Finite-dimensional Complex Research Articles

Morse theoretic signal compression and reconstruction on chain complexes

At the intersection of Topological Data Analysis (TDA) and machine learning, the field of cellular signal processing has advanced rapidly in recent years. In this context, each signal on the cells of a complex is processed using the combinatorial Laplacian, and the resultant Hodge decomposition. Meanwhile, discrete Morse theory has been widely used to speed up computations by reducing the size of complexes while preserving their global topological properties. In this paper, we provide an approach to signal compression and reconstruction on chain complexes that leverages the tools of algebraic discrete Morse theory. The main goal is to reduce and reconstruct a based chain complex together with a set of signals on its cells via deformation retracts, preserving as much as possible the global topological structure of both the complex and the signals. We first prove that any deformation retract of real degree-wise finite-dimensional based chain complexes is equivalent to a Morse matching. We will then study how the signal changes under particular types of Morse matchings, showing its reconstruction error is trivial on specific components of the Hodge decomposition. Furthermore, we provide an algorithm to compute Morse matchings with minimal reconstruction error.

Chromatic fixed point theory and the Balmer spectrum for extraspecial 2-groups

abstract: In the early 1940s, P. A. Smith showed that if a finite $p$-group $G$ acts on a finite dimensional complex $X$ that is mod $p$ acyclic, then its space of fixed points, $X^G$, will also be mod $p$ acyclic. In their recent study of the Balmer spectrum of equivariant stable homotopy theory, Balmer and Sanders were led to study a question that can be shown to be equivalent to the following: if a $G$-space $X$ is a equivariant homotopy retract of the $p$-localization of a based finite $G$-C.W. complex, given $H<G$ and $n$, what is the smallest $r$ such that if $X^H$ is acyclic in the $(n+r)$th Morava $K$-theory, then $X^G$ must be acyclic in the $n$th Morava $K$-theory? Barthel et.~al. then answered this when $G$ is abelian, by finding general lower and upper bounds for these ``blue shift'' numbers which agree in the abelian case. In our paper, we first prove that these potential chromatic versions of Smith's theorem are equivalent to chromatic versions of a 1952 theorem of E. E. Floyd, which replaces acyclicity by bounds on dimensions of mod $p$ homology, and thus applies to all finite dimensional $G$-spaces. This unlocks new techniques and applications in chromatic fixed point theory. Applied to the problem of understanding blue shift numbers, we are able to use classic constructions and representation theory to search for lower bounds. We give a simple new proof of the known lower bound theorem, and then get the first results about nonabelian 2-groups that do not follow from previously known results. In particular, we are able to determine all blue shift numbers for extraspecial 2-groups. Applied in ways analogous to Smith's original applications, we prove new fixed point theorems for $K(n)_*$-homology disks and spheres. Finally, our methods offer a new way of using equivariant results to show the collapsing of certain Atiyah-Hirzebruch spectral sequences in certain cases. Our criterion appears to apply to the calculation of all 2-primary Morava $K$-theories of all real Grassmanians.

Construction of even-level representations of [formula omitted] with residue field of characteristic two

We construct the finite-dimensional continuous complex representations of SL2 over compact discrete valuation rings of even residual characteristic, assuming the level is large enough compared to the ramification index, in the mixed characteristic case. We also prove that the complex group algebras of SL2 over finite quotient rings of such compact discrete valuation rings depend on the characteristic of the ring. In particular, we prove that the group algebras C[SL2(Z/2rZ)] and C[SL2(F2[t]/(tr))] are not isomorphic for any r≥4.

TQFTs and quantum computing

Quantum computing is captured in the formalism of the monoidal subcategory of VectC generated by C2 — in particular, quantum circuits are diagrams in VectC — while topological quantum field theories, in the sense of Atiyah, are diagrams in VectC indexed by cobordisms. We initiate a program to formalize this connection. In doing so, we equip cobordisms with machinery for producing linear maps by parallel transport along curves under a connection and then assemble these structures into a higher category. Finite-dimensional complex vector spaces and linear maps between them are given a suitable higher categorical structure which we call FVectC. We realize quantum circuits as images of cobordisms under monoidal functors from these modified cobordisms to FVectC, which are computed by taking parallel transports of vectors and then combining the results in a pattern encoded in the domain category.

The Weyl law for algebraic tori

We give an asymptotic evaluation for the number of automorphic characters of an algebraic torus T with bounded analytic conductor. The analytic conductor is defined via the local Langlands correspondence for tori by choosing a finite-dimensional complex algebraic representation of the L -group of T . Our results therefore fit into a general framework of counting automorphic representations on reductive groups by analytic conductor.

