Related papers: The Onsager Algebra
Let $\mathfrak{g}(A)$ be the Kac-Moody algebra with respect to a symmetrizable generalized Cartan matrix $A$. We give an explicit presentation of the fix-point Lie subalgebra $\mathfrak{k}(A)$ of $\mathfrak{g}(A)$ with respect to the…
A finite-dimensional unital and associative algebra over $\mathbb{R}$, or what we shall call simply "an algebra" in this paper for short, generalities the construction by which we derive the complex numbers by "adjoining an element $i$" to…
This thesis details the results of four interrelated projects. The first of these presents a new proof of the theorem of Cooper, Danciger and Wienhard classifying the limits under conjugacy of the orthogonal groups in GL(n; R). The second…
Pascal Baseilhac and Stefan Kolb recently introduced the Lusztig automorphism $L$ of the $q$-Onsager algebra $\mathcal O_q$. In this paper, we express each of $L, L^{-1}$ as a formal sum involving some quantum adjoints. In addition, (i) we…
We give a presentation of the centralizer algebras for tensor products of spinor representations of quantum groups via generators and relations. In the even-dimensional case, this can be described in terms of non-standard q-deformations of…
In [Lu6] Lusztig defined a certain algebra $H,$ which is a direct sum of various algebras $H_{\mathfrak{o}}.$ We establish an explicit algebra isomorphism between each algebra $H_{\mathfrak{o}}$ and some matrix algebra with coefficients in…
From a commutative associative algebra $A$, the infinite dimensional unital 3-Lie Poisson algebra~$\mathfrak{L}$~is constructed, which is also a canonical Nambu 3-Lie algebra, and the structure of $\mathfrak{L}$ is discussed. It is proved…
There exists two types of nonassociative algebras whose associator satisfies a symmetric relation associated with a 1-dimensional invariant vector space with respect to the natural action of the symmetric group on three elements. The first…
We introduce and study a new class of algebras, which we name \textit{quantum generalized Heisenberg algebras} and denote by $\mathcal{H}_q (f,g)$, related to generalized Heisenberg algebras, but allowing more parameters of freedom, so as…
We show how the Onsager algebra, used in the original solution of the two-dimensional Ising model, arises as an infinite-dimensional symmetry of certain self-dual models that also have a $U(1)$ symmetry. We describe in detail the example of…
In this paper we study the complete reducibility of representations of infinite-dimensional Lie algebras from the perspective of the representation theory of vertex algebras.
In the framework of operator theory, we investigate a close Lie theoretic relationship between all operator ideals and certain classical groups of invertible operators that can be described as the solution sets of certain algebraic…
Let W be a finite Coxeter group. We define its Hecke-group algebra by gluing together appropriately its group algebra and its 0-Hecke algebra. We describe in detail this algebra (dimension, several bases, conjectural presentation,…
The algebra generated by the down and up operators on a differential partially ordered set (poset) encodes essential enumerative and structural properties of the poset. Motivated by the algebras generated by the down and up operators on…
Poisson algebra is usually defined to be a commutative algebra together with a Lie bracket, and these operations are required to satisfy the Leibniz rule. We describe Poisson structures in terms of a single bilinear operation. This enables…
Revisiting the results by Winternitz [Symmetry in physics, CRM Proc. Lecture Notes 34, American Mathematical Society, Providence, RI, 2004, pp. 215-227], we thoroughly refine his classification of Lie subalgebras of the real order-three…
Since the introduction of the concept of isotopism of algebras by Albert in 1942, a prolific literature on the subject has been developed for distinct types of algebras. Nevertheless, there barely exists any result on the problem of…
In this article we give a short and informal overview of some aspects of the theory of C*- and von Neumann algebras. We also mention some classical results and applications of these families of operator algebras.
We introduce the notion of omni-Lie superalgebra as a super version of the omni-Lie algebra introduced by Weinstein. This algebraic structure gives a nontrivial example of Leibniz superalgebra and Lie 2-superalgebra. We prove that there is…
A well-known result on Lie Theory states that every finite-dimensional complex solvable Lie algebra can be represented as a matrix Lie algebra, with upper-triangular square matrices as elements. However, this result does not specify which…