Exterior Algebra Research Articles

Grassmann time-evolving matrix product operators: An efficient numerical approach for fermionic path integral simulations.

Developing numerical exact solvers for open quantum systems is a challenging task due to the non-perturbative and non-Markovian nature when coupling to structured environments. The Feynman-Vernon influence functional approach is a powerful analytical tool to study the dynamics of open quantum systems. Numerical treatments of the influence functional including the quasi-adiabatic propagator technique and the tensor-network-based time-evolving matrix product operator method have proven to be efficient in studying open quantum systems with bosonic environments. However, the numerical implementation of the fermionic path integral suffers from the Grassmann algebra involved. In this work, we present a detailed introduction to the Grassmann time-evolving matrix product operator method for fermionic open quantum systems. In particular, we introduce the concepts of Grassmann tensor, signed matrix product operator, and Grassmann matrix product state to handle the Grassmann path integral. Using the single-orbital Anderson impurity model as an example, we review the numerical benchmarks for structured fermionic environments for real-time nonequilibrium dynamics, real-time and imaginary-time equilibration dynamics, and its application as an impurity solver. These benchmarks show that our method is a robust and promising numerical approach to study strong coupling physics and non-Markovian dynamics. It can also serve as an alternative impurity solver to study strongly correlated quantum matter with dynamical mean-field theory.

An Observer-Based View of Euclidean Geometry

An influence network of events is a view of the universe based on events that may be related to one another via influence. The network of events forms a partially ordered set which, when quantified consistently via a technique called chain projection, results in the emergence of spacetime and the Minkowski metric as well as the Lorentz transformation through changing an observer from one frame to another. Interestingly, using this approach, the motion of a free electron as well as the Dirac equation can be described. Indeed, the same approach can be employed to show how a discrete version of some of the features of Euclidean geometry including directions, dimensions, subspaces, Pythagorean theorem, and geometric shapes can emerge. In this paper, after reviewing the essentials of the influence network formalism, we build on some of our previous works to further develop aspects of Euclidean geometry. Specifically, we present the emergence of geometric shapes, a discrete version of the parallel postulate, the dot product, and the outer (wedge product) in 2+1 dimensions. Finally, we show that the scalar quantification of two concatenated orthogonal intervals exhibits features that are similar to those of the well-known concept of a geometric product in geometric Clifford algebras.

Non-abelian tensor product and circular orderability of groups

For a group G we consider its tensor square G⊗G and exterior square G∧G. We prove that for a circularly orderable group G, under some assumptions on H1(G) and H2(G), its exterior square and tensor square are left-orderable. This yields an obstruction for a circularly orderable group G to have torsion. We apply these results to study circular orderability of tabulated virtual knot groups.

Free superfermions construction for the coupled super KP and super modified KP hierarchies

In this paper, the Kac–van de Leur version of the coupled super KP (SKP) and super modified KP (SmKP) hierarchies are constructed. We define the tau functions as the orbit of a new group G(g) acting on the weight vectors, where G(g) is generated by elements in the even part of the tensor product of Clifford superalgebra and Grassmann algebra. The super Hirota bilinear equations of the coupled SKP and SmKP hierarchies are obtained by super boson-fermion correspondence of type A, and the bilinear identities with respect to the super Baker functions and tau functions in superbosonic Fock space are also constructed. Finally, the Miura transformation in terms of superfermions is also discussed.

A bound for the exterior product of S-units

A bound for the exterior product of S-units

Commutative Poisson algebras from deformations of noncommutative algebras

It is well-known that a formal deformation of a commutative algebra A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document} leads to a Poisson bracket on A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document} and that the classical limit of a derivation on the deformation leads to a derivation on A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}, which is Hamiltonian with respect to the Poisson bracket. In this paper we present a generalization of it for formal deformations of an arbitrary noncommutative algebra A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}. The deformation leads in this case to a Poisson algebra structure on Π(A):=Z(A)×(A/Z(A))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Pi (\\mathcal {A}){:}{=}Z(\\mathcal {A})\ imes (\\mathcal {A}/Z(\\mathcal {A}))$$\\end{document} and to the structure of a Π(A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Pi (\\mathcal {A})$$\\end{document}-Poisson module on A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}. The limiting derivations are then still derivations of A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}, but with the Hamiltonian belong to Π(A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Pi (\\mathcal {A})$$\\end{document}, rather than to A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}. We illustrate our construction with several cases of formal deformations, coming from known quantum algebras, such as the ones associated with the nonabelian Volterra chains, Kontsevich integrable map, the quantum plane and the quantized Grassmann algebra.

