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In this paper, we prove the existence of small and big Ramsey degrees of classes of finite unary algebras in an arbitrary (not necessarily finite) algebraic language Ω. Our results generalize some Ramsey-type results of M. Sokić concerning finite unary algebras over finite languages. To do so, we develop a completely new strategy that relies on the fact that right adjoints preserve the Ramsey property. We then treat unary algebras as Eilenberg-Moore coalgebras for a functor with comultiplication, and using pre-adjunctions transport the Ramsey properties, we are interested in from the category of finite or countably infinite chains of order type ω. Moreover, we show that finite objects have finite big Ramsey degrees in the corresponding cofree structures over countably many generators. [ABSTRACT FROM AUTHOR]

Let 퓟n := H*((ℝP∞)n) ≅ ℤ2[x1, x2, ..., xn] be the graded polynomial algebra over ℤ2, where ℤ2 denotes the prime field of two elements. We investigate the Peterson hit problem for the polynomial algebra 퓟n, viewed as a graded left module over the mod-2 Steenrod algebra, 퓐. For n > 4, this problem is still unsolved, even in the case of n = 5 with the help of computers. In this article, we study the hit problem for the case n = 6 in the generic degree dr = 6(2r − 1) + 4.2r with r an arbitrary non-negative integer. By considering ℤ2 as a trivial 퓐-module, then the hit problem is equivalent to the problem of finding a basis of ℤ2-vector space ℤ2 ⊗퓐퓟n. The main goal of the current article is to explicitly determine an admissible monomial basis of the ℤ2 vector space ℤ2 ⊗퓐퓟6 in some degrees. As an application, the behavior of the sixth Singer algebraic transfer in the degree 6(2r − 1) + 4.2r is also discussed at the end of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we define and study the variety of tense modal pseudocomplemented De Morgan algebras. This variety is a proper subvariety of the variety of tense tetravalent modal algebras. A tense modal pseudocomplemented De Morgan algebra is a modal pseudocomplemented De Morgan algebra endowed with two tense operators G and H satisfying additional conditions. Also, the variety of tense modal pseudocomplemented De Morgan algebras is intimately connected with some well-known varieties of De Morgan algebras with tense operators. [ABSTRACT FROM AUTHOR]

In 2012, the second author introduced, and initiated the investigations into, the variety 퓘 of implication zroupoids that generalize De Morgan algebras and ∨-semilattices with 0. An algebra A = 〈 A, →, 0 〉, where → is binary and 0 is a constant, is called an implication zroupoid (퓘-zroupoid, for short) if A satisfies: (x → y) → z ≈ [(z′ → x) → (y → z)′]′, where x′ := x → 0, and 0″ ≈ 0. Let 퓘 denote the variety of implication zroupoids and A ∈ 퓘. For x, y ∈ A, let x ∧ y := (x → y′)′ and x ∨ y := (x′ ∧ y′)′. In an earlier paper, we had proved that if A ∈ 퓘, then the algebra Amj = 〈A, ∨, ∧〉 is a bisemigroup. The purpose of this paper is two-fold: First, we generalize the notion of semidistributivity from lattices to bisemigroups and prove that, for every A ∈ 퓘, the bisemigroup Amj is semidistributive. Secondly, we generalize the Whitman Property from lattices to bisemigroups and prove that the subvariety 퓜퓔퓙 of 퓘, defined by the identity: x ∧ y ≈ x ∨ y, satisfies the Whitman Property. We conclude the paper with two open problems. [ABSTRACT FROM AUTHOR]

