Search Results (7)

The q-Heisenberg algebra ${\mathfrak{h}}_n\left(q\right)$ is a significant class of solvable polynomial algebras, and it unifies the canonical commutation relations of Heisenberg algebras and the deformation theory of quantum groups. In this paper, we employ Gröbner-Shirshov basis theory and PBW (Poincar$\acute{e}$-Birkhoff-Witt) basis techniques to systematically investigate ${\mathfrak{h}}_n\left(q\right)$. Our main results establish that: ${\mathfrak{h}}_n\left(q\right)$ possesses an iterated skew-polynomial algebra structure, and it satisfies the important homological regularity properties of being Auslander regular, Artin-Schelter regular, and Cohen-Macaulay. These findings provide deep insights into the algebraic structure of ${\mathfrak{h}}_n\left(q\right)$, while simultaneously bridging the gap between noncommutative algebra and quantum representation theory. Furthermore, our constructive approach yields computable methods for studying modules over ${\mathfrak{h}}_n\left(q\right)$, opening new avenues for further research in deformation quantization and quantum algebra. Full article

The Clifford–Weyl algebra ${𝒜}_q^{\pm }$, as a class of solvable polynomial algebras, combines the anti-commutation relations of Clifford algebras ${𝒜}_q^{+}$ with the differential operator structure of Weyl algebras ${𝒜}_q^{-}$. It exhibits rich algebraic and geometric properties. This paper employs Gröbner–Shirshov basis principles in concert with Poincaré–Birkhoff–Witt (PBW) basis methodology to delineate the iterated skew polynomial structures within ${𝒜}_q^{+}\mathrm{and}{𝒜}_q^{-}$. By constructing explicit PBW generators, we analyze the structural properties of both algebras and their modules using constructive methods. Furthermore, we prove that ${𝒜}_q^{+}\mathrm{and}{𝒜}_q^{-}$ are Auslander regular, Cohen–Macaulay, and Artin–Schelter regular. These results provide new tools for the representation theory in noncommutative geometry. Full article

For the Temperley–Lieb algebras of type $F_n$ with $n\ge 4$, we construct their Gröbner–Shirshov bases. Explicitly, the corresponding finite sets consisting of the standard monomials of type $F_n$, which are exactly the fully commutative elements of $F_n$, are enumerated when n = 4, 5, and 6. Full article

Twenty years ago, Rota posed the problem of finding all possible algebraic identities that can be satisfied by a linear operator on an algebra, named Rota’s Classification Problem later. Rota’s Classification Problem has proceeded two steps to understand it and has been studied actively recently. In particular, the method of Gröbner-Shirshov bases has been used successfully in the study of Rota’s Classification Problem. Quite recently, a new approach introduced to Rota’s Classification Problem and classified some (new) operated polynomial identities. In this paper, we prove that all operated polynomial identities classified via this new approach are Gröbner-Shirshov. This gives a partial answer of Rota’s Classification Problem. Full article

We establish a method of Gröbner–Shirshov bases for trialgebras and show that there is a unique reduced Gröbner–Shirshov basis for every ideal of a free trialgebra. As applications, we give a method for the construction of normal forms of elements of an arbitrary trisemigroup, in particular, A.V. Zhuchok’s (2019) normal forms of the free commutative trisemigroups are rediscovered and some normal forms of the free abelian trisemigroups are first constructed. Moreover, the Gelfand–Kirillov dimension of finitely generated free commutative trialgebra and free abelian trialgebra are calculated, respectively. Full article

This paper surveys results related to well-known works of B. Plotkin and V. Remeslennikov on the edge of algebra, logic and geometry. We start from a brief review of the paper and motivations. The first sections deal with model theory. In the first part of the second section we describe the geometric equivalence, the elementary equivalence, and the isotypicity of algebras. We look at these notions from the positions of universal algebraic geometry and make emphasis on the cases of the first order rigidity. In this setting Plotkin’s problem on the structure of automorphisms of (auto)endomorphisms of free objects, and auto-equivalence of categories is pretty natural and important. The second part of the second section is dedicated to particular cases of Plotkin’s problem. The last part of the second section is devoted to Plotkin’s problem for automorphisms of the group of polynomial symplectomorphisms. This setting has applications to mathematical physics through the use of model theory (non-standard analysis) in the studying of homomorphisms between groups of symplectomorphisms and automorphisms of the Weyl algebra. The last sections deal with algorithmic problems for noncommutative and commutative algebraic geometry.The first part of it is devoted to the Gröbner basis in non-commutative situation. Despite the existence of an algorithm for checking equalities, the zero divisors and nilpotency problems are algorithmically unsolvable. The second part of the last section is connected with the problem of embedding of algebraic varieties; a sketch of the proof of its algorithmic undecidability over a field of characteristic zero is given. Full article

We construct a Gröbner-Shirshov basis of the Temperley-Lieb algebra $\mathcal{T}(d,n)$ of the complex reflection group $G(d,1,n)$ , inducing the standard monomials expressed by the generators $\left\{E_i\right\}$ of $\mathcal{T}(d,n)$ . This result generalizes the one for the Coxeter group of type $B_n$ in the paper by Kim and Lee We also give a combinatorial interpretation of the standard monomials of $\mathcal{T}(d,n)$ , relating to the fully commutative elements of the complex reflection group $G(d,1,n)$ . More generally, the Temperley-Lieb algebra $\mathcal{T}(d,r,n)$ of the complex reflection group $G(d,r,n)$ is defined and its dimension is computed. Full article