Related papers: Three dimensional Sklyanin algebras and Groebner b…
A finite Grobner-Shirshov basis is constructed for the plactic algebra of rank 3 over a field K. It is also shown that plactic algebras of rank exceeding 3 do not have finite Grobner-Shirshov bases associated to the natural…
This work concerns commutative algebras of the form $R=Q/I$, where $Q$ is a standard graded polynomial ring and $I$ is a homogenous ideal in $Q$. It has been proposed that when $R$ is Koszul the $i$th Betti number of $R$ over $Q$ is at most…
We construct a family of connected graded domains of GK-dimension 4 that are birational to P2, and show that the general member of this family is noetherian. This disproves a conjecture of the first author and Stafford. The algebras we…
Let $A$ be a Koszul Artin-Schelter regular algebra with Nakayama automorphism $\xi$. We show that the Yoneda Ext-algebra of the skew polynomial algebra $A[z;\xi]$ is a trivial extension of a Frobenius algebra. Then we prove that $A[z;\xi]$…
We explicitly identify the algebra generated by symplectic Fourier-Deligne transforms (i.e. convolution with Kazhdan-Laumon sheaves) acting on the Grothendieck group of perverse sheaves on the basic affine space $G/U$, answering a question…
The relation between Wilson and para-Racah polynomials and representations of the degenerate rational Sklyanin algebra is established. Second order Heun operators on quadratic grids with no diagonal terms are determined. These special or…
Using the point fusion procedure we obtain the new integrable systems from the Elliptic Schlesinger system (ESS). These new systems have the pole orders higher than one in the matrix of the Lax operator. Quadratic Poisson algebras on the…
The $SU_3$-skein algebra of a surface $F$ is spanned by isotopy classes of certain framed graphs in $F\times I$ called $3$-webs subject to the skein relations encapsulating relations between $U_q(sl(3))$-representations. These skein…
We give a description of the connected graded algebras which are finitely generated and presented of global dimension 2 or 3 and which are Gorenstein. These algebras are constructed from multilinear forms. We generalize the construction by…
We present rational Schur algebra $S(n,r,s)$ over an arbitrary ground field $K$ as a quotient of the distribution algebra $Dist(G)$ of the general linear group $G=GL(n)$ by an ideal $I(n,r,s)$ and provide an explicit description of the…
We construct three compatible quadratic Poisson structures such that generic linear combination of them is associated with Elliptic Sklyanin algebra in n generators. Symplectic leaves of this elliptic Poisson structure is studied. Explicit…
The description of number of dual (quasy)-exactly solvable models with its hidden symmetry algebra has been given at different levels of analysis within the framework of generalized Kustaanheimo-Stiefel (KS)-transformations. It's shown that…
Let $K\langle X\rangle =K\langle X_1,...,X_n\rangle$ be the free algebra of $n$ generators over a field $K$, and let $R\langle X\rangle =R\langle X_1,...,X_n\rangle$ be the free algebra of $n$ generators over an arbitrary commutative ring…
A $Z_3$-graded Hopf algebra structure of exterior algebra with two generators is introduced. Two covariant differential calculus on the $Z_3$-graded exterior algebra are presented. Using the generators and their partial derivatives a…
We provide a new foundational approach to the generalization of terms up to equational theories. We interpret generalization problems in a universal-algebraic setting making a key use of projective and exact algebras in the variety…
A major current goal of noncommutative geometry is the classification of noncommutative projective surfaces. The generic case is to understand algebras birational to the Sklyanin algebra. In this thesis we complete a considerable component…
In this paper we use Kuperberg's $\mathfrak{sl}_3$-webs and Khovanov's $\mathfrak{sl}_3$-foams to define a new algebra $K^S$, which we call the $\mathfrak{sl}_3$-web algebra. It is the $\mathfrak{sl}_3$ analogue of Khovanov's arc algebra.…
For $k\geq 0$, a $k$-generalized Markov number is an integer which appears in some positive integer solution to the $k$-generalized Markov equation $x^2 + y^2 + z^2 + k(yz + zx + xy) = (3 + 3k)xyz$. In this paper, we discuss a combinatorial…
We study the two-sided Guionnet-Jones-Shlyakhtenko construction applied to the group planar algebra $P(\mathcal{G})$ of a finite non-trivial group $\mathcal{G}$. This produces a sequence of von Neumann algebras $M^k$ for $k \geq 0$ with no…
A concise study of ternary and cubic algebras with $Z_3$ grading is presented. We discuss some underlying ideas leading to the conclusion that the discrete symmetry group of permutations of three objects, $S_3$, and its abelian subgroup…