In 1973, Katriňák proved that regular double p-algebras can be regarded as (regular) double Heyting algebras by ingeniously constructing binary terms for the Heyting implication and its dual in terms of pseudocomplement and its dual. In this paper, we prove a converse to Katriňák's theorem, in the sense that in the variety ℝ D ℙ ℂ ℍ of regular dually pseudocomplemented Heyting algebras, the implication operation → satisfies Katriňák's formula. As applications of this result together with the above-mentioned Katriňák's theorem, we show that the varieties ℝ D B L ℙ , ℝ D ℙ ℂ ℍ , ℝ ℙ ℂ ℍ d and ℝ D B L ℍ of regular double p-algebras, regular dually pseudocomplemented Heyting algebras, regular pseudocomplemented dual Heyting algebras, and regular double Heyting algebras, respectively, are term-equivalent to each other and also that the varieties ℝ D M ℙ , ℝ D M ℍ , ℝ D M D B L ℍ , ℝ D M D B L ℙ of regular De Morgan p-algebras, regular De Morgan Heyting algebras, regular De Morgan double Heyting algebras, and regular De Morgan double p-algebras, respectively, are also term-equivalent to each other. From these results and recent results of Adams, Sankappanavar and Vaz de Carvalho on varieties of regular double p-algebras and regular pseudocomplemented De Morgan algebras, we deduce that the lattices of subvarieties of all these varieties have cardinality 2 ℵ 0 . We then define new logics, ℛ D P C ℋ , ℛ P C ℋ d and ℛ D ℳ ℋ , and show that they are algebraizable with ℝ D ℙ ℂ ℍ , ℝ ℙ ℂ ℍ d and ℝ D M ℍ , respectively, as their equivalent algebraic semantics. It is also deduced that the lattices of extensions of all of the above mentioned logics have cardinality 2 ℵ 0 . [ABSTRACT FROM AUTHOR]

Let W ⁢ (n) be the Jacobson–Witt algebra over algebraic closed field 핂 with characteristic p > 2 . In [K. Ou and B. Shu, Borel subalgebras of restricted Cartan-type Lie algebras, J. Algebra Appl. 21 2022, 11, Paper No. 2250210], we introduced the so-called B-subalgebra of W ⁢ (n) , which serves as an analog of the Borel subalgebra of classical Lie algebras. As a sequel, we describe the structure of the variety ℬ consisting of all B-subalgebras of W ⁢ (n) in this paper. This variety presents an analog of the flag variety for classical Lie algebras. It is shown that ℬ is related to the variety of all full flags in 핂 n + 1 . Additionally, we provide a detailed description of the varieties for W ⁢ (1) as an illustrative example. With the above setting-up, one may establish the Springer theory and geometric representations for the Jacobson–Witt algebras. [ABSTRACT FROM AUTHOR]

In this paper, we continue the discussion about relations between exponential polynomials and generalized moment functions on a commutative hypergroup. We are interested in the following problem: is it true that every finite-dimensional variety is spanned by moment functions? Let m be an exponential on X. In our former paper, we have proved that if the linear space of all m-sine functions in the variety of an m-exponential monomial is (at most) one-dimensional, then this variety is spanned by moment functions generated by m. In this paper, we show that this may happen also in cases where the m-sine functions span a more than one-dimensional subspace in the variety. We recall the notion of a polynomial hypergroup in d variables, describe exponentials on it and give the characterization of the so-called m-sine functions. Next we show that the Fourier algebra of a polynomial hypergroup in d variables is the polynomial ring in d variables. Finally, using the Ehrenpreis–Palamodov Theorem, we show that every exponential polynomial on the polynomial hypergroup in d variables is a linear combination of moment functions contained in its variety. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new technique in the study of the * -regular closure of some specific group algebras KG inside 풰 ⁢ (G) , the * -algebra of unbounded operators affiliated to the group von Neumann algebra 풩 ⁢ (G) . The main tool we use for this study is a general approximation result for a class of crossed product algebras of the form C K ⁢ (X) ⋊ T ℤ , where X is a totally disconnected compact metrizable space, T is a homeomorphism of X, and C K ⁢ (X) stands for the algebra of locally constant functions on X with values on an arbitrary field K. The connection between this class of algebras and a suitable class of group algebras is provided by the Fourier transform. Utilizing this machinery, we study an explicit approximation for the lamplighter group algebra. This is used in another paper by the authors to obtain a whole family of ℓ 2 -Betti numbers arising from the lamplighter group, most of them transcendental. [ABSTRACT FROM AUTHOR]

