Exterior Algebra Research Articles

On the L∞ structure of Poisson gauge theory

The Poisson gauge theory is a semi-classical limit of full non-commutative gauge theory. In this work we construct an algebra which governs both the action of gauge symmetries and the dynamics of the Poisson gauge theory. We derive the minimal set of non-vanishing ℓ-brackets and prove that they satisfy the corresponding homotopy relations. On the one hand, it provides new explicit non-trivial examples of L∞ algebras. On the other hand, it can be used as a starting point for bootstrapping the full non-commutative gauge theory. The first few brackets of such a theory are constructed explicitly in the text. In addition we show that the derivation properties of ℓ-brackets on with respect to the truncated product on the exterior algebra are satisfied only for the canonical non-commutativity. In general, does not have a structure of P∞ algebra.

A note on the pp-wave solution of minimal massive 3D gravity coupled with Maxwell–Chern–Simons theory

In this work, we examine a family of pp-wave solutions of minimal massive 3D gravity minimally coupled with the Maxwell–Chern–Simons theory. An elaborate investigation of the field equations shows that the theory admits pp-wave solutions provided that there exist an anti-self duality relation between the electric and the magnetic components of the Maxwell two-form field. By employing Noether–Wald formalism, we also construct Noether charges of the theory within exterior algebra formalism.

On the singular gamma, Wishart, and beta matrix‐variate density functions

AbstractWhen a real positive definite matrix follows a Wishart or, more generally, a matrix‐variate gamma distribution with shape parameter and positive definite scale parameter matrix , one can represent as for some matrix of dimension . When , has a singular distribution whose properties can be studied via the density function of . It will be shown that when follows a matrix‐variate extended Gaussian distribution, the density function of the resulting singular gamma distribution can be obtained by making use of successive transformations and their associated Jacobians. The singular Wishart distribution will then be obtained as a particular case. The marginal and conditional density functions resulting from an arbitrary partitioning of will be considered as well. The same technique will also be applied to the derivation of the density functions of real and complex singular type‐1 and type‐2 beta‐distributed matrices. It so happens that the proposed approach, which is based on manipulations involving the wedge products of certain differential elements, generally proves more efficient than the intricate procedures that have hitherto been employed in the literature.

Abelianization of some groups of interval exchanges

Let IET be the group of bijections from $\mathopen{[}0,1 \mathclose{[}$ to itself that are continuous outside a finite set, right-continuous and piecewise translations. The abelianization homomorphism $f: \text{IET} \to A$, called SAF-homomorphism, was described by Arnoux-Fathi and Sah. The abelian group $A$ is the second exterior power of the reals over the rationals. For every subgroup $\Gamma$ of $\mathbb{R/Z}$ we define $\text{IET}(\Gamma)$ as the subgroup of $\text{IET}$ consisting of all elements $f$ such that $f$ is continuous outside $\Gamma$. Let $\tilde{\Gamma}$ be the preimage of $\Gamma$ in $\mathbb{R}$. We establish an isomorphism between the abelianization of $\text{IET}(\Gamma)$ and the second skew-symmetric power of $\tilde{\Gamma}$ over $\mathbb{Z}$ denoted by ${}^\circleddash\!\!\bigwedge^2_{\mathbb{Z}} \tilde{\Gamma}$. This group often has non-trivial $2$-torsion, which is not detected by the SAF-homomorphism. We then define $\text{IET}^{\bowtie}$ the group of all interval exchange transformations with flips. Arnoux proved that this group is simple thus perfect. However for every subgroup $\text{IET}^{\bowtie}(\Gamma)$ we establish an isomorphism between its abelianization and $\langle \lbrace a \otimes a ~ [\text{mod}~2] \mid a \in \tilde{\Gamma} \rbrace \rangle \times \langle \lbrace \ell \wedge \ell ~ [\text{mod}~2] \mid \ell \in \tilde{\Gamma} \rbrace \rangle$ which is a $2$-elementary abelian subgroup of $\bigotimes^2_{\mathbb{Z}} \tilde{\Gamma} / (2\bigotimes^2_{\mathbb{Z}} \tilde{\Gamma}) \times {}^\circleddash\!\!\bigwedge^2_{\mathbb{Z}} \tilde{\Gamma} / (2 {}^\circleddash\!\!\bigwedge^2_{\mathbb{Z}} \tilde{\Gamma})$.