Graded picture invariants and polynomial invariants for mixed tensor superspaces

In this paper we consider the mixed tensor space of a Z2-graded vector space. We obtain a spanning set of invariants of the associated symmetric algebra under the action of the general linear supergroup as well as the queer supergroup over the Grassmann algebra. As a consequence, we give a generating set of polynomial invariants for the simultaneous adjoint action of the general linear supergroup on several copies of its Lie superalgebra. We show that in this special case, this set turns out to be the set of supertrace monomials, which is analogous to the result of Procesi for the general linear group. A queer supergroup analogue of these results is also obtained.

Unoriented 2-dimensional TQFTs and the category $\mathrm{Rep}(S_{t}\wr\mathbb{Z}_{2})$

We construct a family of unoriented 2-dimensional cobordism theories parametrized by certain triples of sequences. We also prove that some specializations of these sequences yield equivalences with an exterior product of Deligne categories. In particular, one of these specializations recovers the generalized Deligne category \mathrm{Rep}(S_{t}\wr \mathbb{Z}_{2}) .

The Exact Density of the Eigenvalues of the Wishart and Matrix-Variate Gamma and Beta Random Variables

The determination of the distributions of the eigenvalues associated with matrix-variate gamma and beta random variables of either type proves to be a challenging problem. Several of the approaches utilized so far yield unwieldy representations that, for instance, are expressed in terms of multiple integrals, functions of skew symmetric matrices, ratios of determinants, solutions of differential equations, zonal polynomials, and products of incomplete gamma or beta functions. In the present paper, representations of the density functions of the smallest, largest and jth largest eigenvalues of matrix-variate gamma and each type of beta random variables are explicitly provided as finite sums when certain parameters are integers and, as explicit series, in the general situations. In each instance, both the real and complex cases are considered. The derivations initially involve an orthonormal or unitary transformation whereby the wedge products of the differential elements of the eigenvalues can be worked out from those of the original matrix-variate random variables. Some of these results also address the distribution of the eigenvalues of a central Wishart matrix as well as eigenvalue problems arising in connection with the analysis of variance procedure and certain tests of hypotheses in multivariate analysis. Additionally, three numerical examples are provided for illustration purposes.

Averaging property of wedge product and naturality in discrete exterior calculus

Averaging property of wedge product and naturality in discrete exterior calculus

Extension groups of tautological bundles on punctual Quot schemes of curves

We prove formulas for the cohomology and the extension groups of tautological bundles on punctual Quot schemes over complex smooth projective curves. As a corollary, we show that the tautological bundle determines the isomorphism class of the original vector bundle on the curve. We also give a vanishing result for the push-forward along the Quot–Chow morphism of tensor and wedge products of duals of tautological bundles.

Additive decomposability preserving maps between exterior powers of vector spaces

In this paper, we characterize additive maps between exterior power of vector spaces that send decomposable multivectors to decomposable multivectors. As an application, we deduce Kuzma's result regarding additive maps on alternate matrices that do not increase the rank of rank two alternate matrices.

Quantization of pseudo-hermitian systems

This work is a generalization of (Raimundo et al 2021 Phys. Rev. A 103 022201) to Grassmann algebras of arbitrary dimensions. Here we present a covariant quantization scheme for pseudoclassical theories focused on non-hermitian quantum mechanics. The quantization maps canonically related pseudoclassical theories to equivalent quantum realizations in arbitrary dimensions. We apply the formalism to the problem of two coupled spins with Heisenberg interaction.