In this paper, we study representations of non-finitely graded Lie algebras 풲 ⁢ ( ϵ ) {\mathcal{W}(\epsilon)} related to Virasoro algebra, where ϵ = ± 1 {\epsilon=\pm 1} . Precisely speaking, we completely classify the free 풰 ⁢ ( 픥 ) {\mathcal{U}(\mathfrak{h})} -modules of rank one over 풲 ⁢ ( ϵ ) {\mathcal{W}(\epsilon)} , and find that these module structures are rather different from those of other graded Lie algebras. We also determine the simplicity and isomorphism classes of these modules. [ABSTRACT FROM AUTHOR]

We identify certain combinatorially defined rational functions which, under the shuffle to Schiffmann algebra isomorphism, map to LLT polynomials in any of the distinguished copies Λ ⁢ (X m , n) ⊂ E of the algebra of symmetric functions embedded in the elliptic Hall algebra ℰ of Burban and Schiffmann. As a corollary, we deduce an explicit raising operator formula for the ∇ operator applied to any LLT polynomial. In particular, we obtain a formula for ∇ m s λ which serves as a starting point for our proof of the Loehr–Warrington conjecture in a companion paper to this one. [ABSTRACT FROM AUTHOR]

The aim of the paper is to provide a characterization criterion of exponential solvable Frobenius Lie algebras (having open coadjoint orbits), in terms of primitive ideals of the associated enveloping algebra. In the case of complex solvable Lie algebras, we also show that an algebraic adjoint orbit is open if and only if the associated primitive ideal through the Dixmier map is trivial. [ABSTRACT FROM AUTHOR]

Hecke–Kiselman algebras A Θ , over a field 핂 , associated to finite oriented graphs Θ are considered. It has been known that every such algebra is an automaton algebra in the sense of Ufranovskii. In particular, its Gelfand–Kirillov dimension is an integer if it is finite. In this paper, a numerical invariant of the graph Θ that determines the dimension of A Θ is found. Namely, we prove that the Gelfand–Kirillov dimension of A Θ is the sum of the number of cyclic subgraphs of Θ and the number of oriented paths of a special type in the graph, each counted certain specific number of times. [ABSTRACT FROM AUTHOR]

The main achievement of this paper is a geometric characterisation of certain subvarieties of the Cartan variety 퓔6(핂) over an arbitrary field 핂. The characterised varieties arise as Veronese representations of certain ring projective planes over quadratic subalgebras of the split octonions 핆' over 핂 (among which the sextonions, a 6-dimensional non-associative algebra). We describe how these varieties are linked to the Freudenthal–Tits magic square, and discuss how they would even fit in, when also allowing the sextonions and other "degenerate composition algebras" as the algebras used to construct the square. [ABSTRACT FROM AUTHOR]

In this paper, a new categorical crystal structure for the quantum affine algebras is presented. We introduce the notion of extended crystals B ^ 픤 ⁢ (∞) for an arbitrary quantum group U q ⁢ (픤) , which is the product of infinite copies of the crystal B ⁢ (∞) . For a complete duality datum 풟 in the Hernandez–Leclerc category 풞 픤 0 of a quantum affine algebra U q ′ ⁢ (픤) , we prove that the set ℬ 풟 ⁢ (픤) of the isomorphism classes of simple modules in 풞 픤 0 has an extended crystal structure isomorphic to B ^ 픤 fin ⁢ (∞) . An explicit combinatorial description of the extended crystal ℬ 풟 ⁢ (픤) for affine type A n (1) is given in terms of affine highest weights. [ABSTRACT FROM AUTHOR]

The paper contributes to Majorana theory. Among the eight non-trivial Norton–Sakuma algebras, four algebras are closed on the set of Majorana generators. These algebras are 2A, 2B, 3C and 4B. The classification of Majorana representations restricted to the closed shapes was anticipated for a long time. In the present article the classification is achieved for shapes restricted to 2A, 3C and 4B and for the set of generating involutions in the target group forming a single conjugacy class. Timmesfeld's classification of {3, 4}+-transposition groups reduces to consideration of just three groups: L3(2), G2(2)' and 3D4(2). Each of these groups possesses a unique Majorana representation of the required shape. Only the representation of L3(2), known before, is based on an embedding into the Monster. [ABSTRACT FROM AUTHOR]

