arXiv subjects
Publications and source records attributed to Willem A. de Graaf.
Let g be a semisimple Lie algebra over the real numbers. We describe an explicit combinatorial construction of the real Weyl group of g with respect to a given Cartan subalgebra. An efficient computation of this Weyl group is important for the classification of regular semisimple subalgebras, real carrier algebras, and real nilpotent orbits associated with g; the latter have various applications in theoretical physics.
We construct generating pairs of simple Lie algebras in characteristic zero. We apply this construction to exhibit infinite series of 2-generator Zariski dense subgroups that are free of rank 2 of the simple algebraic groups SL(n, C), Sp(n, C), G_2(C).
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.
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.
In the classification of stationary solutions in extended supergravities with symmetric scalar manifolds, the nilpotent orbits of a real symmetric pair play an important role. In this paper we discuss two approaches to determining the nilpotent orbits of a real symmetric pair. We apply our methods to an explicit example, and thereby classify the nilpotent orbits of SL_2(R)^4 acting on the fourth tensor power of the natural 2-dimensional SL_2(R)-module. This makes it possible to classify all stationary solutions of the so-called STU-supergravity model.
For a semisimple Lie algebra over the complex numbers, Dynkin (1952) developed an algorithm to classify the regular semisimple subalgebras, up to conjugacy by the inner automorphism group. For a graded semisimple Lie algebra over the complex numbers, Vinberg (1979) showed that a classification of a certain type of regular subalgebras (called carrier algebras) yields a classification of the nilpotent orbits in a homogeneous component of that Lie algebra. Here we consider these problems for (graded) semisimple Lie algebras over the real numbers. First, we describe an algorithm to classify the regular semisimple subalgebras of a real semisimple Lie algebra. This also yields an algorithm for listing, up to conjugacy, the carrier algebras in a real graded semisimple real algebra. We then discuss what needs to be done to obtain a classification of the nilpotent orbits from that; such classifications have applications in differential geometry and theoretical physics. Our algorithms are implemented in the language of the computer algebra system GAP, using our package CoReLG; we report on example computations.
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.
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.
We describe an algorithm for obtaining generators of the unit group of the integral group ring ZG of a finite abelian group G. We used our implementation in Magma of this algorithm to compute the unit groups of ZG for G of order up to 110. In particular for those cases we obtained the index of the group of Hoechsmann units in the full unit group. At the end of the paper we describe an algorithm for the more general problem of finding generators of an arithmetic group corresponding to a diagonalizable algebraic group.
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.
Let g be a real form of a simple complex Lie algebra. Based on ideas of Djokovic and Vinberg, we describe an algorithm to compute representatives of the nilpotent orbits of g using the Kostant-Sekiguchi correspondence. Our algorithms are implemented for the computer algebra system GAP and, as an application, we have built a database of nilpotent orbits of all real forms of simple complex Lie algebras of rank at most 8. In addition, we consider two real forms g and g' of a complex simple Lie algebra g^c with Cartan decompositions g= k+p and g'=k'+p'. We describe an explicit construction of an isomorphism g -> g', respecting the given Cartan decompositions, which fails if and only if g and g' are not isomorphic. This isomorphism can be used to map the representatives of the nilpotent orbits of g to other realisations of the same algebra.
We describe a method for classifying the Novikov algebras with a given associated Lie algebra. Subsequently we give the classification of the Novikov algebras of dimension 3 over R and C, as well as the classification of the 4-dimensional Novikov algebras over C whose associated Lie algebra is nilpotent. In particular this includes a list of all 4-dimensional commutative associative algebras over C.
Algorithms are described that help with obtaining a classification of the semisimple subalgebras of a given semisimple Lie algebra, up to linear equivalence. The algorithms have been used to obtain classifications of the semisimple subalgebras of the simple Lie algebras of ranks <= 8. These have been made available as a database inside the SLA package of GAP4. The subalgebras in this database are explicitly given, as well as the inclusion relations among them.
Let $Q$ be an algebraic group with $q=\Lie Q$ and $V$ a $Q$-module. The index of $V$ is the minimal codimension of the $Q$-orbits in the dual space $V^*$. There is a general inequality, due to Vinberg, relating the index of $V$ and the index of a $Q_v$-module $V/q.v$ for $v\in V$. A pair $(Q,V)$ is said to have GIB if Vinberg's inequality turns into an equality for all $v\in V$. In this article, we are interested in the GIB property of $θ$-representations, where $θ$ is a finite order automorphism of a simple Lie algebra $g$. An automorphism of order $m$ defines a $Z/mZ$-grading $g=g_0+g_1+...+g_{m-1}$. If $G_0$ is the identity component of $G^θ$, then it acts on $\gt g_1$ and this action is called a $θ$-representation. We classify inner automorphisms of $gl_n$ and all finite order autmorphisms of the exceptional Lie algebras such that $(G_0,g_1)$ has GIB and $g_1$ contains a semisimple element.
We describe two algorithms for finding representatives of the nilpotent orbits of a theta-group. The algorithms have been implemented in the computer algebra system GAP (inside the package SLA). We comment on their performance. We apply the algorithms to study the nilpotent orbits of theta-groups, where theta is an N-regular automorphism of a simple Lie algebra of exceptional type.
For a Del Pezzo surface of degree 8 given over the rationals we decide whether there is a rational parametrization of the surface and construct one in the affirmative case. We define and use the Lie algebra of the surface to reach the aim. The algorithm has been implemented in Magma.
It is well-known that a Severi-Brauer surface has a rational point if and only if it is isomorphic to the projective plane. Given a Severi-Brauer surface, we study the problem to decide whether such an isomorphism to the projective plane, or such a rational point, does exist; and to construct such an isomorphism or such a point in the affirmative case. We give an algorithm using Lie algebra techniques. The algorithm has been implemented in Magma.