Hall Polynomials for Nakayama Algebras

Let \(\mathcal{A}\) be a representation finite algebra over finite field k. In this note we first show that the existence of Hall polynomials for \(\mathcal{A}\) equivalent to the existence of the Hall polynomial \(\varphi^{M}_{N L}\) for each \(M, L \in mod\mathcal{A}\) and \(N\in ind\mathcal{A}\). Then we show that for a basic connected Nakayama algebra \(\mathcal{A}\), \(\mathcal{H}(\mathcal{A})=\mathcal{L}(\mathcal{A})\) and Hall polynomials exist for this algebra. We also provide another proof of the existence of Hall polynomials for the representation directed split algebras.     

Enveloping Algebras of the Nilpotent Malcev Algebra of Dimension Five

Pérez-Izquierdo and Shestakov recently extended the PBW theorem to Malcev algebras. It follows from their construction that for any Malcev algebra M over a field of characteristic ≠ 2, 3 there is a representation of the universal nonassociative enveloping algebra U(M) by linear operators on the polynomial algebra P(M). For the nilpotent non-Lie Malcev algebra \(\mathbb{M}\) of dimension 5, we use this representation to determine explicit structure constants for \(U(\mathbb{M})\); from this it follows that \(U(\mathbb{M})\) is not power-associative. We obtain a finite set of generators for the alternator ideal \(I(\mathbb{M}) \subset U(\mathbb{M})\) and derive structure constants for the universal alternative enveloping algebra \(A(\mathbb{M}) = U(\mathbb{M})/I(\mathbb{M})\), a new infinite dimensional alternative algebra. We verify that the map \(\iota\colon \mathbb{M} \to A(\mathbb{M})\) is injective, and so \(\mathbb{M}\) is special.     

Ideal Connes-Amenability of Dual Banach Algebras

Following Runde, we define the concept of ideal Connes-amenability for dual Banach algebras. For an Arens regular dual Banach algebra \({\mathcal {A}}\), we prove that the ideal Connes-amenability of \(\mathcal {A^{**}}\), the second dual of \({\mathcal {A}}\) necessities ideal Connes-amenability of \({{\mathcal {A}}}\). As a typical example, we show that von Neumann algebras are always ideally Connes-amenable. For a locally compact group G, the Fourier–Stieltjes algebra of G is ideally Connes-amenable, but not ideally amenable.     

Pure States on Cuntz Algebras Arising from Geometric Progressions

Let \(\mathcal {O}_{n}\) denote the Cuntz algebra for n ≥ 2. We introduce an embedding f of \(\mathcal {O}_{m}\) into \(\mathcal {O}_{n}\) arising from a geometric progression of Cuntz generaters of \(\mathcal {O}_{n}\). By identifying \(\mathcal {O}_{m}\) with \(f(\mathcal {O}_{m})\), we extend Cuntz states on \(\mathcal {O}_{m}\) to \(\mathcal {O}_{n}\). We show (i) a necessary and sufficient condition of the uniqueness of the extension, (ii) the complete classification of all such extensions up to unitary equivalence of their GNS representations, and (iii) the decomposition formula of a mixing state into a convex hull of pure states. The complete set of invariants of all GNS representations by such pure states is given as a certain set of complex unit vectors.     

A binomial Laurent phenomenon algebra associated with the complete graph

In this paper, we find the exchange graph of \(\mathcal {A}({\tau _n})\), the rank n binomial Laurent phenomenon algebra associated with the complete graph \(K_n\). More specifically, we prove that the exchange graph is isomorphic to that of \(\mathcal {A}(t_n)\), a rank n linear Laurent phenomenon algebra associated with the complete graph which is discussed in Lam and Pylyavskyy (Linear Laurent phenomenon algebras, arXiv:1206.2612v2, 2012).     

Gradings on Modules Over Lie Algebras of E Types

For any grading by an abelian group G on the exceptional simple Lie algebra \(\mathcal {L}\) of type E ₆ or E ₇ over an algebraically closed field of characteristic zero, we compute the graded Brauer invariants of simple finite-dimensional modules, thus completing the computation of these invariants for simple finite-dimensional Lie algebras. This yields the classification of finite-dimensional G-graded simple \(\mathcal {L}\)-modules, as well as necessary and sufficient conditions for a finite-dimensional \(\mathcal {L}\)-module to admit a G-grading compatible with the given G-grading on \(\mathcal {L}\).     

