Consecutive Support: Better Be Close!

We propose a new measure of support (the number of occur- rences of a pattern), in which instances are more important if they occur with a certain frequency and close after each other in the stream of trans- actions. We will explain this new consecutive support and discuss how patterns can be found faster by pruning the search space, for instance using so-called parent support recalculation. Both consecutiveness and the notion of hypercliques are incorporated into the Eclat algorithm. Synthetic examples show how interesting phenomena can now be discov- ered in the datasets. The new measure can be applied in many areas, ranging from bio-informatics to trade, supermarkets, and even law en- forcement. E.g., in bio-informatics it is important to find patterns con- tained in many individuals, where patterns close together in one chro- mosome are more significant.Comment: 10 page

Constructing homomorphisms between Verma modules

A practical method for constructing a nontrivial homomorphsim between Verma modules is described.Comment: 15 pages, added references, some comments on the affine case, an application concerning the construction of irreducible module

Computations with nilpotent orbits in SLA

We report on some computations with nilpotent orbits in simple Lie algebras of exceptional type within the SLA package of GAP4. Concerning reachable nilpotent orbits our computations firstly confirm the classification of such orbits in Lie algebras of exceptional type by Elashvili and Grelaud, secondly they answer a question by Panyushev, and thirdly they show in what way a recent result of Yakimova for the Lie algebras of classical type extends to the exceptional types. The second topic of this note concerns abelianizations of centralizers of nilpotent elements. We give tables with their dimensions.Comment: This paper contains arXiv:1004.4061 [math.RA] (except the appendix

An algorithm to compute the canonical basis of an irreducible Uq(g)-module

An algorithm is described to compute the canonical basis of an irreducible module over a quantized enveloping algebra of a finite-dimensional semisimple Lie algebra. The algorithm works for modules that are constructed as a submodule of a tensor product of modules with known canonical bases.Comment: 12 page

Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2

First we describe the Skjelbred-Sund method for classifying nilpotent Lie algebras. Then we use it to classify 6-dimensional nilpotent Lie algebras over any field of characteristic not 2. The proof of this classification is essentially constructive: for a given 6-dimensional nilpotent Lie algebra L, following the steps of the proof, it is possible to find a Lie algebra M that occurs in the list, and an isomorphism L --> M.Comment: 23 page

Classification of solvable Lie algebras

We illustrate some simple ideas that can be used for obtaining a classification of small-dimensional solvable Lie algebras.Using these we obtain the classification of 3 and 4 dimensional solvable Lie algebras (over fields of any characteristic). Precise conditions for isomorphism are given.Comment: 16 page

Classification of nilpotent associative algebras of small dimension

We classify nilpotent associative algebras of dimensions up to 4 over any field. This is done by constructing the nilpotent associative algebras as central extensions of algebras of smaller dimension, analogous to methods known for nilpotent Lie algebras

Closed subsets of root systems and regular subalgebras

We describe an algorithm for classifying the closed subsets of a root system, up to conjugation by the associated Weyl group. Such a classification of an irreducible root system is closely related to the classification of the regular subalgebras, up to inner automorphism, of the corresponding simple Lie algebra. We implement our algorithm to classify the closed subsets of the irreducible root systems of ranks 3 through 7. We present a complete description of the classification for the closed subsets of the rank 3 irreducible root system. We employ this root system classification to classify all regular subalgebras of the rank 3 simple Lie algebras. We present only summary data for the classifications in higher ranks due to the large size of these classifications. Our algorithm is implemented in the language of the computer algebra system GAP

Constructing semisimple subalgebras of real semisimple Lie algebras

We consider the problem of constructing semisimple subalgebras of real (semi-) simple Lie algebras. We develop computational methods that help to deal with this problem. Our methods boil down to solving a set of polynomial equations. In many cases the equations turn out to be sufficiently "pleasant" to be able to solve them. In particular this is the case for S-subalgebras

E7⊂Sp(56,R)E_7 \subset Sp(56,R) irrep decompositions of interest for physical models

In this note we show how to obtain the projection matrix for the $E_7 \subset C_{28}$ chain and we tabulate some decompositions of the symplectic algebra $C_{28}$ representations into irreps of the $E_7$ subalgebra that are important for various physical models