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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…
After a short introduction on Clifford algebras of polynomials, we give a general method of constructing a matrix representation. This process of linearization leads naturally to two fundamental structures: the generalized Clifford algebra…
We define the Higgs algebra $\mathcal{H}_\P1$ of the projective line, as a convolution algebra of constructible functions on the global nilpotent cone $\underline{\Lambda}_\P1$, a lagrangian substack of the Higgs bundle $T^*\Coh_\P1$, where…
These are significantly expanded lecture notes for the author's minicourse at MSRI in June 2012, as published in the MSRI lecture note series, with some minor additional corrections. In these notes, we give an example-motivated review of…
Using constructions of Hirsch and Hodkinson, we show that the class of strongly atom structures for various cylindric-like algebras is not elementary. This applies to diagonal free reducts and polyadic algebras with and without equality.…
We discuss multiplicative properties of the binary quadratic form $a x^2 + b x y + c y^2$ by considering a ring of matrices which is closed under a triple product. We prove that the ring forms a ternary algebra in the sense of Hestenes, and…
Quantum theory (QT), namely in terms of Schr\"odinger's 1926 wave functions in general requires complex numbers to be formulated. However, it soon turned out to even require some hypercomplex algebra. Incorporating Special Relativity leads…
We introduce C-Algebras (quantum analogues of compact Riemann surfaces), defined by polynomial relations in non-commutative variables and containing a real parameter that, when taken to zero, provides a classical non-linear,…
We introduce two classes of algebras coming from partial triangulations of marked surfaces. The first one, called frozen algebra of a partial triangulation, is generally of infinite rank and contains frozen Jacobian algebras of…
The recent focus on deformations of algebras called quantum algebras can be attributed to the fact that they appear to be the basic algebraic structures underlying an amazingly diverse set of physical situations. To date many interesting…
By using the approach in \cite{XX2006} to Hall algebras arising in homologically finite triangulated categories, we find an `almost' associative multiplication structure for indecomposable objects in a 2-periodic triangulated category. As…
Nichols algebras of group type with many cubic relations are classified under a technical assumption on the structure of Hurwitz orbits of the third power of the underlying indecomposable rack. All such Nichols algebras are…
To any finite group G of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, H_k, of the smash product of G with the polynomial algebra on V. The algebra H_k, called a symplectic reflection algebra,…
This work introduces a novel $q$-$\hbar$ deformation of the Heisenberg algebra, designed to unify and extend several existing $q$-deformed formulations. Starting from the canonical Heisenberg algebra defined by the commutation relation…
The relation between nonlinear algebras and linear ones is established. For one-dimensional nonlinear deformed Heisenberg algebra with two operators we find the function of deformation for which this nonlinear algebra can be transformed to…
We examine classes of quantum algebras emerging from involutive, non-degenerate set-theoretic solutions of the Yang-Baxter equation and their q-analogues. After providing some universal results on quasi-bialgebras and admissible Drinfeld…
Hermitian cubic norm structures were recently introduced in order to study the class of skew-dimension one structurable algebras (which are typically only defined over fields of characteristic different from $2$ and $3$) over arbitrary…
We introduce the notion of quasi-triangular Novikov bialgebras, which constructed from solutions of the Novikov Yang-Baxter equation whose symmetric parts are invariant. Triangular Novikov bialgebras and factorizable Novikov bialgebras are…
We investigate the deformations and rigidity of boundary Heisenberg-like algebras. In particular, we focus on the Heisenberg and $\text{Heisenberg}\oplus\mathfrak{witt}$ algebras which arise as symmetry algebras in three-dimensional gravity…
Commutative Hilbertian Frobenius algebras are those commutative semi-group objects in the monoidal category of Hilbert spaces, for which the Hilbert adjoint of the multiplication satisfies the Frobenius compatibility relation, that is, this…