise
What Can Be Computed in Algebraic Geometry?
 IN COMPUTATIONAL ALGEBRAIC GEOMETRY AND COMMUTATIVE ALGEBRA
, 1992
"... This paper evolved from a long series ofd;qIS3k);[ between the two authors, going back to around 1980, on the problems of making effective computations in algebraic geometry,and it took more dmorek shape in a survey talk given by the second author at a conference on Computer Algebra in 1984. The goa ..."
Abstract

Cited by 41 (0 self)
 Add to MetaCart
This paper evolved from a long series ofd;qIS3k);[ between the two authors, going back to around 1980, on the problems of making effective computations in algebraic geometry,and it took more dmorek shape in a survey talk given by the second author at a conference on Computer Algebra in 1984. The goal at that time was to bring together the perspectives of theoretical computer scientists and of working algebraic geometers, while laying out what we consid)S to be the main computational problems and bound on their complexity. Only part of the talk was written d wnand since that time there has been a good dW of progress. However, the material that was written up may still serve as a useful introdok;I; to some of theidkq and estimates used in thisfield (at least the edk;;; of this volume think so), even though most of the results includk here are either published elsewhere, or exist as "folktheorems" by now. The article has four sections. The first two parts are concerned with the theory of Gröbner bases; their construction provid; the foundqIk) for most computations,and their complexity d;q;k) the complexity of most techniques in this area. The first part introdok; Gröbner bases from a geometric point of view, relating them to a number ofidS which we take up in more drekG in subsequent sections. The second part dr elops the theory of Gröbner bases more carefully, from an algebraic point of view. It could be read ind end tly,and requires less background The third part is an investigation into bound in algebraic geometry of relevance to
On free conformal and vertex algebras
 J. Algebra
, 1999
"... Vertex algebras and conformal algebras have recently attracted a lot of attention due to their connections with physics and Moonshine representations of the Monster. See, for example, [6], [10], [15], [17], [19]. In this paper we describe bases of free conformal and free vertex algebras (as introduc ..."
Abstract

Cited by 27 (8 self)
 Add to MetaCart
Vertex algebras and conformal algebras have recently attracted a lot of attention due to their connections with physics and Moonshine representations of the Monster. See, for example, [6], [10], [15], [17], [19]. In this paper we describe bases of free conformal and free vertex algebras (as introduced in [6], see also [20]). All linear spaces are over a field k of characteristic 0. Throughout this paper Z+ will stand for the set of nonnegative integers. In §1 and §2 we give a review of conformal and vertex algebra theory. All statements is these sections are either in [9], [15], [16], [17], [18], [20] or easily follow from results therein. In §3 we investigate free conformal and vertex algebras. 1. Conformal algebras 1.1. Definition of conformal algebras. We first recall some basic definitions and constructions, see [16], [17], [18], [20]. The main object of investigation is defined as follows: Definition 1.1. A Conformal algebra is a linear space C endowed with a linear operator D: C → C and a sequence of bilinear products ○n: C ⊗ C → C, n ∈ Z+, such that for any a, b ∈ C one has (i) (locality) There is a nonnegative integer N = N(a, b) such that a ○n b = 0 for any n � N; (ii) D(a ○n b) = (Da) ○n b + a ○n (Db); (iii) (Da) ○n b = −na n−1 b. 1.2. Spaces of power series. Now let us discuss the main motivation for the Definition 1.1. We closely follow [14] and [18]. 1.2.1. Circle products. Let A be an algebra. Consider the space of power series A[[z, z −1]]. We will write series a ∈ A[[z, z −1]] in the form a(z) = ∑ a(n)z −n−1, a(n) ∈ A. n∈Z On A[[z, z−1]] there is an infinite sequence of bilinear products ○n, n ∈ Z+, given by n a ○n b (z) = Resw a(w)b(z)(z − w) ). (1.1) Explicitly, for a pair of series a(z) = ∑ n∈Z a(n)z−n−1 and b(z) = ∑ n∈Z b(n)z−n−1 we have −m−1 a ○n b (z) = a ○n b (m)z, where
Graded Calabi Yau algebras of dimension 3
, 2006
"... In this paper we prove that Graded Calabi Yau Algebras of dimension 3 are isomorphic to path algebras of quivers with relations derived from a superpotential. We show that for a given quiver Q and a degree d, the set of good superpotentials of degree d, i.e. those that give rise to Calabi Yau alge ..."
Abstract