We investigate the role of gentle algebras in higher homological algebra. In the first part of the paper, we show that if the module category of a gentle algebra Λ contains a d-cluster tilting subcategory for some d ≥ 2 , then Λ is a radical square zero Nakayama algebra. This gives a complete classification of weakly d-representation finite gentle algebras. In the second part, we use a geometric model of the derived category to prove a similar result in the triangulated setup. More precisely, we show that if 풟 b ⁡ (Λ) contains a d-cluster tilting subcategory that is closed under [ d ] , then Λ is derived equivalent to an algebra of Dynkin type A. Furthermore, our approach gives a geometric characterization of all d-cluster tilting subcategories of 풟 b ⁡ (Λ) that are closed under [ d ] . [ABSTRACT FROM AUTHOR]

The broadly applied notions of Lie bialgebras, Manin triples, classical r-matrices and 풪 -operators of Lie algebras owe their importance to the close relationships among them. Yet these notions and their correspondences are mostly understood as classes of objects and maps among the classes. To gain categorical insight, this paper introduces, for each of the classes, a notion of homomorphisms, uniformly called coherent homomorphisms, so that the classes of objects become categories and the maps among the classes become functors or category equivalences. For this purpose, we start with the notion of an endo Lie algebra, consisting of a Lie algebra equipped with a Lie algebra endomorphism. We then generalize the above classical notions for Lie algebras to endo Lie algebras. As a result, we obtain the notion of coherent endomorphisms for each of the classes, which then generalizes to the notion of coherent homomorphisms by a polarization process. The coherent homomorphisms are compatible with the correspondences among the various constructions, as well as with the category of pre-Lie algebras. [ABSTRACT FROM AUTHOR]

Let F be an algebraically closed field and let R be a locally finite algebra over F. This paper aims to show that any element of R is a product of at most three unipotent elements from R if and only if the element lies in the first derived subgroup of the unit group of R. In addition, this necessary and sufficient condition is applied to twisted group algebras of locally finite groups over a field of either zero characteristic or characteristic p ≠ 0 for some prime p. Moreover, we explore some crucial properties satisfied by certain algebras like the conditions concerned with the connections between unipotent elements of index 2 and commutators, as well as we investigate the unipotent radical of some subgroups of a finite-dimensional algebra R over a field F with at least four elements. In particular, we again apply these results to twisted group algebras. [ABSTRACT FROM AUTHOR]

The Levi theorem tells us that every finite-dimensional Lie algebra is the semi-direct product of a semi-simple Lie algebra and a solvable Lie algebra. Brieskorn gave the connection between simple Lie algebras and simple singularities. Simple Lie algebras have been well understood, but not the solvable (nilpotent) Lie algebras. Therefore, it is important to establish connections between singularities and solvable (nilpotent) Lie algebras. In this paper, we give a new connection between nilpotent Lie algebras and nilradicals of derivation Lie algebras of isolated complete intersection singularities. As an application, we obtain the correspondence between the nilpotent Lie algebras of dimension less than or equal to 7 and the nilradicals of derivation Lie algebras of isolated complete intersection singularities with modality less than or equal to 1. Moreover, we give a new characterization theorem for zero-dimensional simple complete intersection singularities. [ABSTRACT FROM AUTHOR]

Let R be an associative ring. A multiplicative semi-derivation d is a map on R satisfying d ⁢ (x ⁢ y) = d ⁢ (x) ⁢ g ⁢ (y) + x ⁢ d ⁢ (y) = d ⁢ (x) ⁢ y + g ⁢ (x) ⁢ d ⁢ (y) and d ⁢ (g ⁢ (x)) = g ⁢ (d ⁢ (x)) {d(xy)=d(x)g(y)+xd(y)=d(x)y+g(x)d(y)\quad\text{and}\quad d(g(x))=g(d(x))} for all x , y ∈ R {x,y\in R} , where g is any map on R. In this paper, we have obtained some conditions on R, which make d additive. Finally, we have also shown that every multiplicative semi-derivation on M n ⁢ (ℂ) {M_{n}(\mathbb{C})} , the algebra of all n × n {n\times n} matrices over the field ℂ {\mathbb{C}} of complex numbers, is an additive derivation. [ABSTRACT FROM AUTHOR]

