Setting up a symbolic algebraic system is the first step in mathematics mechanization of any branch of mathematics. In this paper, we establish a compact symbolic algebraic framework for local geometric computing in intrinsic differential geometry, by choosing only the Lie derivative and the covariant derivative as basic local differential operators. In this framework, not only geometric entiti...

Simple finite group schemes over an algebraically closed field of positive characteristic p 6= 2, 3 have been classified. We consider the problem of determining their infinitesimal deformations. In particular, we compute the infinitesimal deformations of the simple finite group schemes of height one corresponding to the restricted simple Lie algebras.

Let g be a complex simple Lie algebra, and h ⊂ g be a Cartan subalgebra. In the end of 1990s, B. Kostant defined two filtrations on h, one using the Clifford algebras and the odd analogue of the Harish-Chandra projection hcodd : Cl(g) → Cl(h), and the other one using the canonical isomorphism ȟ = h∗ (here ȟ is the Cartan subalgebra in the simple Lie algebra ǧ corresponding to the dual root syst...

We present the construction of an associative, commutative algebra Ĝ(X) of generalized functions on a manifold X satisfying the following optimal set of permanence properties: (i) D(X) is linearly embedded into Ĝ(X), f(p) ≡ 1 is the unity in Ĝ(X). (ii) For every smooth vector field ξ on X there exists a derivation operator L̂ξ : Ĝ(X) → Ĝ(X) which is linear and satisfies the Leibniz rule. (iii) L...

We consider the problem of decomposing a semisimple Lie algebra deened over a eld of characteristic zero as a direct sum of its simple ideals. The method is based on the decomposition of the action of a Cartan subalgebra. An implementation of the algorithm in the system ELIAS is discussed at the end of the paper.

This paper concerns the algebraic structure of finite-dimensional complex Leibniz algebras. In particular, we introduce left central and symmetric Leibniz algebras, and study the poset of Lie subalgebras using an associative bilinear pairing taking values in the Leibniz kernel.

We study symplectic structures on characteristically nilpotent Lie algebras (CNLAs) by computing the cohomology space H(g, k) for certain Lie algebras g. Among these Lie algebras are filiform CNLAs of dimension n ≤ 14. It turns out that there are many examples of CNLAs which admit a symplectic structure. A generalization of a sympletic structure is an affine structure on a Lie algebra.

The main point of the construction of spin Calogero type classical integrable systems based on dynamical r-matrices, developed by L.-C. Li and P. Xu, is reviewed. It is shown that non-Abelian dynamical r-matrices with variables in a reductive Lie algebra F and their Abelian counterparts with variables in a Cartan subalgebra of F lead essentially to the same models.

It is an initially surprising fact how much of the geometry and arithmetic of Shimura varieties (e.g., moduli spaces of abelian varieties) is governed by the theory of linear algebraic groups. This is in some sense unfortunate, because the theory of algebraic groups (even over the complex numbers, and still more over a nonalgebraically closed field like Q) is rich and complicated, containing fo...

Let G be a connected and reductive group over the algebraically closed field K. J-P. Serre has introduced the notion of a G-completely reducible subgroup H ≤ G. In this note, we give a notion of G-complete reducibility – G-cr for short – for Lie subalgebras of Lie(G), and we show that if the smooth subgroup H < G is G-cr, then Lie(H) is G-cr as well.