Characteristic cohomology II: Matrix singularities

AbstractFor a germ of a variety , a singularity of “type ” is given by a germ , which is transverse to in an appropriate sense, such that . In part I of this paper, we introduced for such singularities the Characteristic Cohomology for the Milnor fiber (for a hypersurface), and complement and link (for the general case). It captures the cohomology of inherited from and is given by subalgebras of the cohomology for for the Milnor fiber and complements, and is a subgroup for the cohomology of the link. We showed these cohomologies are functorial and invariant under diffeomorphism groups of equivalences for Milnor fibers and for complements and links. We also gave geometric criteria for detecting the nonvanishing of the characteristic cohomology.In this paper, we apply these methods in the case denotes any of the varieties of singular complex matrices, which may be either general, symmetric, or skew‐symmetric (with even). For these varieties, we have shown in another paper that their Milnor fibers and complements have compact “model submanifolds” for their homotopy types, which are classical symmetric spaces in the sense of Cartan. As a result, we first give the structure of the characteristic cohomology subalgebras for the Milnor fibers and complements as images of exterior algebras (or in one case a module on two generators over an exterior algebra). For links, the characteristic cohomology group is the image of a shifted upper truncated exterior algebra. In addition, we extend these results for the complement and link to the case of general complex matrices.Second, we then apply the geometric detection methods introduced in Part I to detect when specific characteristic cohomology classes for the Milnor fiber or complement are nonzero. We identify an exterior subalgebra on a specific set of generators and for the link that it contains an appropriate shifted upper truncated exterior subalgebra. The detection criterion involves a special type of “kite map germ of size ” based on a given flag of subspaces. The general criterion that detects such nonvanishing characteristic cohomology is then given in terms of the defining germ containing such a kite map germ of size . Furthermore, we use a restricted form of kite spaces to give a cohomological relation between the cohomology of local links and the global link for the varieties of singular matrices.

$$\textbf{GL}$$-algebras in positive characteristic I: the exterior algebra

$$\textbf{GL}$$-algebras in positive characteristic I: the exterior algebra

On a generalized notion of metrics

In these notes we generalize the notion of a (pseudo) metric measuring the distance of two points, to a (pseudo) n-metric which assigns a value to a tuple of n≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\ge 2$$\\end{document} points. Some elementary properties of pseudo n-metrics are provided and their construction via exterior products is investigated. We discuss some examples from the geometry of Euclidean vector spaces leading to pseudo n-metrics on the unit sphere, on the Stiefel manifold, and on the Grassmann manifold. Further, we construct a pseudo n-metric on hypergraphs and discuss the problem of generalizing the Hausdorff metric for closed sets to a pseudo n-metric.

Simple Harish-Chandra modules over the superconformal current algebra

In this paper, we classify the simple Harish-Chandra modules over the superconformal current algebra gˆ, which is the semi-direct sum of the N=1 superconformal algebra with the affine Lie superalgebra g˙⊗A⊕CC1, where g˙ is a finite-dimensional simple Lie algebra, and A is the tensor product of the Laurent polynomial algebra and the Grassmann algebra. As an application, we can directly get the classification of the simple Harish-Chandra modules over the N=1 Heisenberg-Virasoro algebra.

The Witt rings of many flag varieties are exterior algebras

The Witt ring of a complex flag variety describes the interesting – i.e. torsion – part of its topological KO-theory. We show that for a large class of flag varieties, these Witt rings are exterior algebras, and that the degrees of the generators can be determined by Dynkin diagram combinatorics. Besides a few well-studied examples such as full flag varieties and projective spaces, this class includes many flag varieties whose Witt rings were previously unknown, including many flag varieties of exceptional types. In particular, it includes all flag varieties of types G 2 G_2 and F 4 F_4 . The results also extend to flag varieties over other algebraically closed fields.

Characterization of Aluminium Anodizing for Coffee Industry Exterior

Abstract Aluminium is a lightweight and malleable metal. So it is widely used for several exterior applications including coffee industry storefronts. The surface of these applications needs hardness characteristics to be more scratch resistant and have attractive aesthetics. One method to improve these characteristics is anodizing. This research aims to produce a harder and more attractive exterior. The method in this research is to conduct aluminium experiments and then anodizing using time variations (12, 20, 28, 36 and 44 minutes). In order to make the hardness safe and durable when used, the anodized aluminium is tested using a hardness test. The optimum value of the test results will be compared with the quality of exterior products on the market. This information will be useful for the aluminium industry to improve the quality of the products produced.

Equivariant log-concavity and equivariant Kähler packages

We show that the exterior algebra ΛR[α1,⋯,αn], which is the cohomology of the torus T=(S1)n, and the polynomial ring R[t1,…,tn], which is the cohomology of the classifying space B(S1)n=(CP∞)n, are Sn-equivariantly log-concave. We do so by explicitly giving the Sn-representation maps on the appropriate sequences of tensor products of polynomials or exterior powers and proving that these maps satisfy the hard Lefschetz theorem. Furthermore, we prove that the whole Kähler package, including the Poincaré duality, the hard Lefschetz theorem, and the Hodge–Riemann bilinear relations, holds on the corresponding sequences in an equivariant setting.