Finite-dimensional Complex Research Articles

On the normalizer of the reflexive cover of a unital algebra of linear transformations

Given a unital algebra A of linear transformations on a finite-dimensional complex vector space V, in this paper we study the set Col(A) consisting of those invertible linear transformations S on V which map every subspace M∈Lat(A) to a subspace SM∈Lat(A). We show that Col(A) is the normalizer of the group of invertible linear transformations in the reflexive cover of A. For the unital algebra (A) which is generated by a linear transformation A, we give the complete description of Col(A). By using the primary decomposition of A, we first reduce the problem of characterizing Col(A) to the problem of characterizing the group Col(N) of a given nilpotent linear transformation N. While Col(N) always contains all invertible linear transformations of the commutant (N)′ of N, it is always contained in the reflexive cover of (N)′. We prove that Col(N) is a proper subgroup of (AlgLat(N)′)−1 if and only if at least two Jordan blocks in the Jordan decomposition of N are of dimension 2 or more. We also determine the group Col(J2⊕J2).

Weighted Composition Operators between the Bloch Type Space and the Hardy Space on the Unit Ball of Complex Banach Spaces

Let X be a finite or infinite dimensional complex Banach space. We characterize the bounded weighted composition operators between the Bloch type space and the Hardy space on the unit ball of X, extending several known results for finite dimensional domains.

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.

A commutative algebra approach to multiplicative Hom-Lie algebras

Let be a finite-dimensional complex Lie algebra and be the affine variety of all multiplicative Hom-Lie algebras on . We use a method of computational ideal theory to describe , showing that consists of two 1-dimensional and one 3-dimensional irreducible components and for . We construct a new family of multiplicative Hom-Lie algebras on the Heisenberg Lie algebra and characterize the affine varieties and . We also study the derivation algebra of a multiplicative Hom-Lie algebra D on and, under some hypotheses on D, we prove that the Hilbert series is a rational function.

Finite dimensional systems of free fermions and diffusion processes on spin groups

In this article, we are concerned with “finite dimensional fermions,” by which we mean vectors in a finite dimensional complex space embedded in the exterior algebra over itself. These fermions are spinless but possess the characterizing anticommutativity property. We associate invariant complex vector fields on the Lie group Spin(2n + 1) to the fermionic creation and annihilation operators. These vector fields are elements of the complexification of the regular representation of the Lie algebra so(2n+1). As such, they do not satisfy the canonical anticommutation relations; however, once they have been projected onto an appropriate subspace of L2(Spin(2n + 1)), these relations are satisfied. We define a free time evolution of this system of fermions in terms of a symmetric positive-definite quadratic form in the creation–annihilation operators. The realization of fermionic creation and annihilation operators brought by the (invariant) vector fields allows us to interpret this time evolution in terms of a positive self-adjoint operator that is the sum of a second order operator, which generates a stochastic diffusion process, and a first order complex operator, which strongly commutes with the second order operator. A probabilistic interpretation is given in terms of a Feynman–Kac-like formula with respect to the diffusion process associated with the second order operator.

Kähler–Ricci solitons induced by infinite-dimensional complex space forms

We exhibit families of non trivial (i.e. not Kaehler-Einstein) radial Kaehler-Ricci solitons (KRS), both complete and not complete, which can be Kaehler immersed into infinite dimensional complex space forms. This result shows that the triviality of a KRS induced by a finite dimensional complex space form proved in [12] does not hold when the ambient space is allowed to be infinite dimensional. Moreover, we show that the radial potential of a radial KRS induced by a non-elliptic complex space form is necessarily defined at the origin.

Approximate orthogonality in complex normed spaces: a directional approach

We introduce and study the notion of approximate directional orthogonality in a complex normed space. We further explore the relevant scenario in the topological setting, which allows us to characterize approximate Birkhoff–James orthogonality of operators on a finite-dimensional complex Banach space. Our investigation extends some existing results in the literature to a broader scope.

Cohomology rings of finite-dimensional pointed Hopf algebras over abelian groups

We show that the cohomology ring of a finite-dimensional complex pointed Hopf algebra with an abelian group of group-like elements is finitely generated. Our strategy has three major steps. We first reduce the problem to the finite generation of cohomology of finite dimensional Nichols algebras of diagonal type. For the Nichols algebras, we do a detailed analysis of cohomology via the Anick resolution reducing the problem further to specific combinatorial properties. Finally, to check these properties, we turn to the classification of Nichols algebras of diagonal type due to Heckenberger. In this paper, we complete the verification of these combinatorial properties for major parametric families, including Nichols algebras of Cartan and super types and develop all the theoretical foundations necessary for the case-by-case analysis. The remaining discrete families are addressed in a separate publication. As an application of the main theorem, we deduce finite generation of cohomology for other classes of finite-dimensional Hopf algebras, including basic Hopf algebras with abelian groups of characters and finite quotients of quantum groups at roots of one.

