Related papers: Reality of Unipotent elements in Classical Lie Gro…

In the introduction of this paper we discuss a possible approach to the unitarizability problem for classical p-adic groups. In this paper we give some very limited support that such approach is not without chance. In a forthcoming paper we…

One of the basic observations of the classical world is that physical entities are real and can be distinguished from each other. However, within quantum theory, the idea of physical realism is not well established. A framework to analyse…

We generalize the classical construction principles of infinite-dimensional real (and complex) Lie groups to the case of Lie groups over non-discrete topological fields. In particular, we discuss linear Lie groups, mapping groups, test…

The unipotent groups are an important class of algebraic groups. We show that techniques used to compute with finitely generated nilpotent groups carry over to unipotent groups. We concentrate particularly on the maximal unipotent subgroup…

In a unitary ring with involution, we prove that each element has at most one weak group inverse if and only if each idempotent element has a unique weak group inverse. Furthermore, we define the $m$-weak group inverse and show some…

Exploiting particular features of classical groups, simple constructions are given for the irreducible constituents of the tensor square of the adjoint modules and the leading terms in higher tensor powers. This provides an independent…

We show that every finite ring has a partition, where each block corresponds to one idempotent. Remarkably, this partition provides a way to \emph{lift} a wide variety of special elements such as idempotents, nilpotents, unipotents, roots…

Underlying any theory of physics is a layer of conceptual frames. They connect the mathematical structures used in theoretical models with physical phenomena, but they also constitute our fundamental assumptions about reality. Many of the…

Using the notion of existentially closed structures, we obtain embedding theorems for groups and Lie algebras. We also prove the existence of some groups and Lie algebras with prescribed properties.

Let $G$ be a real reductive group in Harish-Chandra's class. We derive some consequences of theory of coherent continuation representations to the counting of irreducible representations of $G$ with a given infinitesimal character and a…

Let $\mathcal{N}_{\mathfrak{g}^*}$ be the variety of nilpotent elements in the dual of the Lie algebra of a reductive algebraic group over an algebraically closed field. In \cite{Lu2} Lusztig proposes a definition of a partition of…

In physics, Lie groups represent the algebraic structure that describes symmetry transformations of a given system. Then, the descending Lie algebra of those groups are necessarily real. In most cases, the complexification of those Lie…

We investigate the notion of real form of complex Lie superalgebras and supergroups, both in the standard and graded version. Our functorial approach allows most naturally to go from the superalgebra to the supergroup and retrieve the real…

Let $R$ be a commutative ring of characteristic zero and $G$ an arbitrary group. In the present paper we classify the groups $G$ for which the set of symmetric elements with respect to the classical involution of the group ring $RG$ is Lie…

For any ring \(R\), some characterizations are obtained for unit regular elements in a corner ring \(eRe\) in terms of unit regular elements in \(R\). \noindent {\bf Key Words}: von Neumann regular rings, unit regular rings, corner rings,…

In this paper, we formulate the notion of split elements of a unipotent class in a connected reductive group $G$. Generalized Green functions of $G$ can be computed by using Lusztig's algorithm, if split elements exist for any unipotent…

Lie algebras are an important class of algebras which arise throughout mathematics and physics. We report on the formalisation of Lie algebras in Lean's Mathlib library. Although basic knowledge of Lie theory will benefit the reader, none…

We call a finite group irrational if none of its elements is conjugate to a distinct power of itself. We prove that those groups are solvable and describe certain classes of these groups, where the above property is only required for…

The Joseph ideal is a special ideal in the universal enveloping algebra of a simple Lie algebra. For the classical groups, its uniqueness is equivalent to the existence of tensors with special properties. The existence of these tensors is…

In order to tackle the problem of generically determining the character tables of the finite groups of Lie type $\mathbf{G}(q)$ associated to a connected reductive group $\mathbf{G}$ over $\overline{\mathbb F}_p$, Lusztig developed the…