Singularities of Serial Robots: Identification and Distance Computation Using Geometric Algebra

The singularities of serial robotic manipulators are those configurations in which the robot loses the ability to move in at least one direction. Hence, their identification is fundamental to enhance the performance of current control and motion planning strategies. While classical approaches entail the computation of the determinant of either a 6×n or n×n matrix for an n-degrees-of-freedom serial robot, this work addresses a novel singularity identification method based on modelling the twists defined by the joint axes of the robot as vectors of the six-dimensional and three-dimensional geometric algebras. In particular, it consists of identifying which configurations cause the exterior product of these twists to vanish. In addition, since rotors represent rotations in geometric algebra, once these singularities have been identified, a distance function is defined in the configuration space C, such that its restriction to the set of singular configurations S allows us to compute the distance of any configuration to a given singularity. This distance function is used to enhance how the singularities are handled in three different scenarios, namely, motion planning, motion control and bilateral teleoperation.

Numerically non-special varieties

Campana introduced the class of special varieties as the varieties admitting no Bogomolov sheaves, i.e. rank-one coherent subsheaves of maximal Kodaira dimension in some exterior power of the cotangent bundle. Campana raised the question of whether one can replace the Kodaira dimension by the numerical dimension in this characterization. We answer partially this question showing that a projective manifold admitting a rank-one coherent subsheaf of the cotangent bundle with numerical dimension one is not special. We also establish the analytic characterization with the non-existence of Zariski dense entire curve and the arithmetic version with non-potential density in the (split) function field setting. Finally, we conclude with a few comments for higher codimensional foliations which may provide some evidence towards a generalization of the aforementioned results.

Equivariant higher twisted K$K$‐theory of SU(n)$SU(n)$ for exponential functor twists

We prove that each exponential functor on the category of finite-dimensional complex inner product spaces and isomorphisms gives rise to an equivariant higher (that is, non-classical) twist of K $K$ -theory over G = S U ( n ) $G=SU(n)$ . This twist is represented by a Fell bundle E → G $\mathcal {E}\rightarrow \mathcal {G}$ , which reduces to the basic gerbe for the top exterior power functor. The groupoid G $\mathcal {G}$ comes equipped with a G $G$ -action and an augmentation map G → G $\mathcal {G}\rightarrow G$ , that is an equivariant equivalence. The C ∗ $C^*$ -algebra C ∗ ( E ) $C^*(\mathcal {E})$ associated to E $\mathcal {E}$ is stably isomorphic to the section algebra of a locally trivial bundle with stabilised strongly self-absorbing fibres. Using a version of the Mayer–Vietoris spectral sequence, we compute the equivariant higher twisted K $K$ -groups K ∗ G ( C ∗ ( E ) ) $K^G_*(C^*(\mathcal {E}))$ for arbitrary exponential functor twists over S U ( 2 ) $SU(2)$ , and also over S U ( 3 ) $SU(3)$ after rationalisation.

Generalized entanglement measure for continuous-variable systems

Concurrence, introduced by Hill and Wootters [Hill and Wootters, Phys. Rev. Lett. 78, 5022 (1997)], provides an important measure of entanglement for a general pair of qubits that is strictly positive for entangled states and vanishes for all separable states. We present an extension of the entanglement measure to general pure continuous variable states of multiple degrees of freedom by generalizing the Lagrange's identity and wedge product framework proposed by Bhaskara and Panigrahi [Bhaskara and Panigrahi, Quantum Inf. Process. 16, 118 (2017)] for pure discrete-variable systems in arbitrary dimensions and extending the concept to mixed continuous-variable states. A family of faithful entanglement measures is constructed that admits necessary and sufficient conditions for separability across arbitrary bipartitions presented by Vedral et al. [Vedral, Plenio, Rippin, and Knight, Phys. Rev. Lett. 78, 2275 (1997)]. The computed entanglement measure in the present approach for general Gaussian states, pair-coherent states, and non-Gaussian continuous-variable Bell states matches with known results. We also quantify entanglement of phase-randomized squeezed states and superposition of squeezed states. Our results also simplify several results in quantum entanglement theory.