U0-free quantum group representations

Let g be a finite dimensional complex simple Lie algebra, and let U=Uq(g) be its quantized enveloping algebra with a triangular decomposition U=U−U0U+. We classify all U-module structures on U0 with the regular action of U0 on itself, which are shown to have direct connections with the q-Weyl algebra and the bounded weight representations. We obtain that the necessary and sufficient condition for the existence of such modules is that U has to be of type An(n≥1), Bn(n≥2) or Cn(n≥3). Moreover, we study their module structures and associated weight modules for each type.

Characterization of contact Lie superalgebras

We present the concept of finite-dimensional complex homogeneous contact Lie superalgebra. The Z2-graded homogeneous cases are studied in detail producing some relevant examples. We characterize homogeneous contact Lie superalgebras in terms of their Berezinian and their structure matrix. These Lie superalgebras are also characterized by means of deformation theory, and as an application we obtain the complete classification of low dimensional Lie superalgebras of this type.

Dirac cohomology for the BGG category [formula omitted

We study Dirac cohomology HDg,h(M) for modules belonging to category O of a finite dimensional complex semisimple Lie algebra. We start by studying the generalized infinitesimal character decomposition of M⊗S, with S being a spin module of h⊥. As a consequence, “Vogan’s conjecture” holds, and we prove a nonvanishing result for HDg,h(M) while we show that in the case of a Hermitian symmetric pair (g,k) and an irreducible unitary module M∈O, Dirac cohomology coincides with the nilpotent Lie algebra cohomology with coefficients in M. In the last part, we show that the higher Dirac cohomology and index introduced by Pandžić and Somberg satisfy nice homological properties for M∈O.

Quotient gradings and the intrinsic fundamental group

Quotient grading classes are essential participants in the computation of the intrinsic fundamental group π1(A) of an algebra A. In order to study quotient gradings of a finite-dimensional semisimple complex algebra A it is sufficient to understand the quotient gradings of twisted group algebra gradings. We establish the graded structure of such quotients using Mackey's obstruction class. Then, for matrix algebras A=Mn(C) we tie up the concepts of braces, group-theoretic Lagrangians and elementary crossed products. We also manage to compute the intrinsic fundamental group of the diagonal algebras A=C4 and A=C5.

Biderivations of complete Lie algebras

The authors of this article intend to present some results obtained in the study of biderivations of complete Lie algebras. Firstly they present a matricial approach to do this, which was a useful and explanatory tool not only in the study of biderivations but also in the synthesis of these results. Then they study all biderivations of a Lie algebra [Formula: see text] with [Formula: see text] and [Formula: see text], called complete. Moreover, as an application of the previous result, they describe all biderivations of a semisimple Lie algebra (that are complete), extending a result obtained by Tang in [X. Tang, Biderivations of finite-dimensional complex simple Lie algebras, Linear Multilinear Algebra 66(2) (2018) 250–259] that describes all biderivations of a complex simple Lie algebra. And thirdly, results on symmetric and skew-symmetric biderivations are also presented.

Decay of scalar curvature on uniformly contractible manifolds with finite asymptotic dimension

AbstractGromov proved a quadratic decay inequality of scalar curvature for a class of complete manifolds. In this paper, we prove that for any uniformly contractible manifold with finite asymptotic dimension, its scalar curvature decays to zero at a rate depending only on the contractibility radius of the manifold and the diameter control of the asymptotic dimension. We construct examples of uniformly contractible manifolds with finite asymptotic dimension whose scalar curvature functions decay arbitrarily slowly. This shows that our result is the best possible. We prove our result by studying the index pairing between Dirac operators and compactly supported vector bundles with Lipschitz control. A key technical ingredient for the proof of our main result is a Lipschitz control for the topological K‐theory of finite dimensional simplicial complexes.

Sympathetic Lie algebras and adjoint cohomology for Lie algebras

We study sympathetic (i.e., perfect and complete) Lie algebras. Among other topics they arise in the study of adjoint Lie algebra cohomology. Here a motivation comes from a conjecture of Pirashvili, which says that a finite-dimensional complex perfect Lie algebra is semisimple if and only if its adjoint cohomology vanishes. We prove several general results for sympathetic Lie algebras and for the adjoint Lie algebra cohomology of arbitrary finite-dimensional Lie algebras in characteristic zero using a result of Hochschild and Serre. Moreover, for certain semidirect products we obtain explicit results for the adjoint cohomology.