Cited by 27 (3 self)
 Add to MetaCart
In this paper we prove that Graded Calabi Yau Algebras of dimension 3 are isomorphic to path algebras of quivers with relations derived from a superpotential. We show that for a given quiver Q and a degree d, the set of good superpotentials of degree d, i.e. those that give rise to Calabi Yau algebras is either empty or almost everything (in the measure theoretic sense). We also give some constraints on the structure of quivers that allow good superpotentials, and for the simplest quivers we give a complete list of the degrees for which good superpotentials exist.
A family of elliptic algebras
 Internat. Math. Res. Notices
, 1997
"... The survey is devoted to associative Z≥0graded algebras presented by n generators and n(n−1) 2 quadratic relations and satisfying the socalled PoincareBirkhoffWitt condition (PBWalgebras). We consider examples of such algebras depending on two continuous parameters (namely, on an elliptic curve ..."
Abstract

Cited by 25 (4 self)
 Add to MetaCart
The survey is devoted to associative Z≥0graded algebras presented by n generators and n(n−1) 2 quadratic relations and satisfying the socalled PoincareBirkhoffWitt condition (PBWalgebras). We consider examples of such algebras depending on two continuous parameters (namely, on an elliptic curve and a point on this curve) which are flat deformations of the polynomial ring in n variables. Diverse properties of these algebras are described, together with their relations to integrable systems, deformation quantization, moduli spaces and other directions of modern investigations.
Downup algebras
 J. Algebra
, 1998
"... Abstract. The algebra generated by the down and up operators on a differential partially ordered set (poset) encodes essential enumerative and structural properties of the poset. Motivated by the algebras generated by the down and up operators on posets, we introduce here a family of infinitedimens ..."
Abstract

Cited by 22 (3 self)
 Add to MetaCart
Abstract. The algebra generated by the down and up operators on a differential partially ordered set (poset) encodes essential enumerative and structural properties of the poset. Motivated by the algebras generated by the down and up operators on posets, we introduce here a family of infinitedimensional associative algebras called downup algebras. We show that downup algebras exhibit many of the important features of the universal enveloping algebra U(sl2) of the Lie algebra sl2 including a PoincaréBirkhoffWitt type basis and a wellbehaved representation theory. We investigate the structure and representations of downup algebras and focus especially on Verma modules, highest weight representations, and category O modules for them. We calculate the exact expressions for all the weights, since that information has proven to be particularly useful in determining structural results about posets. Combinatorial source of downup algebras. Assume P is a partially ordered set (poset), and let CP denote the complex vector space whose basis is the set P. For many posets there are two welldefined
Ideal classes of the Weyl algebra and noncommutative projective geometry
, 2001
"... Let R be the set of isomorphism classes of ideals in the Weyl algebra A = A1(C), and let C be the set of isomorphism classes of triples (V, X, Y), where V is a finitedimensional (complex) vector space, and X, Y are endomorphisms of V such that [X, Y]+I has rank 1. Following a suggestion of L. Le B ..."
Abstract

Cited by 21 (2 self)
 Add to MetaCart
Let R be the set of isomorphism classes of ideals in the Weyl algebra A = A1(C), and let C be the set of isomorphism classes of triples (V, X, Y), where V is a finitedimensional (complex) vector space, and X, Y are endomorphisms of V such that [X, Y]+I has rank 1. Following a suggestion of L. Le Bruyn, we define a map θ: R → C by appropriately extending an ideal of A to a sheaf over a quantum projective plane, and then using standard methods of homological algebra. We prove that θ is inverse to a bijection ω: C → R constructed in [BW] by a completely different method. The main step in the proof is to show that θ is equivariant with respect to natural actions of the group G = Aut(A) on R and C: for that we have to study also the extensions of an ideal to certain weighted quantum projective planes. Along the way, we find an elementary description of θ.