Abstract We introduce an $$L^p$$ L p -operator algebraic analogue of Hilbert C*-modules. We present the theory of concrete $$L^p$$ L p -modules, their morphisms, and basic constructions including countable direct sums and tensor products. We then define $$L^p$$ L p -correspondences and the interior tensor product of these.

Abstract We propose a concept of truncation for arbitrary smooth projective toric varieties and construct explicit cellular resolutions for nef truncations of their total coordinate rings. We show that these resolutions agree with the short resolutions of Hanlon, Hicks, and Lazarev, which were motivated by symplectic geometry, and we use our definition to exhibit nontrivial homology in the commutative algebraic analogue of their construction.

Abstract These notes were prepared for the Lefschetz Preparatory School , a graduate summer course held in Krakow, May 6–10, 2024. They present the story of the algebraic Lefschetz properties from their origin in algebraic geometry to some recent developments in commutative algebra. The common thread of the notes is a bias towards topics surrounding the algebraic Lefschetz properties that have a topological flavor. These range from the Hard Lefschetz Theorem for cohomology rings to commutative algebraic analogues of these rings, namely artinian Gorenstein rings, and topologically motivated operations among such rings.

A phenomenon of "algebraic self-similarity" on 3d cubic lattice, providing what can be called an algebraic analogue of Kadanoff--Wilson theory, is shown to possess a 4d version as well. Namely, if there is a $4\times 4$ matrix $A$ whose entries are indeterminates over the field $\mathbb F_2$, then the $2\times 2\times 2\times 2$ block made of sixteen copies of $A$ reveals the existence of four direct "block spin" summands corresponding to the same matrix $A$. Moreover, these summands can be written out in quite an elegant way. Somewhat strikingly, if the entries of $A$ are just zeros and ones -- elements of $\mathbb F_2$ -- then there are examples where two more "block spins" split out, and this time with different $A$'s.Comment: 12 pages, 3 figures

Let E / Q E/\mathbf {Q} be an elliptic curve and let p p be an odd prime of good reduction for E E . Let K K be an imaginary quadratic field satisfying the classical Heegner hypothesis and in which p p splits. The goal of this paper is two-fold: (1) we formulate a p p -adic BSD conjecture for the p p -adic L L -function L p B D P L_\mathfrak {p}^{\mathrm {BDP}} introduced by Bertolini–Darmon–Prasanna [Duke Math. J. 162 (2013), pp. 1033–1148]; and (2) for an algebraic analogue F p ¯ B D P F_{\overline {\mathfrak {p}}}^{\mathrm {BDP}} of L p B D P L_\mathfrak {p}^{\mathrm {BDP}} , we show that the “leading coefficient” part of our conjecture holds, and that the “order of vanishing” part follows from the expected “maximal non-degeneracy” of an anticyclotomic p p -adic height. In particular, when the Iwasawa–Greenberg Main Conjecture ( F p ¯ B D P ) = ( L p B D P ) (F_{\overline {\mathfrak {p}}}^{\mathrm {BDP}})=(L_\mathfrak {p}^{\mathrm {BDP}}) is known, our results determine the leading coefficient of L p B D P L_{\mathfrak {p}}^{\mathrm {BDP}} at T = 0 T=0 up to a p p -adic unit. Moreover, by adapting the approach of Burungale–Castella–Kim [Algebra Number Theory 15 (2021), pp. 1627–1653], we prove the main conjecture for supersingular primes p p under mild hypotheses. In the p p -ordinary case, and under some additional hypotheses, similar results were obtained by Agboola–Castella [J. Théor. Nombres Bordeaux 33 (2021), pp 629–658], but our method is new and completely independent from theirs, and apply to all good primes.

Algebraic analogues of results of Alladi–Johnson using the Chebotarev Density Theorem

This is the continuation of the study of differential graded (dg) vertex algebras defined in our previous paper [Caradot et al., “Differential graded vertex operator algebras and their Poisson algebras,” J. Math. Phys. 64, 121702 (2023)]. The goal of this paper is to construct a functor from the category of dg vertex Lie algebras to the category of dg vertex algebras which is left adjoint to the forgetful functor. This functor not only provides an abundant number of examples of dg vertex algebras, but it is also an important step in constructing a homotopy theory [see Avramov and Halperin, “Through the looking glass: A dictionary between rational homotopy theory and local algebra,” in Algebra, Algebraic Topology and their Interactions, Lecture Notes in Mathematics, edited by J. E. Roos (Springer, Berlin, Heidelberg, 1986), Vol. 1183, pp. 1–27 and D. G. Quillen, Homotopical Algebra, Lecture Notes in Mathematics (Springer, Berlin, Heidelberg, 1967), Vol. 43] in the category of vertex algebras. Vertex Lie algebras were introduced as analogues of vertex algebras, but in which we only consider the singular part of the vertex operator map and the equalities it satisfies. In this paper, we extend the definition of vertex Lie algebras to the dg setting. We construct a pair of adjoint functors between the categories of dg vertex algebras and dg vertex Lie algebras, which leads to the explicit construction of dg vertex (operator) algebras. We will give examples based on the Virasoro algebra, the Neveu–Schwarz algebra, and dg Lie algebras.

