Tag - Non-associative rings

In my talk I would like to discuss my joint articles with S. Sierra about the primitive ideals of universal enveloping U(W) and the symmetric algebra S(W) of Witt Lie algebra W and similar Lie algebras (including Virasoro Lie algebra). The key theorem in this setting is that every nontrivial quotient by a two-sided ideal of U(W) or S(W) has finite Gelfand-Kirillov dimension. Together with Sierra we enhanced this statement to the description of primitive Poisson ideals of S(W) in terms of certain points on the complex plane plus a few parameters attached to these points. In the end I will try to explain how all these concepts works for the ideals whose quotient has Gelfand-Kirillov dimension 2.

We say that an element x in a ring R is nilpotent last-regular if it is nilpotent of certain index n+1 and its last nonzero power xⁿ is regular von Neumann, i.e., there exists another element y∈R such that xⁿyxⁿ=xⁿ. This type of elements naturally arise when studying certain inner derivations in the Lie algebra Skew(R,∗) of a ring R with involution ∗ whose indices of nilpotence differ when considering them acting as derivations on Skew(R,∗) and on the whole R. When moving to the symmetric Martindale ring of quotients Q_m^s(R) of R we still obtain inner derivations with the same indices of nilpotence on Q_m^s(R) and on the skew-symmetric elements Skew(Q_m^s(R),∗) of Q_m^s(R), but with the extra condition of being generated by a nilpotent last-regular element. This condition strongly determines the structure of Q_m^s(R) and of Skew(Q_m^s(R),∗). We will review the Jordan canonical form of nilpotent last-regular elements and show how to get gradings in associative algebras (with and without involution) when they have such elements.

The problem of determining centralizers in the enveloping algebras of Lie algebras is considered from both the algebraic and analytical perspectives. Applications of the procedure, such as the decomposition problem of the enveloping algebra of a simple Lie algebra, the labelling problem, and the construction of orthonormal bases of states are considered.

We say that a Lie ring R is called a unique addition Lie ring, or briefly a UA-Lie ring, if any commutator-preserving bijection on R preserves the addition as well. We prove that any semisimple Lie algebra and any its parabolic subalgebra is a UA-Lie ring. Also we describe wide classes of solvable UA-Lie rings.

In recent joint work with D. Nakano, an analogue 𝒩 of the nilpotent cone was constructed for classical simple Lie superalgebras, and 𝒩 was shown to consist of only finitely many nilpotent orbits. In this talk, we determine several geometric properties such as the dimension and irreducibility of 𝒩 for the Lie superalgebra 𝔤𝔩(m|n), and we give a more detailed description of the geometry and structure of the nilpotent orbits in this case. We also demonstrate connections between our nilpotent orbit representatives and certain signed Young diagrams appearing in the work of Kraft and Procesi.