Uniform probability distribution over all density matrices

Let mathscr {H} be a finite-dimensional complex Hilbert space and mathscr {D} the set of density matrices on mathscr {H}, i.e., the positive operators with trace 1. Our goal in this note is to identify a probability measure u on mathscr {D} that can be regarded as the uniform distribution over mathscr {D}. We propose a measure on mathscr {D}, argue that it can be so regarded, discuss its properties, and compute the joint distribution of the eigenvalues of a random density matrix distributed according to this measure.

Representation growth of compact special linear groups of degree two

We study the finite-dimensional continuous complex representations of SL2 over the ring of integers of non-Archimedean local fields of even residual characteristic. We prove that for characteristic two, the abscissa of convergence of the representation zeta function is 1, resolving the last remaining open case of this problem. We additionally prove that, contrary to the expectation, the group algebras of C[SL2(Z/(22rZ))] and C[SL2(F2[t]/(t2r))] are not isomorphic for any r>1. This is the first known class of reductive groups over finite rings wherein the representation theory in the equal and mixed characteristic settings is genuinely different. From our methods, we explicitly obtain the primitive representation zeta polynomials of SL2(F2[t]/(t2r)) and SL2(Z/22rZ) for 1≤r≤3.

Categories of weight modules for unrolled restricted quantum groups at roots of unity

Motivated by connections to the singlet vertex operator algebra in the $\mathfrak{g}=\mathfrak{sl}_2$ case, we study the unrolled restricted quantum group $\overline{U}_q^{ H}(\mathfrak{g})$ for any finite dimensional complex simple Lie algebra $\mathfrak{g}$ at arbitrary roots of unity with a focus on its category of weight modules. We show that the braid group action naturally extends to the unrolled quantum groups and that the category of weight modules is a generically semi-simple ribbon category (previously known only for odd roots) with trivial Müger center and self-dual projective modules. Bibliography: 44 titles.

Categories of weight modules for unrolled restricted quantum groups at roots of unity

Motivated by connections to the singlet vertex operator algebra in the $\mathfrak{g}=\mathfrak{sl}_2$ case, we study the unrolled restricted quantum group $\overline{U}_q^{ H}(\mathfrak{g})$ for any finite dimensional complex simple Lie algebra $\mathfrak{g}$ at arbitrary roots of unity with a focus on its category of weight modules. We show that the braid group action naturally extends to the unrolled quantum groups and that the category of weight modules is a generically semi-simple ribbon category (previously known only for odd roots) with trivial Müger center and self-dual projective modules.

Open Subsets in a Stein Space with Singularities

Serre proved that a domain $Y$ in $\mathbb{C}^n$ is Stein if and only if $H^i(Y,\mathcal{O}_Y) = 0$ for all $i \gt 0$. Laufer showed that if $Y$ is an open subset of a Stein manifold of dimension $n$ and $H^i(Y,\mathcal{O}_Y)$ is a finite dimensional complex vector space for every $i \gt 0$, then $Y$ is Stein. Vâjâitu generalized these theorems to singular Stein space of dimension $n$. In this paper, we consider singular Stein spaces $X$ with arbitrary dimension and give necessary and sufficient conditions for an open subset $Y$ in $X$ to be Stein. We show that if $Y$ is an open subset of a reduced Stein space $X$ with arbitrary dimension and singularities, then $Y$ is Stein if and only if $H^i(Y,\mathcal{O}_Y)$ is a finite dimensional complex vector space for every $i \gt 0$. Without cohomology condition, if $X-Y$ is a closed subspace of $X$, then we show that the geometric condition of the boundary $X-Y$ determines the Steinness of $Y$. More precisely, we show that if $X$ is normal and the boundary $X-Y$ is the support of an effective $\mathbb{Q}$-Cartier divisor, or $X-Y$ is of pure codimension $1$ and does not contain any singular points of $X$, then $Y$ is Stein.

Doubly transitive lines I: Higman pairs and roux

We study lines through the origin of finite-dimensional complex vector spaces that enjoy a doubly transitive automorphism group. In doing so, we make fundamental connections with both discrete geometry and algebraic combinatorics. In particular, we show that doubly transitive lines are necessarily optimal packings in complex projective space, and we introduce a fruitful generalization of regular abelian distance-regular antipodal covers of the complete graph.