The connected sum construction, which takes as input Gorenstein rings and produces new Gorenstein rings, can be considered as an algebraic analogue for the topological construction having the same name. We determine the graded Betti numbers for connected sums of graded Artinian Gorenstein algebras. Along the way, we find the graded Betti numbers for fiber products of graded rings; an analogous result was obtained in the local case by Geller [Proc. Amer. Math. Soc. 150 (2022), pp. 4159–4172]. We relate the connected sum construction to the doubling construction, which also produces Gorenstein rings. Specifically, we show that, for any number of summands, a connected sum of doublings is the doubling of a fiber product ring.

Inspired by recent developments in the theory of Banach and operator algebras of locally compact groups, we construct several new classes of bifunctors ( A , B ) ↦ A ⊗ α B (A,B)\mapsto A\otimes _{\alpha } B , where A ⊗ α B A\otimes _\alpha B is a cross norm completion of A ⊙ B A\odot B for each pair of C*-algebras A A and B B . For the first class of bifunctors considered ( A , B ) ↦ A ⊗ p B (A,B)\mapsto A{\otimes _p} B ( 1 ≤ p ≤ ∞ 1\leq p\leq \infty ), A ⊗ p B A{\otimes _p} B is a Banach algebra cross-norm completion of A ⊙ B A\odot B constructed in a fashion similar to p p -pseudofunctions PF p ∗ ( G ) \text {PF}^*_p(G) of a locally compact group. Taking a cue from the recently introduced symmetrized p p -pseudofunctions due to Liao and Yu and later by the second and the third named authors, we also consider ⊗ p , q {\otimes _{p,q}} for Hölder conjugate p , q ∈ [ 1 , ∞ ] p,q\in [1,\infty ] – a Banach ∗ * -algebra analogue of the tensor product ⊗ p , q {\otimes _{p,q}} . By taking enveloping C*-algebras of A ⊗ p , q B A{\otimes _{p,q}} B , we arrive at a third bifunctor ( A , B ) ↦ A ⊗ C p , q ∗ B (A,B)\mapsto A{\otimes _{\mathrm C^*_{p,q}}} B where the resulting algebra A ⊗ C p , q ∗ B A{\otimes _{\mathrm C^*_{p,q}}} B is a C*-algebra. For G 1 G_1 and G 2 G_2 belonging to a large class of discrete groups, we show that the tensor products C r ∗ ( G 1 ) ⊗ C p , q ∗ C r ∗ ( G 2 ) \mathrm C^*_{\mathrm r}(G_1){\otimes _{\mathrm C^*_{p,q}}}\mathrm C^*_{\mathrm r}(G_2) coincide with a Brown-Guentner type C*-completion of ℓ 1 ( G 1 × G 2 ) \mathrm \ell ^1(G_1\times G_2) and conclude that if 2 ≤ p ′ > p ≤ ∞ 2\leq p’>p\leq \infty , then the canonical quotient map C r ∗ ( G ) ⊗ C p , q ∗ C r ∗ ( G ) → C r ∗ ( G ) ⊗ C p , q ∗ C r ∗ ( G ) \mathrm C^*_{\mathrm r}(G){\otimes _{\mathrm C^*_{p,q}}}\mathrm C^*_{\mathrm r}(G)\to \mathrm C^*_{\mathrm r}(G){\otimes _{\mathrm C^*_{p,q}}}\mathrm C^*_{\mathrm r}(G) is not injective for a large class of non-amenable discrete groups possessing both the rapid decay property and Haagerup’s approximation property. A Banach ∗ * -algebra A A is symmetric if the spectrum S p A ( a ∗ a ) \mathrm {Sp}_A(a^*a) is contained in [ 0 , ∞ ) [0,\infty ) for every a ∈ A a\in A , and rigidly symmetric if A ⊗ γ B A\otimes _{\gamma } B is symmetric for every C*-algebra B B . A theorem of Kügler asserts that every type I C*-algebra is rigidly symmetric. Leveraging our new constructions, we establish the converse of Kügler’s theorem by showing for C*-algebras A A and B B that A ⊗ γ B A\otimes _{\gamma }B is symmetric if and only if A A or B B is type I. In particular, a C*-algebra is rigidly symmetric if and only if it is type I. This strongly settles a question of Leptin and Poguntke from 1979 [J. Functional Analysis 33 (1979), pp. 119—134] and corrects an error in the literature.

In this paper, a short review of the calculus of exact finite-differences of integer order is proposed. The finite-difference operators are called the exact finite-differences of integer orders, if these operators satisfy the same characteristic algebraic relations as standard differential operators of the same order on some function space. In this paper, we prove theorem that this property of the exact finite-differences is satisfies for the space of simple entire functions on the real axis (i.e., functions that can be expanded into power series on the real axis). In addition, new results that describe the exact finite-differences beyond the set of entire functions are proposed. A generalized expression of exact finite-differences for non-entire functions is suggested. As an example, the exact finite-differences of the square root function is considered. The use of exact finite-differences for numerical and computer simulations is not discussed in this paper. Exact finite-differences are considered as an algebraic analog of standard derivatives of integer order.

