Related papers: Finite-Dimensional Calculus
Given a map $\Xi\colon U(\mathfrak{g})\rightarrow A$ of associative algebras, with $U(\mathfrak{g})$ the universal enveloping algebra of a (complex) finite-dimensional reductive Lie algebra $\mathfrak{g}$, the restriction functor from…
We introduce a twisted quantum affine algebra associated to each simply laced finite dimensional simple Lie algebra. This new algebra is a Hopf algebra with a Drinfeld-type comultiplication. We obtain this algebra by considering its vertex…
Roughly speaking, RA(=real analysis) means to study properties of (smooth) functions defined on real space, and CA(=complex analysis) stands for studying properties of (holomorphic) functions defined on spaces with complex structure. But to…
For any arbitrary algebraic curve, we define an infinite sequence of invariants. We study their properties, in particular their variation under a variation of the curve, and their modular properties. We also study their limits when the…
In this work we are motivated by factorization of bosonic quantum dynamics and we study the corresponding Lie algebras, which can potentially be infinite dimensional. To characterize such factorization, we identify conditions for these Lie…
In this survey, we first present basic facts on A-infinity algebras and modules including their use in describing triangulated categories. Then we describe the Quillen model approach to A-infinity structures following K. Lefevre's thesis.…
In pursuit of a noncommutative spectrum functor, we argue that the Heyneman-Sweedler finite dual coalgebra can be viewed as a quantization of the maximal spectrum of a commutative affine algebra, integrating prior perspectives of Takeuchi,…
We consider deformations of quantum exterior algebras extended by finite groups. Among these deformations are a class of algebras which we call truncated quantum Drinfeld Hecke algebras in view of their relation to classical Drinfeld Hecke…
The notion of the eigenvalue problem in the Fock space with polynomial eigenfunctions is introduced. This problem is classified by using the finite-dimensional representations of the $\mathfrak{sl}(2)$-algebra in Fock space. In the complex…
The Hirzebuch-Riemann-Roch (HRR) and Lefschetz type formulas for finite dimensional elementary algebras of finite global dimension are explicitly given. They have cohomological, homological, Hochschild cohomological and Hochschild…
We study finite-dimensional representations of hyper loop algebras over non-algebraically closed fields. The main results concern the classification of the irreducible representations, the construction of the Weyl modules, base change,…
Multivariate extensions of the Krawtchouk polynomials have been studied by numerous authors in recent decades by exploring new connections to probability, representation theory and quantum integrability. We develop a theory of multivariate…
This paper is based on the introduction to the monograph ``Double affine Hecke algebras'' to be published by Cambridge University Press. The connections with Knizhnik-Zamolodchikov equations, Kac-Moody algebras, tau-function, harmonic…
The Hecke-Kiselman algebra of a finite oriented graph $\Theta$ over a field $K$ is studied. If $\Theta$ is an oriented cycle, it is shown that the algebra is semiprime and its central localization is a finite direct product of matrix…
Let $K$ be a field, $Q$ a quiver, and $\mathcal{A}$ the ideal of the path algebra $KQ$ that is generated by the arrows of $Q$. We present old and new results about the representation theories of the truncations $KQ/\mathcal{A}^L$, $L \in…
The notion of trigonometric spin double affine Hecke algebras (tsDaHa) and trigonometric double affine Hecke-Clifford algebras (tDaHCa) associated to classical Weyl groups are introduced. The PBW basis property is established. An algebra…
We consider the polynomial representation of Double Affine Hecke Algebras (DAHAs) and construct its submodules as ideals of functions vanishing on the special collections of affine planes. This generalizes certain results of Kasatani in…
This is an overview article on finite-dimensional algebras and quivers, written for the Encyclopedia of Mathematical Physics. We cover path algebras, Ringel-Hall algebras and the quiver varieties of Lusztig and Nakajima.
An adaptation of Kieu's hypercomputational quantum algorithm (KHQA) is presented. The method that was used was to replace the Weyl-Heisenberg algebra by other dynamical algebra of low dimension that admits infinite-dimensional irreducible…
We construct a wide class of finite W-algebras as truncations of Yangians. These truncations correspond to algebra homomorphisms and allow to construct the W-algebras as exchange algebras, the R-matrix being the Yangian's one. As an…