Extending Structures II: The Quantum Version

Let A be a Hopf algebra and H a coalgebra. We shall describe and classify up to an isomorphism all Hopf algebras E that factorize through A and H: that is E is a Hopf algebra such that A is a Hopf subalgebra of E, H is a subcoalgebra in E with 1_{E} \in H and the multiplication map $A\otimes H \to E$ is bijective. The tool we use is a new product, we call it the unified product, in the construction of which A and H are connected by three coalgebra maps: two actions and a generalized cocycle. Both the crossed product of an Hopf algebra acting on an algebra and the bicrossed product of two Hopf algebras are special cases of the unified product. A Hopf algebra E factorizes through A and H if and only if E is isomorphic to a unified product of A and H. All such Hopf algebras E are classified up to an isomorphism that stabilizes A and H by a Schreier type classification theorem. A coalgebra version of lazy 1-cocycles as defined by Bichon and Kassel plays the key role in the classification theorem.Comment: 24 pages, 3 figures. Final version, to appear in Journal of Algebr

A review of associative classification mining

Associative classification mining is a promising approach in data mining that utilizes the association rule discovery techniques to construct classification systems, also known as associative classifiers. In the last few years, a number of associative classification algorithms have been proposed, i.e. CPAR, CMAR, MCAR, MMAC and others. These algorithms employ several different rule discovery, rule ranking, rule pruning, rule prediction and rule evaluation methods. This paper focuses on surveying and comparing the state-of-the-art associative classification techniques with regards to the above criteria. Finally, future directions in associative classification, such as incremental learning and mining low-quality data sets, are also highlighted in this paper

Cocycle Twists and Extensions of Braided Doubles

It is well known that central extensions of a group G correspond to 2-cocycles on G. Cocycles can be used to construct extensions of G-graded algebras via a version of the Drinfeld twist introduced by Majid. We show how 2-cocycles can be defined for an abstract monoidal category C, following Panaite, Staic and Van Oystaeyen. A braiding on C leads to analogues of Nichols algebras in C, and we explain how the recent work on twists of Nichols algebras by Andruskiewitsch, Fantino, Garcia and Vendramin fits in this context. Furthermore, we propose an approach to twisting the multiplication in braided doubles, which are a class of algebras with triangular decomposition over G. Braided doubles are not G-graded, but may be embedded in a double of a Nichols algebra, where a twist may be carried out if careful choices are made. This is a source of new algebras with triangular decomposition. As an example, we show how to twist the rational Cherednik algebra of the symmetric group by the cocycle arising from the Schur covering group, obtaining the spin Cherednik algebra introduced by Wang.Comment: 60 pages, LaTeX; v2: references added, misprints correcte

The group of bi-Galois objects over the coordinate algebra of the Frobenius-Lusztig kernel of SL(2)

We construct, for q a root of unity of odd order, an embedding of the projective special linear group PSL(n) into the group of bi-Galois objects over u_q(sl(n))*, the coordinate algebra of the Frobenius-Lusztig kernel of SL(n), which is shown to be an isomorphism at n=2

Speech Recognition by Composition of Weighted Finite Automata

We present a general framework based on weighted finite automata and weighted finite-state transducers for describing and implementing speech recognizers. The framework allows us to represent uniformly the information sources and data structures used in recognition, including context-dependent units, pronunciation dictionaries, language models and lattices. Furthermore, general but efficient algorithms can used for combining information sources in actual recognizers and for optimizing their application. In particular, a single composition algorithm is used both to combine in advance information sources such as language models and dictionaries, and to combine acoustic observations and information sources dynamically during recognition.Comment: 24 pages, uses psfig.st