Isometric multipliers of a vector valued Beurling algebra on a discrete semigroup

Let (S,ω) be a weighted abelian semigroup, let M _ω (S) be the semigroup of ω-bounded multipliers of S, and let \(\mathcal {A}\) be a strictly convex commutative Banach algebra with identity. It is shown that T is an onto isometric multiplier of \(\ell ^{1}(S,\omega , \mathcal {A})\) if and only if there exists an invertible σ ∈ M _ω (S), a unitary point \(a \in \mathcal {A}\), and a k>0 such that \(T(f)= ka{\sum }_{x \in S} f(x)\delta _{\sigma (x)}\) for each \(f={\sum }_{x \in S}f(x)\delta _{x} \in \ell ^{1}(S,\omega ,\mathcal {A})\). It is also shown that an isomorphism from \(\ell ^{1}(S_{1},\omega _{1},\mathcal {A})\) onto \(\ell ^{1}(S_{2},\omega _{2}, \mathcal {B})\) induces an isomorphism from \(M(\ell ^{1}(S_{1},\omega _{1},\mathcal {A}))\), the set of all multipliers of \(\ell ^{1}(S_{1},\omega _{1},\mathcal {A})\), onto \(M(\ell ^{1}(S_{2},\omega _{2},\mathcal {B}))\).     

Compact right multipliers on a Banach algebra related to locally compact semigroups

For a locally compact semigroup \({\mathcal{S}}\), let \(L_{0}^{\infty}({\mathcal{S}},M_{a}({\mathcal{S}}))\) be the Banach space of all μ-measurable (\(\mu\in M_{a}({\mathcal{S}})\)) functions vanishing at infinity, where \(M_{a}({\mathcal{S}})\) denotes the algebra of all measures in the measure algebra \(M({\mathcal{S}})\) of \({\mathcal{S}}\) with continuous translations. Here, we study right compact multipliers on the Banach algebra \(L_{0}^{\infty}({\mathcal{S}},M_{a}({\mathcal{S}}))^{*}\) equipped with an Arens product.     

Lie Algebras of Slow Growth and Klein-Gordon PDE

We discuss the notion of characteristic Lie algebra of a hyperbolic PDE. The integrability of a hyperbolic PDE is closely related to the properties of the corresponding characteristic Lie algebra χ. We establish two explicit isomorphisms:

1. 1)
   the first one is between the characteristic Lie algebra \(\chi (\sinh {u})\) of the sinh-Gordon equation \(u_{xy}=\sinh {u}\) and the non-negative part \({\mathcal {L}}({\mathfrak {sl}}(2,{\mathbb {C}}))^{\ge 0}\) of the loop algebra of \({\mathfrak {sl}}(2,{\mathbb {C}})\) that corresponds to the Kac-Moody algebra \(A_{1}^{(1)}\)

   $$\chi(\sinh{u})\cong {\mathcal{L}}({\mathfrak{s}\mathfrak{l}}(2,{\mathbb{C}}))^{\ge 0}={\mathfrak{s}\mathfrak{l}}(2, {\mathbb{C}}) \otimes {\mathbb{C}}[t]. $$
    
2. 2)
   the second isomorphism is for the Tzitzeica equation u_xy = e^u + e^??2u

   $$\chi(e^{u}{+}e^{-2u}) \cong {\mathcal{L}}({\mathfrak{s}\mathfrak{l}}(3,{\mathbb{C}}), \mu)^{\ge0}=\bigoplus_{j = 0}^{+\infty}{\mathfrak{g}}_{j (\text{mod} \; 2)} \otimes t^{j}, $$

   where \({\mathcal {L}}({\mathfrak {sl}}(3,{\mathbb {C}}), \mu )=\bigoplus _{j \in {\mathbb {Z}}}{\mathfrak {g}}_{j (\text {mod} \; 2)} \otimes t^{j}\) is the twisted loop algebra of the simple Lie algebra \({\mathfrak {sl}}(3,{\mathbb {C}})\) that corresponds to the Kac-Moody algebra \(A_{2}^{(2)}\).
    

