Power boundedness in the Fourier and Fourier–Stieltjes algebras on homogeneous spaces

We study power boundedness in the Fourier and Fourier–Stieltjes algebras, Open image in new window and Open image in new window of a homogeneous space Open image in new window The main results characterizes when all elements with spectral radius at most one, in any of these algebras, are power bounded.     

Segal–Bargmann and Weyl transforms on compact Lie groups

We present an elementary derivation of the reproducing kernel for invariant Fock spaces associated with compact Lie groups which, as Ólafsson and Ørsted showed in (Lie Theory and its Applicaitons in Physics. World Scientific, 1996), yields a simple proof of the unitarity of Hall’s Segal–Bargmann transform for compact Lie groups K. Further, we prove certain Hermite and character expansions for the heat and reproducing kernels on K and \({K_{\mathbb C}}\) . Finally, we introduce a Toeplitz (or Wick) calculus as an attempt to have a quantization of the functions on \({K_{\mathbb C}}\) as operators on the Hilbert space L ²(K).     

Segal–Bargmann and Weyl transforms on compact Lie groups

We present an elementary derivation of the reproducing kernel for invariant Fock spaces associated with compact Lie groups which, as ólafsson and ?rsted showed in (Lie Theory and its Applicaitons in Physics. World Scientific, 1996), yields a simple proof of the unitarity of Hall’s Segal–Bargmann transform for compact Lie groups K. Further, we prove certain Hermite and character expansions for the heat and reproducing kernels on K and K_{\mathbb C}{K_{\mathbb C}} . Finally, we introduce a Toeplitz (or Wick) calculus as an attempt to have a quantization of the functions on K_{\mathbb C}{K_{\mathbb C}} as operators on the Hilbert space L ²(K).     

Givental group action on topological field theories and homotopy Batalin–Vilkovisky algebras

In this paper, we initiate the study of the Givental group action on Cohomological Field Theories in terms of homotopical algebra. More precisely, we show that the stabilisers of Topological Field Theories in genus 0 (respectively in genera 0 and 1) are in one-to-one correspondence with commutative homotopy Batalin–Vilkovisky algebras (respectively wheeled commutative homotopy BV-algebras).     

Mixed Stieltjes–Hilbert and Fourier Sine and Cosine Convolutions

We introduce generalized convolutions of Stieltjes, Hilbert, and Fourier sine and cosine transforms and consider their applications to integral equations.     

A Fourier–Mukai approach to the K-theory of compact Lie groups

Let G be a compact, connected, simply-connected Lie group. We use the Fourier–Mukai transform in twisted K-theory to give a new proof of the ring structure of the K-theory of G.     

Relative trace formulae toward Bessel and Fourier–Jacobi periods on unitary groups

We propose an approach, via relative trace formulae, toward the global restriction problem involving Bessel or Fourier–Jacobi periods on unitary groups \({{\rm U}_n \times {\rm U}_m}\) , generalizing the work of Jacquet–Rallis for m = n ? 1 (which is a Bessel period). In particular, when m = 0, we recover a relative trace formula proposed by Flicker concerning Kloosterman/Fourier integrals on quasi-split unitary groups. As evidences for our approach, we prove the vanishing part of the fundamental lemmas in all cases, and the full lemma for \({{\rm U}_n \times {\rm U}_n}\) .     

Endomorphism algebras and Igusa–Todorov algebras

Let A be an Artin algebra. If $V\in \operatorname{mod} A$ such that the global dimension of  $\operatorname{End}_{A}V$ is at most 3, then for any ${M\in \operatorname{add}_{A}V}$ , both B and B ^op are 2-Igusa–Todorov algebras, where ${B=\operatorname{End}_{A}M}$ . Let ${P\in \operatorname{mod} A}$ be projective and ${B=\operatorname{End}_{A}P}$ such that the projective dimension of P as a right B-module is at most n(<∞). If A is an m-syzygy-finite algebra (resp. an m-Igusa–Todorov algebra), then B is an (m+n)-syzygy-finite algebra (resp. an (m+n)-Igusa–Todorov algebra); in particular, the finitistic dimension of B is finite in both cases. Some applications of these results are given.     

Lefschetz fixed point theorems for Fourier–Mukai functors and DG algebras

We propose some variants of Lefschetz fixed point theorem for Fourier–Mukai functors on a smooth projective algebraic variety. Independently we also suggest a similar theorem for endo-functors on the category of perfect modules over a smooth and proper DG algebra.     

Open subgroups of locally compact Kac–Moody groups

Let $G$ be a complete Kac–Moody group over a finite field. It is known that $G$ possesses a BN-pair structure, all of whose parabolic subgroups are open in $G$ . We show that, conversely, every open subgroup of $G$ is contained with finite index in some parabolic subgroup; moreover there are only finitely many such parabolic subgroups. The proof uses some new results on parabolic closures in Coxeter groups. In particular, we give conditions ensuring that the parabolic closure of the product of two elements in a Coxeter group contains the respective parabolic closures of those elements.     

On Fomin–Kirillov algebras for complex reflection groups

In this note, we apply classification results for finite-dimensional Nichols algebras to generalizations of Fomin–Kirillov algebras to complex reflection groups. First, we focus on the case of cyclic groups where the corresponding Nichols algebras are only finite-dimensional up to order four, and we include results about the existence of Weyl groupoids and finite-dimensional Nichols subalgebras for this class. Second, recent results by Heckenberger–Vendramin [ArXiv e-prints, 1412.0857 (December 2014)] on the classification of Nichols algebras of semisimple group type can be used to find that these algebras are infinite-dimensional for many non-exceptional complex reflection groups in the Shephard–Todd classification.     

Gröbner-Shirshov bases for free Gelfand-Dorfman-Novokov algebras and for right ideals of free right Leibniz algebras

In this paper, we first found a magmatic (i.e., absolutely non-associative) Gröbner-Shirshov basis of a free Gelfand-Dorfman-Novikov algebra GDN(X) such that the corresponding set of irreducible magmatic words is the Dzhumadildaev-Löfwall linear basis of the GDN(X). Then, we prove a Composition-Diamond lemma for right ideals of a free right Leibniz algebra Lei(X).