Chevalley Bases Research Articles

Canonical structure constants for simple Lie algebras

AbstractLet $$\mathfrak {g}$$ g be a finite-dimensional simple Lie algebra over $$\mathbb {C}$$ C . In the 1950s Chevalley showed that $$\mathfrak {g}$$ g admits particular bases, now called “Chevalley bases”, for which the corresponding structure constants are integers. Such bases are not unique but, using Lusztig’s theory of canonical bases, one can single out a “canonical” Chevalley basis which is unique up to a global sign. In this paper, we give explicit formulae for the structure constants with respect to such a basis.

Integral Basis of Affine Vertex Algebra <i>V<sub>k</sub></i> (sl<sub>2</sub>) and Virasoro Vertex Algebra <i>V</i><sub><i>vir</i></sub> (2<i>k</i>,0)

In this paper, we consider an integral basis for affine vertex algebra Vk (sl2) when the level k is integral by a direct calculation, then use the similar way to analyze an integral basis for Virasoro vertex algebra Vvir (2k,0). Finally, we take the combination of affine algebras and Virasoro Lie algebras into consideration. By analogy with the construction of Lie algebras over Z using Chevalley bases, we utilize the Z-basis of Lav whose structure constants are integral to find an integral basis for the universal enveloping algebra of it.

On the construction of semisimple Lie algebras and Chevalley groups

Let g \mathfrak {g} be a semisimple complex Lie algebra. Recently, Lusztig simplified the traditional construction of the corresponding Chevalley groups (of adjoint type) using the “canonical basis” of the adjoint representation of g \mathfrak {g} . Here, we present a variation of this idea which leads to a new, and quite elementary, construction of g \mathfrak {g} itself from its root system. An additional feature of this set-up is that it also gives rise to explicit Chevalley bases of g \mathfrak {g} .

Algorithmic construction of Chevalley bases

AbstractWe present a new algorithm for constructing a Chevalley basis for any Chevalley Lie algebra over a finite field. This is a necessary component for some constructive recognition algorithms of exceptional quasisimple groups of Lie type. When applied to a simple Chevalley Lie algebra in characteristic p⩾5, our algorithm has complexity involving the seventh power of the Lie rank, which is likely to be close to best possible.

Chevalley supergroups

In the framework of algebraic supergeometry, we give a construction of the scheme-theoretic supergeometric analogue of Chevalley groups, namely affine algebraic supergroups associated to simple Lie superalgebras of classical type. In particular both basic (even exceptional) and strange cases are covered. This provides a unified approach to most of the algebraic supergroups considered so far in literature, and an effective method to construct new ones. As an intermediate step, we prove an existence theorem for Chevalley bases of simple classical Lie superalgebras and a PBW-like theorem for their associated Kostant superalgebras.

Computing Chevalley bases in small characteristics

Let L be the Lie algebra of a simple algebraic group defined over a field F and let H be a split maximal toral subalgebra of L. Then L has a Chevalley basis with respect to H. If char ( F ) ≠ 2 , 3 , it is known how to find it. In this paper, we treat the remaining two characteristics. To this end, we present a few new methods, implemented in Magma, which vary from the computation of centralizers of one root space in another to the computation of a specific part of the Lie algebra of derivations of L.

Blocks in weight spaces in the root lattice

It is useful to think of the p-restricted modular Weyl problem in eigenvalueeeigenvector terms. This problem asks for the character of the irreducible p-version of an irreducible module V(A) for a complex semisimple Lie algebra, cf. Humphreys [6]. The eigenvalue results concern the weight lattice of the root system and especially n(A), the saturated set of weights occuring in the module. The eigenvector results are better indexed by 17,(A), the root weights. If p E D(A), then ,~l= ,I T, where r E A:, the positive cone of the root lattice of the root system. So we have p E Z7(A) iff T E n,(n). Consider, for instance, the problem of finding p-modular maximal vectors, cf. [4]. The main eigenvalue result, Kac-Weisfeiler [S], finds that a necessary condition for p E n(A) to be the weight of such a vector is that ,u be p-linked to A. This condition is ultimately descended from the shape of the Weyl character formula. The main p-restricted eigenvector result, [4], yields that a necessary condition for r E n,(A) to be the root weight of such a vector is that r be in the Kostant cone of A,‘, i.e., that r can be written as a sum of positive nonsimple roots. This condition comes from some combinatorial properties of Kostant’s partition function and Chevalley bases. For another kind of eigenvalue-eigenvector result, see Franklin [lo]. This paper, a sequel to [3], provides further foundational results for the eigenvector approach to this restricted modular Weyl problem, as well as deep insights into the combinatorial and geometrical structure of P(z), Kostant’s partition function, and its relations with structures on weight spaces in the root lattice. Let R be a root system and A:, the positive cone of the root lattice with