Hence the Lie algebras \(\chi (\sinh {u})\) and χ(e^u + e^??2u) are slowly linearly growing Lie algebras with average growth rates \(\frac {3}{2}\) and \(\frac {4}{3}\) respectively.     

Representations and Corresponding Operators Induced by Hecke Algebras

In this paper, by establishing free-probabilistic models on the Hecke algebras \(\mathcal {H}(G_{p})\), we construct canonical free probability spaces \((\mathcal {H}(G_{p}), \psi _{p})\), where \(G_{p} = GL_{2}(\mathbb {Q} _{p})\), for primes \(p\). Dependent upon such free-probabilistic structures, we study corresponding representations of \(\mathcal {H}(G_{p})\), and consider spectral properties of operators realized under representations.     

Geometry of Coadjoint Orbits and Multiplicity-one Branching Laws for Symmetric Pairs

Consider the restriction of an irreducible unitary representation π of a Lie group G to its subgroup H. Kirillov’s revolutionary idea on the orbit method suggests that the multiplicity of an irreducible H-module ν occurring in the restriction π|_H could be read from the coadjoint action of H on \(\mathcal {O}^{G} \cap \text {pr}^{-1}({\mathcal {O}}^{H})\), provided π and ν are ‘geometric quantizations’ of a G-coadjoint orbit \(\mathcal {O}^{G}\) and an H-coadjoint orbit \(\mathcal {O}^{H}\), respectively, where \(\text {pr} \colon \sqrt {-1}\mathfrak {g}^{\ast } \to \sqrt {-1}\mathfrak {h}^{\ast }\) is the projection dual to the inclusion \(\mathfrak {h} \subset \mathfrak {g}\) of Lie algebras. Such results were previously established by Kirillov, Corwin and Greenleaf for nilpotent Lie groups. In this article, we highlight specific elliptic orbits \(\mathcal {O}^{G}\) of a semisimple Lie group G corresponding to highest weight modules of scalar type. We prove that the Corwin–Greenleaf number \(\sharp (\mathcal {O}^{G} \cap \text {pr}^{-1}({\mathcal {O}}^{H}))/H\) is either zero or one for any H-coadjoint orbit \(\mathcal {O}^{H}\), whenever (G,H) is a symmetric pair of holomorphic type. Furthermore, we determine the coadjoint orbits \(\mathcal {O}^{H}\) with nonzero Corwin–Greenleaf number. Our results coincide with the prediction of the orbit philosophy, and can be seen as ‘classical limits’ of the multiplicity-free branching laws of holomorphic discrete series representations (Kobayashi [Progr. Math. 2007]).     