Let G be a locally compact group (not necessarily abelian) and B be a homogeneous Banach space on G, which is in a good situation with respect to a homogeneous function algebra on G. Feichtinger showed that there exists a minimal Banach space Bmin in the family of all homogenous Banach spaces C on G, containing all elements of B with compact support. In this paper, the amenability and super amenability of Bmin with respect to the convolution product or with respect to the pointwise product are showed to correspond to amenability, discreteness or finiteness of the group G and conversely. We also prove among other things that Bmin is a symmetric Segal subalgebra of L1(G) on an IN-group G, under certain conditions, and we apply our results to study pseudo-amenability and some other homological properties of Bmin on IN-groups. Furthermore, we determine necessary and sufficient conditions on A under which Amin with the pointwise product is an abstract Segal algebra or Segal algebra in A, whenever A is a homogeneous function algebra with an approximate identity. We apply these results to study amenability of some Feichtinger algebras with respect to the pointwise product. [ABSTRACT FROM AUTHOR]

Let ℜ be a ring with center Z(ℜ). In this paper, we study the higher-order commutators with power central values on rings and algebras involving generalized derivations. Motivated by [A. Alahmadi, S. Ali, A. N. Khan and M. Salahuddin Khan, A characterization of generalized derivations on prime rings, Comm. Algebra 44 2016, 8, 3201–3210], we characterize generalized derivations and related maps that satisfy certain differential identities on prime rings. Precisely, we prove that if a prime ring of characteristic different from two admitting generalized derivation 𝔉 such that ([𝔉(sm)sn + sn𝔉(sm), sr]k)l ∈ Z(ℜ) or every s ∈ ℜ, then either 𝔉(s) = ps for every s ∈ ℜ or ℜ satisfies s4 and 𝔉(s) = sp for every s ∈ ℜ and p ∈ 𝔘, the Utumi quotient ring of ℜ. As an application, we prove that any spectrally generalized derivation on a semisimple Banach algebra satisfying the above mentioned differential identity must be a left multiplication map. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to provide a cohomology of n-Hom-Leibniz algebras. Moreover, we study some higher operations on cohomology spaces and deformations. [ABSTRACT FROM AUTHOR]

Let G be a group, let A = ⊕ g ∈ G A g be an epsilon-strongly graded ring over G, let R := A 1 be the homogeneous component associated with the identity of G, and let 홿횒회횂 ⁢ (R) be the Picard semigroup of R. In the first part of this paper, we prove that the isomorphism class [ A g ] is an element of 홿횒회횂 ⁢ (R) for all g ∈ G . Moreover, the association g ↦ [ A g ] determines a partial representation of G on 홿횒회횂 ⁢ (R) which induces a partial action γ of G on the center Z ⁢ (R) of R. Sufficient conditions for A to be an Azumaya R γ -algebra are presented if R is commutative. In the second part, we study when B is a partial crossed product in the following cases: B = M n ⁡ (A) is the ring of matrices with entries in A, or B = END A (M) = ⊕ l ∈ G Mor A (M , M) l is the direct sum of graded endomorphisms of graded left A-modules M with degree l, or B = END A ⁡ (M) where M = A ⊗ R N is the induced module of a left R-module N. Assuming that R is semiperfect, we prove that there exists a subring of A which is an epsilon-strongly graded ring over a subgroup of G and it is graded equivalent to a partial crossed product. [ABSTRACT FROM AUTHOR]

In this paper, a class of non-weight modules over N = 1 Lie superalgebras of Block type (without center) is studied. These modules over Ramond algebras of Block type are free of rank 1 when restricted to the universal enveloping algebra of the Cartan subalgebra, while such modules are free of rank 2 over the Neveu–Schwarz algebras of Block type. We present the classification of these modules. Furthermore, the simplicity of these modules is determined. [ABSTRACT FROM AUTHOR]

To a strongly 퐺-graded algebra 퐴 with 1-component 퐵, we associate the group Picent gr ⁢ (A) of isomorphism classes of invertible 퐺-graded (A , A) -bimodules over the centralizer of 퐵 in 퐴. Our main result is a Picent version of the Beattie–del Río exact sequence, involving Dade's group G ⁢ [ B ] , which relates Picent gr ⁢ (A) , Picent ⁢ (B) , and group cohomology. [ABSTRACT FROM AUTHOR]