We introduce an algebraic analogue of dynamical systems, based on term rewriting. We show that a recursive function applied to the output of an iterated rewriting system defines a formal class of models into which all the main architectures for dynamic machine learning models (including recurrent neural networks, graph neural networks, and diffusion models) can be embedded. Considered in category theory, we also show that these algebraic models are a natural language for describing the compositionality of dynamic models. Furthermore, we propose that these models provide a template for the generalisation of the above dynamic models to learning problems on structured or non-numerical data, including ‘hybrid symbolic-numeric’ models.

We present a proof-producing integration of ACL2 and Imandra for proving nonlinear inequalities. This leverages a new Imandra interface exposing its nonlinear decision procedures. The reasoning takes place over the reals, but the proofs produced are valid over the rationals and may be run in both ACL2 and ACL2(r). The ACL2 proofs Imandra constructs are extracted from Positivstellensatz refutations, a real algebraic analogue of the Nullstellensatz, and are found using convex optimization.

Vector valued Beurling algebra analogues of Wiener’s theorem

Diassociative family algebras and averaging family operators

The aim of this work is to introduce and study the notions of Hom-pre-Jordan algebra and Hom-J-dendriform algebra which generalize Hom-Jordan algebras. Hom-pre-Jordan algebras are regarded as the underlying algebraic structures of the Hom-Jordan algebras behind the Rota-Baxter operators and O-operators introduced in this paper. Hom-pre-Jordan algebras are also analogues of Hom-pre-Lie algebras for Hom-Jordan algebras. The anti-commutator of a Hom-pre-Jordan algebra is a Hom-Jordan algebra and the left multiplication operator gives a representation of a Hom-Jordan algebra. On the other hand, a Hom-J-dendriform algebra is a Hom-Jordan algebraic analogue of a Hom-dendriform algebra such that the anti-commutator of the sum of the two operations is a Hom-pre-Jordan algebra.

We say that a Lie (super)algebra is ''symmetric'' if with every root (with respect to the maximal torus) it has the opposite root of the same multiplicity. Over algebraically closed fields of positive characteristics (up to 7 or 11, enough to formulate a general conjecture), we computed the cohomology corresponding to the infinitesimal deformations of all known simple finite-dimensional symmetric Lie (super)algebras of rank <9, except for superizations of the Lie algebras with ADE root systems, and queerified Lie algebras, considered only partly. The moduli of deformations of any Lie superalgebra constitute a supervariety. Any infinitesimal deformation given by any odd cocycleis integrable. All deformations corresponding to odd cocycles are new. Among new results are classifications of the cocycles describing deforms (results of deformations) of the 29-dimensional Brown algebra in characteristic 3, of Weisfeiler-Kac algebras and orthogonal Lie algebras without Cartan matrix in characteristic 2. Open problems: describe non-isomorphic deforms and equivalence classes of cohomology theories. Appendix: For several modular analogs of complex simple Lie algebras, and simple Lie algebras indigenous to characteristics 3 and 2, we describe the space of cohomology with trivial coefficients. We show that the natural multiplication in this space is very complicated.

A flow of quantum genetic Lotka-Volterra algebras on [formula omitted

Association schemes on triples (ASTs) are ternary analogues of classical association schemes, whose relations and adjacency algebras are ternary instead of binary. We provide a survey of the current progress in the study of ASTs, highlighting open questions, suggesting research directions, and producing some related results. We review properties of the ternary adjacency algebras of ASTs, ASTs whose relations are invariant under some group action, and ASTs obtained from 2-designs and two-graphs. We also provide a notion of fusion and fission ASTs, using the AST obtained from the affine special linear group A S L (2, q ) as an example.

In view of a well-known theorem of Dixmier, its is natural to consider primitive quotients of Uq+(g) as quantum analogues of Weyl algebras. In this work, we study primitive quotients of Uq+(G2) and compute their Lie algebra of derivations. In particular, we show that, in some cases, all derivations are inner showing that the corresponding primitive quotients of Uq+(G2) should be considered as deformations of Weyl algebras.

We obtain a complete characterisation of factorial multiparameter Hecke von Neumann algebras associated with right-angled Coxeter groups. Considering their ℓp-convolution algebra analogues, we exhibit an interesting parameter dependence, contrasting phenomena observed earlier for group Banach algebras. Translated to Iwahori-Hecke von Neumann algebras, these results allow us to draw conclusions on spherical representation theory of groups acting on right-angled buildings, which are in strong contrast to behaviour of spherical representations in the affine case. We also investigate certain graph product representations of right-angled Coxeter groups and note that our von Neumann algebraic structure results show that these are finite factor representations. Further classifying a suitable family of them up to unitary equivalence allows us to reveal high-dimensional Euclidean subspaces of the space of extremal characters of right-angled Coxeter groups.