Related papers: Sklyanin algebras and a cubic root of 1

We compute the superpotentials for known families of Koszul Artin-Schelter regular algebras of dimension four using Magma, and apply Schur-Weyl duality from representation theory to determine the relevant invariants. Through the Borel-Weil…

The class of $N$-Koszul graded algebras of finite global dimension has gained lots of attention in recent years, especially in the study of Artin-Schelter regular algebras. While structurally rich and concrete, the only known examples of…

We introduce a class of finite dimensional nonlinear superalgebras $L = L_{\bar{0}} + L_{\bar{1}}$ providing gradings of $L_{\bar{0}} = gl(n) \simeq sl(n) + gl(1)$. Odd generators close by anticommutation on polynomials (of degree $>1$) in…

We consider integrable vertex models whose Boltzmann weights (R-matrices) are trigonometric solutions to the graded Yang-Baxter equation. As is well known the latter can be generically constructed from quantum affine superalgebras…

Let $X= \{x_1, x_2, \cdots, x_n\}$ be a finite alphabet, and let $K$ be a field. We study classes $\mathfrak{C}(X, W)$ of graded $K$-algebras $A = K\langle X\rangle / I$, generated by $X$ and with a fixed set of obstructions $W$. Initially…

We study the defining equations of the Rees algebra of square-free monomial ideals in a polynomial ring over a field. We determine that when an ideal $I$ is generated by $n$ square-free monomials of the same degree then $I$ has relation…

Let $S$ be an $\mathbb N$-graded Koszul Artin-Schelter regular algebra and let $\sigma$ be a graded algebra automorphism of $S$. We study the stable category of graded maximal Cohen-Macaulay modules over the trivial extension algebra…

Let $\mathcal{G}$ denote the variety generated by infinite dimensional Grassmann algebras; i.e., the collection of all unitary associative algebras satisfying the identity $[[z_1,z_2],z_3]=0$, where $[z_i,z_j]=z_iz_j-z_jz_i$. Consider the…

We discuss certain homological properties of graded algebras whose trivial modules admit non-pure resolutions. Such algebras include both of Artin-Schelter regular algebras of types (12221) and (13431). Under certain conditions, a module…

We define and study the ternary analogues of Clifford algebras. It is proved that the ternary Clifford algebra with $N$ generators is isomorphic to the subalgebra of the elements of grade zero of the ternary Clifford algebra with $N+1$…

We study the algebraic and geometric properties of stated skein algebras of surfaces with punctured boundary. We prove that the skein algebra of the bigon is isomorphic to the quantum group ${\mathcal O}_{q^2}(\mathrm{SL}(2))$ providing a…

In this paper, using pseudo path algebras, we generalize Gabriel's Theorem on elementary algebras to left Artinian algebras over a field $k$ when it is splitting over its radical, in particular, when the dimension of the quotient algebra…

We show that each irreducible tensor representation of weight 2 of the rotation group of three-dimensional space in the space of rank 3 covariant tensors gives rise to an associative algebra with unity. We find the algebraic relations that…

Let $K$ be an arbitrary field of characteristic zero, $P_n:= K[ x_1, ..., x_n]$ be a polynomial algebra, and $P_{n, x_1}:= K[x_1^{-1}, x_1, ..., x_n]$, for $n\geq 2$. Let $\s' \in {\rm Aut}_K(P_n)$ be given by $$ x_1\mapsto x_1-1, \quad…

In this paper, which is a follow-up to my paper with Yonezawa "sl(N)-web categories", I define and study sl(N)-web algebras for any N greater than one. For N=2 these algebras are isomorphic to Khovanov's arc algebras and for N=3 they are…

The $12$-dimensional Fomin-Kirillov algebra $FK_3$ is defined as the quadratic algebra with generators $a$, $b$ and $c$ which satisfy the relations $a^2=b^2=c^2=0$ and $ab+bc+ca=0=ba+cb+ac$. By a result of A. Milinski and H.-J. Schneider,…

We consider the double affine Hecke algebra $H=H(k_0,k_1,k^\vee_0,k^\vee_1;q)$ associated with the root system $(C^\vee_1,C_1)$. We display three elements $x$, $y$, $z$ in $H$ that satisfy essentially the $Z_3$-symmetric Askey-Wilson…

Starting from Sklyanin's separation of variables for the sl3 Yangian model, we derive the separation of variables for the quantum sl3 Gaudin model. We use the resulting new variables for rewriting the sl3 Knizhnik-Zamolodchikov equations,…

This paper improves several previously known results. First, the results describing the R-skewsymmetric algebra and the quadratic dual of the R-symmetric algebra as Frobenius algebras are shown to be true with any restriction on the…

In this paper, we compute the Hochschild homology of elliptic Sklyanin algebras. These algebras are deformations of polynomial algebra with a Poisson bracket called the Sklyanin Poisson bracket.