We consider algebras that can be realized as PBW deformations of (Artin-Schelter) regular algebras. This is equivalent to the homogenization of the algebra being regular. It is shown that the homogenization, when it is a geometric algebra, contains a component whose points are in 1-1 correspondence with the simple modules of the deformation. We classify all PBW deformations of 2-dimensional regular algebras and give examples of 3-dimensional deformations. Other properties, such as the skew Calabi-Yau property and closure under tensor products, are considered.

When A = k[x 1 , . . . , xn] and G is a small subgroup of GLn(k), Auslander's Theorem says that the skew group algebra A#G is isomorphic to End A G (A) as graded algebras. We prove a generalization of Auslander's Theorem for permutation actions on (−1)-skew polynomial rings, (−1)-quantum Weyl algebras, three-dimensional Sklyanin algebras, and a certain homogeneous down-up algebra. We also show that certain fixed rings A G are graded isolated singularities in the sense of Ueyama.

Matrix congruence can be used to mimic linear maps between homogeneous quadratic polynomials in n variables. We introduce a generalization, called standardform congruence, which mimics affine maps between non-homogeneous quadratic polynomials. Canonical forms under standard-form congruence for three-by-three matrices are derived. This is then used to give a classification of algebras defined by two generators and one degree two relation. We also apply standard-form congruence to classify homogenizations of these algebras.

Abstract. We provide formulas for computing the discriminant of noncommutative algebras over central subalgebras in the case of Ore extensions and skew group extensions. The formulas follow from a more general result regarding the discriminants of certain twisted tensor products. We employ our formulas to compute automorphism groups for examples in each case.

We study the Zariski cancellation problem for Poisson algebras asking whether A[t] ∼ = B[t] implies A ∼ = B when A and B are Poisson algebras. We resolve this affirmatively in the cases when A and B are both connected graded Poisson algebras finitely generated in degree one without degree one Poisson central elements and when A is a Poisson integral domain of Krull dimension two with nontrivial Poisson bracket. We further introduce Poisson analogues of the Makar-Limanov invariant and the discriminant to deal with the Zariski cancellation problem for other families of Poisson algebras.The interested reader is directed to the survey by Gupta for further background on this problem [18].In recent years, several works have extended the Zariski cancellation problem from commutative algebraic geometry to noncommutative projective algebraic geometry, where the cancellation property is studied for certain types of Artin-Schelter regular algebras, which are noncommutative graded analogues of commutative polynomial rings; see [4, 5,25,26].In this paper, we extend the original Zariski cancellation problem in a different direction. Instead of generalizing the affine variety Y into a noncommutative algebraic variety, which is represented by a noncommutative algebra as its coordinate ring, we assume Y to have extra structure, namely we assume that there exists a bivector π ∈ 2 (T Y ) satisfying a vanishing Schouten-Nijenhuis bracket [π, π] = 0. In algebraic terms, Y is an affine Poisson variety whose coordinate ring turns out to be a commutative Poisson algebra.Therefore, we are interested in the following question. Question 1.2 ((Zariski Cancellation Problem for Poisson Algebras)). When is a Poisson algebra A cancellative? That is, when does an isomorphism of Poisson algebras A[t] ∼ = B[t] for another Poisson algebra B imply an isomorphism A ∼ = B as Poisson algebras? The notion of the Poisson bracket, first introduced by Siméon Denis Poisson, arises naturally in Hamiltonian mechanics and differential geometry. Poisson algebras have become deeply entangled with noncommutative geometry, integrable systems, and topological field theories. They are essential in the study of the noncommutative discriminant [6, 33] and representation theory of noncommutative algebras [43, 42]. In addition, there has been renewed interest in enveloping algebras of Poisson algebras [27, 28]. Recently, Adjamagbo and van den Essen [2] proved that the famous Jacobian conjecture for polynomial algebras has an equivalent statement for Poisson algebras. As pointed out by van den Essen [40], the Zariski cancellation problem (especially in dimension two) is closely related to the Jacobian conjecture. Therefore, one of our motivations is to study the Zariski cancellation problem for Poisson algebras with a potential link to the above-mentioned Poisson version of the Jacobian conjecture. While we are searching for general methods and techniques in this direction, we find many theories developed by Bell and Zhang in the (associative) noncommutative setting [4, ...