We define four 3×3 commuting contractions which do not dilate to commuting isometries. However they do satisfy the scalar von Neumann inequality. These matrices are all nilpotent of order 2. We also show that any three 3×3 commuting contractions which are scalar plus nilpotent of order 2 do dilate to commuting isometries.

We provide an explicit construction for a complete set of orthogonal primitive idempotents of finite group algebras over nilpotent groups. Furthermore, we give a complete set of matrix units in each simple epimorphic image of a finite group algebra of a nilpotent group.

The Leibniz algebras appeared as a generalization of the Lie algebras. In this work we deal with the classification of nilpotent complex Leibniz algebras of low dimensions. Namely, the classification of nilpotent complex Leibniz algebras dimensions less than 3 is extended to the dimension four.

We describe several approaches for realizing the Mal’cev correspondence between Q-powered nilpotent groups and nilpotent Lie algebras over Q. We apply it to fast collection in polycyclic groups. Our methods are fully implemented and publicly available. We report on the implementation and give runtimes for some example groups.

Let G be an adjoint algebraic group of type B, C, or D over an algebraically closed field of characteristic 2. We construct a Springer correspondence for the Lie algebra of G. In particular, for orthogonal Lie algebras in characteristic 2, the structure of component groups of nilpotent centralizers is determined and the number of nilpotent orbits over finite fields is obtained.

This paper completes the classification of admissible nilpotent orbits of the noncompact simple exceptional real Lie algebras. The author has previoulsly determined such orbits for exceptional real simple Lie algebras of inner type. Here he uses the same techniques, with some modifications, to classify the admissible nilpotent orbits of E6(6) and E6(−26) under their simply connected Lie groups.

We show that almost nonnegatively curved m -manifolds are, up to finite cover, nilpotent spaces in the sense of homotopy theory and have C(m)-nilpotent fundamental groups. We also show that up to a finite cover almost nonnegatively curved manifolds are fiber bundles with simply connected fibers over nilmanifolds.

For any non-torsion group G with identity e, we construct a strongly G-graded ring R such that the Jacobson radical J(Re) is locally nilpotent, but J(R) is not locally nilpotent. This answers a question posed by Puczy lowski.

Recently Havas and Vaughan-Lee proved that 4-Engel groups are locally nilpotent. Their proof relies on the fact that a certain 4-Engel group T is nilpotent and this they prove using a computer and the Knuth-Bendix algorithm. In this paper we give a short hand-proof of the nilpotency of T .

We construct the BRST operator for the non-linear WB2 and W4 algebras. Contrary to the general belief, the nilpotent condition of the BRST operator doesn’t determine all the coefficients. We find a three and seven parameter family of nilpotent BRST operator for WB2 and W4 respectively. These free parameters are related to the canonical transformation of the ghost antighost fields.