Grassmannian stochastic analysis and the stochastic quantization of Euclidean fermions

We introduce a stochastic analysis of Grassmann random variables suitable for the stochastic quantization of Euclidean fermionic quantum field theories. Analysis on Grassmann algebras is developed here from the point of view of quantum probability: a Grassmann random variable is an homomorphism of an abstract Grassmann algebra into a quantum probability space, i.e. a C^{*}-algebra endowed with a suitable state. We define the notion of Gaussian processes, Brownian motion and stochastic (partial) differential equations taking values in Grassmann algebras. We use them to study the long time behavior of finite and infinite dimensional Langevin Grassmann stochastic differential equations driven by Gaussian space-time white noise and to describe their invariant measures. As an application we give a proof of the stochastic quantization and of the removal of the space cut-off for the Euclidean Yukawa model.

Set Partitions, Fermions, and Skein Relations

Abstract Let $\Theta _n = (\theta _1, \dots , \theta _n)$ and $\Xi _n = (\xi _1, \dots , \xi _n)$ be two lists of $n$ variables, and consider the diagonal action of ${{\mathfrak {S}}}_n$ on the exterior algebra $\wedge \{ \Theta _n, \Xi _n \}$ generated by these variables. Jongwon Kim and the 2nd author defined and studied the fermionic diagonal coinvariant ring$FDR_n$ obtained from $\wedge \{ \Theta _n, \Xi _n \}$ by modding out by the ideal generated by the ${{\mathfrak {S}}}_n$-invariants with vanishing constant term. On the other hand, the 2nd author described an action of ${{\mathfrak {S}}}_n$ on the vector space with basis given by noncrossing set partitions of $\{1,\dots ,n\}$ using a novel family of skein relations that resolve crossings in set partitions. We give an isomorphism between a natural Catalan-dimensional submodule of $FDR_n$ and the skein representation. To do this, we show that set partition skein relations arise naturally in the context of exterior algebras. Our approach yields an ${{\mathfrak {S}}}_n$-equivariant way to resolve crossings in set partitions. We use fermions to clarify, sharpen, and extend the theory of set partition crossing resolution.

Partitions of the complete hypergraph $$K_6^3$$ and a determinant-like function

In this paper, we introduce a determinant-like map $$\mathrm{det}^{S^3}$$ and study some of its properties. For this, we define a graded vector space $$\Lambda ^{S^3}_V$$ that has similar properties with the exterior algebra $$\Lambda _V$$ and the exterior GSC-operad $$\Lambda ^{S^2}_V$$ from Staic. When $$\mathrm{dim}(V_2)=2$$ , we show that $$\mathrm{dim}_k(\Lambda ^{S^3}_{V_2}[6])=1$$ , which gives the existence and uniqueness of the map $$\mathrm{det}^{S^3}$$ . We also give an explicit formula for $$\mathrm{det}^{S^3}$$ as a sum over certain 2-partitions of the complete hypergraph $$K_6^3$$ .

An asymptotic for the Hall–Paige conjecture

Hall and Paige conjectured in 1955 that a finite group G has a complete mapping if and only if its Sylow 2-subgroups are trivial or noncyclic. This conjecture was proved in 2009 by Wilcox, Evans, and Bray using the classification of finite simple groups and extensive computer algebra. Using a completely different approach motivated by the circle method from analytic number theory, we prove that the number of complete mappings of any group G of order n satisfying the Hall–Paige condition is (e−1/2+o(1))|Gab|n!2/nn.

Formalizing Geometric Algebra in Lean

This paper explores formalizing Geometric (or Clifford) algebras into the Lean 3 theorem prover, building upon the substantial body of work that is the Lean mathematics library, mathlib. As we use Lean source code to demonstrate many of our ideas, we include a brief introduction to the Lean language targeted at a reader with no prior experience with Lean or theorem provers in general. We formalize the multivectors as the quotient of the tensor algebra by a suitable relation, which provides the ring structure automatically, then go on to establish the universal property of the Clifford algebra. We show that this is quite different to the approach taken by existing formalizations of Geometric algebra in other theorem provers; most notably, our approach does not require a choice of basis. We go on to show how operations and structure such as involutions, versors, and the mathbb {Z}_2-grading can be defined using the universal property alone, and how to recover an induction principle from the universal property suitable for proving statements about these definitions. We outline the steps needed to formalize the wedge product and mathbb {N}-grading, and some of the gaps in mathlib that currently make this challenging.