Affine Super Schur Duality

Schur duality is an equivalence, for d\leq n , between the category of finite-dimensional representations over \mathbb{C} of the symmetric group S_d on d letters, and the category of finite-dimensional representations over \mathbb{C} of \operatorname{GL}(n,\mathbb{C}) whose irreducible subquotients are subquotients of \smash{\overline{\mathbb{E}}^{\otimes d}} , \overline{\mathbb{E}}=\mathbb{C}^n . The latter are called polynomial representations homogeneous of degree d . It is based on decomposing \smash{\overline{\mathbb{E}}^{\otimes d}} as a \mathbb{C}[S_d]\times\operatorname{GL}(n,\mathbb{C}) -bimodule. It was used by Schur to conclude the semisimplicity of the category of finite-dimensional complex \operatorname{GL}(n,\mathbb{C}) -modules from the corresponding result for S_d that had been obtained by Young. Here we extend this duality to the affine super case by constructing a functor \mathcal{F}\colon M\mapsto M\otimes_{\mathbb{C}[S_d]}\mathbb{E}^{\otimes d} , \mathbb{E} now being the super vector space \mathbb{C}^{m|n} , from the category of finite-dimensional \mathbb{C}[S_d\ltimes\mathbb{Z}^d] -modules, or representations of the affine Weyl, or symmetric, group S_d^a=S_d\ltimes\mathbb{Z}^d , \emph{to} the category of finite-dimensional representations of the universal enveloping algebra of the affine Lie superalgebra \mathfrak{U}(\widehat{\operatorname{sl}}(m|n)) that are \mathbb{E}^{\otimes d} -compatible, namely the subquotients of whose restriction to \mathfrak{U}(\operatorname{sl}](m|n)) are constituents of \mathbb{E}^{\otimes d} . Both categories are not semisimple. When d<m+n the functor defines an equivalence of categories. As an application we conclude that the irreducible finite-dimensional \mathbb{E}^{\otimes d} -compatible representations of the affine superalgebra \widehat{\operatorname{sl}}(m|n) are tensor products of evaluation representations at distinct points of \mathbb{C}^\times .

Transitive irreducible Lie superalgebras of vector fields

Let $\mathfrak{d}$ be the Lie superalgebra of superderivations of the sheaf of sections of the exterior algebra of the homogeneous vector bundle $E$ over the flag variety $G/P$, where $G$ is a simple finite-dimensional complex Lie group and $P$ its parabolic subgroup. Then, $\mathfrak{d}$ is transitive and irreducible whenever $E$ is defined by an irreducible $P$-module $V$ such that the highest weight of $V^*$ is dominant. Moreover, $\mathfrak{d}$ is simple; it is isomorphic to the Lie superalgebra of vector fields on the superpoint, i.e., on a $0|n$-dimensional supervariety.

The Polynomial Modules over Quantum Group Uq(sl3)

Let [Formula: see text] be a finite dimensional complex simple Lie algebra with Cartan subalgebra [Formula: see text]. Then [Formula: see text] has a [Formula: see text]-module structure if and only if [Formula: see text] is of type [Formula: see text] or of type [Formula: see text]; this is called the polynomial module of rank one. In the quantum version, the rank one polynomial modules over [Formula: see text] have been classified. In this paper, we prove that the quantum group [Formula: see text] has no rank one polynomial module.

Twisted Ruelle zeta function at zero for compact hyperbolic surfaces

Let X be an orientable compact connected hyperbolic surface of genus g. In this paper, we prove that the twisted Selberg and Ruelle zeta functions, associated with an arbitrary finite-dimensional complex representation χ of π1(X) admit a meromorphic continuation to C. Moreover, we study the behavior of the twisted Ruelle zeta function at s=0 and prove that at this point it has a zero of order dim⁡(χ)(2g−2).

On Frobenius-Perron dimension

We propose a notion of Frobenius-Perron dimension for certain free Z \mathbb {Z} -modules of infinite rank and compute it for the Z \mathbb {Z} -modules of finite dimensional complex representations of unitary groups with nonnegative dominant weights. The definition of Frobenius-Perron dimension that we are introducing naturally generalizes the well-known Frobenius-Perron dimension on the category of finite dimensional complex representations of a finite group.

WHEN IS A COMPLETION OF THE UNIVERSAL ENVELOPING ALGEBRA A BANACH PI-ALGEBRA?

AbstractWe prove that a Banach algebra B that is a completion of the universal enveloping algebra of a finite-dimensional complex Lie algebra $\mathfrak {g}$ satisfies a polynomial identity if and only if the nilpotent radical $\mathfrak {n}$ of $\mathfrak {g}$ is associatively nilpotent in B. Furthermore, this holds if and only if a certain polynomial growth condition is satisfied on $\mathfrak {n}$ .

Generalized Perfect Codes for Symmetric Classical-Quantum Channels

We define a new family of codes for symmetric classical-quantum channels and establish their optimality. To this end, we extend the classical notion of generalized perfect and quasi-perfect codes to channels defined over some finite dimensional complex Hilbert output space. The resulting optimality conditions depend on the channel considered and on an auxiliary state defined on the output space of the channel. For certain <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> -qubit classical-quantum channels, we show that codes based on a generalization of Bell states are quasi-perfect and, therefore, they feature the smallest error probability among all codes of the same blocklength and cardinality.

ON 2-TRANSITIVE SETS OF EQUIANGULAR LINES

AbstractWe determine all finite sets of equiangular lines spanning finite-dimensional complex unitary spaces for which the action on the lines of the set-stabiliser in the unitary group is 2-transitive with a regular normal subgroup.