Stringy K-theory and the Chern character

We construct two new G-equivariant rings: \(\mathcal{K}(X,G)\), called the stringy K-theory of the G-variety X, and \(\mathcal{H}(X,G)\), called the stringy cohomology of the G-variety X, for any smooth, projective variety X with an action of a finite group G. For a smooth Deligne–Mumford stack \(\mathcal{X}\), we also construct a new ring \(\mathsf{K}_{\mathrm{orb}}(\mathcal{X})\) called the full orbifold K-theory of \(\mathcal{X}\). We show that for a global quotient \(\mathcal{X} = [X/G]\), the ring of G-invariants \(K_{\mathrm{orb}}(\mathcal{X})\) of \(\mathcal{K}(X,G)\) is a subalgebra of \(\mathsf{K}_{\mathrm{orb}}([X/G])\) and is linearly isomorphic to the “orbifold K-theory” of Adem-Ruan [AR] (and hence Atiyah-Segal), but carries a different “quantum” product which respects the natural group grading.We prove that there is a ring isomorphism \(\mathcal{C}\mathbf{h}:\mathcal{K}(X,G)\to\mathcal{H}(X,G)\), which we call the stringy Chern character. We also show that there is a ring homomorphism \(\mathfrak{C}\mathfrak{h}_\mathrm{orb}:\mathsf{K}_{\mathrm{orb}}(\mathcal{X}) \rightarrow H^\bullet_{\mathrm{orb}}(\mathcal{X})\), which we call the orbifold Chern character, which induces an isomorphism \(Ch_{\mathrm{orb}}:K_{\mathrm{orb}}(\mathcal{X})\rightarrow H^\bullet_{\mathrm{orb}}(\mathcal{X})\) when restricted to the sub-algebra \(K_{\mathrm{orb}}(\mathcal{X})\). Here \(H_{\mathrm{orb}}^\bullet(\mathcal{X})\) is the Chen–Ruan orbifold cohomology. We further show that \(\mathcal{C}\mathbf{h}\) and \(\mathfrak{C}\mathfrak{h}_\mathrm{orb}\) preserve many properties of these algebras and satisfy the Grothendieck–Riemann–Roch theorem with respect to étale maps. All of these results hold both in the algebro-geometric category and in the topological category for equivariant almost complex manifolds.We further prove that \(\mathcal{H}(X,G)\) is isomorphic to Fantechi and Göttsche’s construction [FG, JKK]. Since our constructions do not use complex curves, stable maps, admissible covers, or moduli spaces, our results greatly simplify the definitions of the Fantechi–Göttsche ring, Chen–Ruan orbifold cohomology, and the Abramovich–Graber–Vistoli orbifold Chow ring.We conclude by showing that a K-theoretic version of Ruan’s Hyper-Kähler Resolution Conjecture holds for the symmetric product of a complex projective surface with trivial first Chern class.     

Largest initial segments pointwise fixed by automorphisms of models of set theory

Given a model \(\mathcal {M}\) of set theory, and a nontrivial automorphism j of \(\mathcal {M}\), let \(\mathcal {I}_{\mathrm {fix}}(j)\) be the submodel of \(\mathcal {M}\) whose universe consists of elements m of \(\mathcal {M}\) such that \(j(x)=x\) for every x in the transitive closure of m (where the transitive closure of m is computed within \(\mathcal {M}\)). Here we study the class \(\mathcal {C}\) of structures of the form \(\mathcal {I}_{\mathrm {fix}}(j)\), where the ambient model \(\mathcal {M}\) satisfies a frugal yet robust fragment of \(\mathrm {ZFC}\) known as \(\mathrm {MOST}\), and \(j(m)=m\) whenever m is a finite ordinal in the sense of \(\mathcal {M}.\) Our main achievement is the calculation of the theory of \(\mathcal {C}\) as precisely \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\). The following theorems encapsulate our principal results: Theorem A. Every structure in \(\mathcal {C}\) satisfies \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm { Collection}\). Theorem B. Each of the following three conditions is sufficient for a countable structure \(\mathcal {N}\) to be in \(\mathcal {C}\):(a) \(\mathcal {N}\) is a transitive model of \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\).(b) \(\mathcal {N}\) is a recursively saturated model of \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\).(c) \(\mathcal {N}\) is a model of \(\mathrm {ZFC}\). Theorem C. Suppose \(\mathcal {M}\) is a countable recursively saturated model of \(\mathrm {ZFC}\) and I is a proper initial segment of \(\mathrm {Ord}^{\mathcal {M}}\) that is closed under exponentiation and contains \(\omega ^\mathcal {M}\) . There is a group embedding \(j\longmapsto \check{j}\) from \(\mathrm {Aut}(\mathbb {Q})\) into \(\mathrm {Aut}(\mathcal {M})\) such that I is the longest initial segment of \(\mathrm {Ord}^{\mathcal {M}}\) that is pointwise fixed by \(\check{j}\) for every nontrivial \(j\in \mathrm {Aut}(\mathbb {Q}).\) In Theorem C, \(\mathrm {Aut}(X)\) is the group of automorphisms of the structure X, and \(\mathbb {Q}\) is the ordered set of rationals.     

Dual Lie bialgebra structures of Poisson types

Let \(\mathcal{A} = \mathbb{F}[x,y]\) be the polynomial algebra on two variables x, y over an algebraically closed field \(\mathbb{F}\) of characteristic zero. Under the Poisson bracket, \(\mathcal{A}\) is equipped with a natural Lie algebra structure. It is proven that the maximal good subspace of \(\mathcal{A}*\) induced from the multiplication of the associative commutative algebra \(\mathcal{A}\) coincides with the maximal good subspace of \(\mathcal{A}*\) induced from the Poisson bracket of the Poisson Lie algebra \(\mathcal{A}\). Based on this, structures of dual Lie bialgebras of the Poisson type are investigated. As by-products, five classes of new infinite-dimensional Lie algebras are obtained.     