Cohomological supports of tensor products of modules over commutative rings

This works concerns cohomological support varieties of modules over commutative local rings. The main result is that the support of a derived tensor product of a pair of differential graded modules over a Koszul complex is the join of the supports of the modules. This generalizes, and gives another proof of, a result of Dao and the third author dealing with Tor-independent modules over complete intersection rings. The result for Koszul complexes has a broader applicability, including to exterior algebras over local rings.

A note on -gradings on the Grassmann algebra and elementary number theory

Let E be the Grassmann algebra of an infinite-dimensional vector space L over a field of characteristic zero. In this paper, we study the -gradings on E having the form , in which each element of a basis of L has -degree , or . We provide a criterion for the support of this structure to coincide with a subgroup of the group , and we describe the graded identities for the corresponding gradings. We strongly use Elementary Number Theory as a tool, providing an interesting connection between this classical part of Mathematics, and PI Theory. Our results are generalizations of the approach presented in Brandão A, Fidelis C, Guimarães A. -gradings of full support on the Grassmann algebra. J Algebra. 2022;601:332–353. DOI:10.1016/j.jalgebra.2022.03.014. See also in arXiv preprint, arXiv:2009.01870v1, 2020].

The zonoid algebra, generalized mixed volumes, and random determinants

We show that every multilinear map between Euclidean spaces induces a unique, continuous, Minkowski multilinear map of the corresponding real cones of zonoids. Applied to the wedge product of the exterior algebra of a Euclidean space, this yields a multiplication of zonoids, defining the structure of a commutative, associative, and partially ordered ring, which we call the zonoid algebra. This framework gives a new perspective on classical objects in convex geometry, and it allows to introduce new functionals on zonoids, in particular generalizing the notion of mixed volume. We also analyze a similar construction based on the complex wedge product, which leads to the new notion of mixed J–volume. These ideas connect to the theory of random determinants.

Formula omitted]-gradings of full support on the Grassmann algebra

Let E be the infinite dimensional Grassmann algebra over a field F of characteristic zero. In this paper we investigate the structures of Z-gradings on E of full support. Using methods of elementary number theory, we describe the Z-graded polynomial identities for the so-called 2-induced Z-gradings on E of full support. As a consequence of this fact we provide examples of Z-gradings on E which are PI-equivalent but not Z-isomorphic. This is the first example of graded algebras with infinite support that are PI-equivalent, but non-isomorphic as graded algebras. We also present the notion of central Z-gradings on E and we show that their Z-graded polynomial identities are closely related to the Z2-graded polynomial identities of E.

A criterion for the existence of periodic points based on the eigenvalues of maps induced in cohomology

We present a criterion for the existence of periodic points based on the eigenvalues of maps induced in cohomology for spaces with rational cohomology isomorphic to a tensor product of a graded exterior algebra with generators in odd dimensions and a graded algebra with all elements of even degree. We give a number of natural examples of such spaces and provide some non-trivial ones. We also give a counterexample to a claim in [3] given there without proof.

Superfiltered A∞$A_\infty$‐deformations of the exterior algebra, and local mirror symmetry

The exterior algebra $E$ on a finite-rank free module $V$ carries a $\mathbb{Z}/2$-grading and an increasing filtration, and the $\mathbb{Z}/2$-graded filtered deformations of $E$ as an associative algebra are the familiar Clifford algebras, classified by quadratic forms on $V$. We extend this result to $A_\infty$-algebra deformations $\mathcal{A}$, showing that they are classified by formal functions on $V$. The proof translates the problem into the language of matrix factorisations, using the localised mirror functor construction of Cho-Hong-Lau, and works over an arbitrary ground ring. We also compute the Hochschild cohomology algebras of such $\mathcal{A}$. By applying these ideas to a related construction of Cho-Hong-Lau we prove a local form of homological mirror symmetry: the Floer $A_\infty$-algebra of a monotone Lagrangian torus is quasi-isomorphic to the endomorphism algebra of the expected matrix factorisation of its superpotential.

On the smoothness of lexicographic points on Hilbert schemes

We study the geometry of standard graded Hilbert schemes of polynomial rings and exterior algebras. Our investigation is motivated by a famous theorem of Reeves and Stillman for the Grothendieck Hilbert scheme, which states that the lexicographic point is smooth. By contrast, we show that, in standard graded Hilbert schemes of polynomial rings and exterior algebras, the lexicographic point can be singular, and it can lie in multiple irreducible components. We answer questions of Peeva-Stillman and of Maclagan-Smith.