Perverse coherent t-structures through torsion theories

Bezrukavnikov (later together with Arinkin) recovered the work of Deligne defining perverse $t$-structures for the derived category of coherent sheaves on a projective variety. In this text we prove that these $t$-structures can be obtained through tilting torsion theories as in the work of Happel, Reiten and Smal\o. This approach proves to be slightly more general as it allows us to define, in the quasi-coherent setting, similar perverse $t$-structures for certain noncommutative projective planes.Comment: New revised version with important correction

On the Notion of Complete Intersection outside the Setting of Skew Polynomial Rings

In recent work of T. Cassidy and the author, a notion of complete intersection was defined for (non-commutative) regular skew polynomial rings, defining it using both algebraic and geometric tools, where the commutative definition is a special case. In this article, we extend the definition to a larger class of algebras that contains regular graded skew Clifford algebras, the coordinate ring of quantum matrices and homogenizations of universal enveloping algebras. Regular algebras are often considered to be non-commutative analogues of polynomial rings, so the results herein support that viewpoint.Comment: This paper replaces the preprint "Defining the Notion of Complete Intersection for Regular Graded Skew Clifford Algebras", and also has a paragraph written correctly that is garbled by the publisher in the published version (paragraph after Example 3.8

Causal sites as quantum geometry

We propose a structure called a causal site to use as a setting for quantum geometry, replacing the underlying point set. The structure has an interesting categorical form, and a natural "tangent 2-bundle," analogous to the tangent bundle of a smooth manifold. Examples with reasonable finiteness conditions have an intrinsic geometry, which can approximate classical solutions to general relativity. We propose an approach to quantization of causal sites as well.Comment: 21 pages, 3 eps figures; v2: added references; to appear in JM

Torsion functors with monomial support

The dependence of torsion functors on their supporting ideals is investigated, especially in the case of monomial ideals of certain subrings of polynomial algebras over not necessarily Noetherian rings. As an application it is shown how flatness of quasicoherent sheaves on toric schemes is related to graded local cohomology.Comment: updated reference

Non-liftable Calabi-Yau spaces

We construct many new non-liftable three-dimensional Calabi-Yau spaces in positive characteristic. The technique relies on lifting a nodal model to a smooth rigid Calabi-Yau space over some number field as introduced by the first author and D. van Straten.Comment: 16 pages, 5 tables; v2: minor corrections and addition

Minimal Universal Two-qubit Quantum Circuits

We give quantum circuits that simulate an arbitrary two-qubit unitary operator up to global phase. For several quantum gate libraries we prove that gate counts are optimal in worst and average cases. Our lower and upper bounds compare favorably to previously published results. Temporary storage is not used because it tends to be expensive in physical implementations. For each gate library, best gate counts can be achieved by a single universal circuit. To compute gate parameters in universal circuits, we only use closed-form algebraic expressions, and in particular do not rely on matrix exponentials. Our algorithm has been coded in C++.Comment: 8 pages, 2 tables and 4 figures. v3 adds a discussion of asymetry between Rx, Ry and Rz gates and describes a subtle circuit design problem arising when Ry gates are not available. v2 sharpens one of the loose bounds in v1. Proof techniques in v2 are noticeably revamped: they now rely less on circuit identities and more on directly-computed invariants of two-qubit operators. This makes proofs more constructive and easier to interpret as algorithm

Existence of Universal Entangler

A gate is called entangler if it transforms some (pure) product states to entangled states. A universal entangler is a gate which transforms all product states to entangled states. In practice, a universal entangler is a very powerful device for generating entanglements, and thus provides important physical resources for accomplishing many tasks in quantum computing and quantum information. This Letter demonstrates that a universal entangler always exists except for a degenerated case. Nevertheless, the problem how to find a universal entangler remains open.Comment: 4 page

Zeroth Poisson homology of symmetric powers of isolated quasihomogeneous surface singularities

Let X be a surface with an isolated singularity at the origin, given by the equation Q(x,y,z)=0, where Q is a weighted-homogeneous polynomial. In particular, this includes the Kleinian surfaces X = C^2/G for G < SL(2,C) finite. Let Y be the n-th symmetric power of X. We compute the zeroth Poisson homology of Y, as a graded vector space with respect to the weight grading. In the Kleinian case, this confirms a conjecture of Alev, that the zeroth Poisson homology of the n-th symmetric power of C^2/G is isomorphic to the zeroth Hochschild homology of the n-th symmetric power of the algebra of G-invariant differential operators on C. That is, the Brylinski spectral sequence degenerates in this case. In the elliptic case, this yields the zeroth Hochschild homology of symmetric powers of the elliptic algebras with three generators modulo their center, for the parameter equal to all but countably many points of the elliptic curve.Comment: 17 page

Monoids, Embedding Functors and Quantum Groups

We show that the left regular representation \pi_l of a discrete quantum group (A,\Delta) has the absorbing property and forms a monoid (\pi_l,\tilde{m},\tilde{\eta}) in the representation category Rep(A,\Delta). Next we show that an absorbing monoid in an abstract tensor *-category C gives rise to an embedding functor E:C->Vect_C, and we identify conditions on the monoid, satisfied by (\pi_l,\tilde{m},\tilde{\eta}), implying that E is *-preserving. As is well-known, from an embedding functor E: C->\mathrm{Hilb} the generalized Tannaka theorem produces a discrete quantum group (A,\Delta) such that C is equivalent to Rep_f(A,\Delta). Thus, for a C^*-tensor category C with conjugates and irreducible unit the following are equivalent: (1) C is equivalent to the representation category of a discrete quantum group (A,\Delta), (2) C admits an absorbing monoid, (3) there exists a *-preserving embedding functor E: C->\mathrm{Hilb}.Comment: Final version, to appear in Int. Journ. Math. (Added some references and Subsection 1.2.) Latex2e, 21 page