Recurrent Dirichlet Forms and Markov Property of Associated Gaussian Fields

For the extended Dirichlet space \(\mathcal {F}_{e}\) of a general irreducible recurrent regular Dirichlet form \((\mathcal {E},\mathcal {F})\) on L ²(E;m), we consider the family \(\mathbb {G}(\mathcal {E})=\{X_{u};u\in \mathcal {F}_{e}\}\) of centered Gaussian random variables defined on a probability space \(({\Omega }, \mathcal {B}, \mathbb {P})\) indexed by the elements of \(\mathcal {F}_{e}\) and possessing the Dirichlet form \(\mathcal {E}\) as its covariance. We formulate the Markov property of the Gaussian field \(\mathbb {G}(\mathcal {E})\) by associating with each set A ? E the sub-σ-field σ(A) of \(\mathcal {B}\) generated by X _u for every \(u\in \mathcal {F}_{e}\) whose spectrum s(u) is contained in A. Under a mild absolute continuity condition on the transition function of the Hunt process associated with \((\mathcal {E}, \mathcal {F})\), we prove the equivalence of the Markov property of \(\mathbb {G}(\mathcal {E})\) and the local property of \((\mathcal {E},\mathcal {F})\). One of the key ingredients in the proof is in that we construct potentials of finite signed measures of zero total mass and show that, for any Borel set B with m(B) >?0, any function \(u\in \mathcal {F}_{e}\) with s(u) ? B can be approximated by a sequence of potentials of measures supported by B.     

Lie Triple Derivations on von Neumann Algebras

Let A be a von Neumann algebra with no central abelian projections. It is proved that if an additive map δ :A → A satisfies δ([[a, b], c]) = [[δ(a), b], c] + [[a, δ(b)], c] +[[a, b], δ(c)] for any a, b, c∈ A with ab = 0(resp. ab = P, where P is a fixed nontrivial projection in A), then there exist an additive derivation d from A into itself and an additive map f :A → ZA vanishing at every second commutator [[a, b], c] with ab = 0(resp.ab = P) such that δ(a) = d(a) + f(a) for any a∈ A.     

On the topological center of a Banach algebra related to a foundation topological semigroup

Let \({\mathcal{S}}\) be a locally compact semigroup and \(L_{0}^{\infty}({\mathcal{S}},M_{a}({\mathcal{S}}))\) be the Banach space of all μ-measurable (\(\mu\in M_{a}({\mathcal{S}})\)) functions vanishing at infinity, where \(M_{a}({\mathcal{S}})\) denotes the algebra of all measures with continuous translations. Recently, we have shown that \(L_{0}^{\infty}({\mathcal{S}},M_{a}({\mathcal{S}}))^{*}\) can be equipped with an Arens type product. Here, we show that the topological center of \(L_{0}^{\infty}({\mathcal{S}},M_{a}({\mathcal{S}}))^{*}\) coincides with \(M_{a}({\mathcal{S}})\) for a class of locally compact semigroups \({\mathcal{S}}\): this gives a partial solution to a conjecture raised by the authors.     

On Homomorphisms from Ringel-Hall Algebras to Quantum Cluster Algebras

In Berenstein and Rupel (2015), the authors defined algebra homomorphisms from the dual Ringel-Hall algebra of certain hereditary abelian category \(\mathcal {A}\) to an appropriate q-polynomial algebra. In the case that \(\mathcal {A}\) is the representation category of an acyclic quiver, we give an alternative proof by using the cluster multiplication formulas in (Ding and Xu, Sci. China Math. 55(10) 2045–2066, 2012). Moreover, if the underlying graph of Q associated with \(\mathcal {A}\) is bipartite and the matrix B associated to the quiver Q is of full rank, we show that the image of the algebra homomorphism is in the corresponding quantum cluster algebra.