Orthogonal projections on hyperplanes intertwined with unitaries

Fix a point in a finite-dimensional complex vector space and consider the sequence of iterates of this point under the composition of a unitary map with the orthogonal projection on the hyperplane orthogonal to the starting point. We prove that, generically, the series of the squared norms of these iterates sums to the dimension of the underlying space. This leads us to construct a (device-dependent) dimension witness for quantum systems which involves the probabilities of obtaining certain strings of outcomes in a sequential yes–no measurement. The exact formula for this series in non-generic cases is provided as well as its analogue in the real case.

Deligne Categories and the Periplectic Lie Superalgebra

We study stabilization of finite-dimensional representations of the periplectic Lie superalgebras $\mathfrak{p}(n)$ as $n \to \infty$. The paper gives a construction of the tensor category $Rep(\underline{P})$, possessing nice universal properties among tensor categories over the category $\mathtt{sVect}$ of finite-dimensional complex vector superspaces. First, it is the envelope of the Deligne category corresponding to the periplectic Lie superalgebra, in the sense of arXiv:1511.07699. Secondly, given a tensor category $\mathcal{C}$ over $\mathtt{sVect}$, exact tensor functors $Rep(\underline{P})\longrightarrow \mathcal{C}$ classify pairs $(X, \omega)$ in $\mathcal{C}$ where $\omega: X \otimes X \to \Pi \mathbf{1}$ is a non-degenerate symmetric form and $X$ not annihilated by any Schur functor. The category $Rep(\underline{P})$ is constructed in two ways. The first construction is through an explicit limit of the tensor categories $Rep(\mathfrak{p}(n))$ ($n\geq 1$) under Duflo-Serganova functors. The second construction (inspired by P. Etingof) describes $Rep(\underline{P})$ as the category of representations of a periplectic Lie supergroup in the Deligne category $\mathtt{sVect} \boxtimes Rep(\underline{GL}_t)$. An upcoming paper by the authors will give results on the abelian and tensor structure of $Rep(\underline{P})$.

Arens–Michael envelopes of nilpotent Lie algebras, holomorphic functions of exponential type, and homological epimorphisms

Our aim is to give an explicit description of the Arens-Michael envelope for the universal enveloping algebra of a finite-dimensional nilpotent complex Lie algebra. It turns out that the Arens-Michael envelope belongs to a class of completions introduced by R.~Goodman in 70s. To find a precise form of this algebra we preliminary characterize the set of holomorphic functions of exponential type on a simply connected nilpotent complex Lie group. This approach leads to unexpected connections to Riemannian geometry and the theory of order and type for entire functions. As a corollary, it is shown that the Arens-Michael envelope considered above is a homological epimorphism. So we get a positive answer to a question investigated earlier by Dosi and Pirkovskii.

The polarization constant of finite dimensional complex spaces is one

AbstractThe polarization constant of a Banach spaceXis defined as\[{\text{c}}(X){\text{ }}{\text{ }}\mathop {\lim }\limits_{k \to \infty } {\text{ }}\sup {\text{c}}{(k,X)^{\frac{1}{k}}},\]where\[{\text{c}}(k,X)\]stands for the best constant\[C > 0\]such that\[\mathop P\limits^ \vee \leqslant CP\]for everyk-homogeneous polynomial\[P \in \mathcal{P}{(^k}X)\]. We show that ifXis a finite dimensional complex space then\[{\text{c}}(X) = 1\]. We derive some consequences of this fact regarding the convergence of analytic functions on such spaces.The result is no longer true in the real setting. Here we relate this constant with the so-called Bochnak’s complexification procedure.We also study some other properties connected with polarization. Namely, we provide necessary conditions related to the geometry ofXfor\[c(2,X) = 1\]to hold. Additionally we link polarization constants with certain estimates of the nuclear norm of the product of polynomials.

Bloch functions on the unit ball of a Banach space

The space of Bloch functions on bounded symmetric domains is extended by considering Bloch functions f f on the unit ball B E B_E of finite and infinite dimensional complex Banach spaces in two different ways: by extending the classical Bloch space considering the boundness of ( 1 − ‖ x ‖ 2 ) ‖ f ′ ( x ) ‖ (1-\|x\|^2) \|f’(x)\| on B E B_E and by preserving the invariance of the correspondiing seminorm when we compose with automorphisms φ \varphi of B E B_E . We study the connection between these spaces proving that they are different in general and prove that all bounded analytic functions on B E B_{E} are Bloch functions in both ways.

A simple construction of basic polynomials invariant under the Weyl group of the simple finite-dimensional complex Lie algebra

Abstract For every simple finite-dimensional complex Lie algebra, I give a simple construction of all (except for the Pfaffian) basic polynomials invariant under the Weyl group. The answer is given in terms of the two basic polynomials of smallest degree.