For n a power of two, we give a complete description of the cohomology algebra H*(G͠n,3; ℤ2) of the Grassmann manifold G͠n,3 of oriented 3-planes in ℝn. We do this by finding a reduced Gröbner basis for an ideal closely related to this cohomology algebra. Using this Gröbner basis we also present an additive basis for H*(G͠n,3; ℤ2). [ABSTRACT FROM AUTHOR]

We address Peirce decompositions for evolution algebras at idempotent elements contained in the associative nucleus of the evolution algebras. If the idempotent elements are natural vectors, the requirement of being nuclear is then proved to be equivalent to the evolution algebra to be baric. A description of baric evolution algebras is also provided. [ABSTRACT FROM AUTHOR]

We bound the Gelfand–Kirillov dimension of unitary Banach space representations of p-adic reductive groups, whose locally analytic vectors afford an infinitesimal character. We use the bound to study Hecke eigenspaces in completed cohomology of Shimura curves and p-adic Banach space representations of the group of units of a quaternion algebra over ℚ p appearing in the p-adic Jacquet–Langlands correspondence, deducing finiteness results in favorable cases. [ABSTRACT FROM AUTHOR]

Let a reductive group G act on a smooth variety X such that a good quotient X / / G exists. We show that the derived category of a noncommutative crepant resolution (NCCR) of X / / G , obtained from a G-equivariant vector bundle on X, can be embedded in the derived category of the (canonical, stacky) Kirwan resolution of X / / G . In fact, the embedding can be completed to a semi-orthogonal decomposition in which the other parts are all derived categories of Azumaya algebras over smooth Deligne–Mumford stacks. [ABSTRACT FROM AUTHOR]

We show that all amenable, minimal actions of a large class of nonamenable countable groups on compact metric spaces have dynamical comparison. This class includes all nonamenable hyperbolic groups, many HNN-extensions, nonamenable Baumslag–Solitar groups, a large class of amalgamated free products, lattices in many Lie groups, A ~ 2 -groups, as well as direct products of the above with arbitrary countable groups. As a consequence, crossed products by amenable, minimal and topologically free actions of such groups on compact metric spaces are Kirchberg algebras in the UCT class, and are therefore classified by K-theory. [ABSTRACT FROM AUTHOR]

We show that the shuffle algebra associated to a doubled quiver (determined by 3-variable wheel conditions) is generated by elements of minimal degree. Together with results of Varagnolo–Vasserot and Yu Zhao, this implies that the aforementioned shuffle algebra is isomorphic to the localized 퐾-theoretic Hall algebra associated to the quiver by Grojnowski, Schiffmann–Vasserot and Yang–Zhao. With small modifications, our theorems also hold under certain specializations of the equivariant parameters, which will allow us in joint work with Sala and Schiffmann to give a generators-and-relations description of the Hall algebra of any curve over a finite field (which is a shuffle algebra due to Kapranov–Schiffmann–Vasserot). When the quiver has no edge loops or multiple edges, we show that the shuffle algebra, localized 퐾-theoretic Hall algebra, and the positive half of the corresponding quantum loop group are all isomorphic; we also obtain the non-degeneracy of the Hopf pairing on the latter quantum loop group. [ABSTRACT FROM AUTHOR]

Plactic key agreement is a new type of cryptographic key agreement that uses Knuth's multiplication of semistandard tableaux from combinatorial algebra. The security of plactic key agreement relies on the difficulty of some computational problems, particularly the division of semistandard tableaux. Tableau division can be used to find the private key from its public key or to find the shared secret from the two exchanged public keys. Monico found a fast division algorithm, which could be a polynomial time in the length of the tableaux. Monico's algorithm solved a challenge that had been previously estimated to cost 2128 steps to break, which is an infeasibly large number for any foreseeable computing power on earth. Monico's algorithm solves this challenge in only a few minutes. Therefore, Monico's attack likely makes the plactic key agreement insecure. If it were not for Monico's attack, plactic key agreement with 1,000-byte public keys might perhaps have provided 128-bit security, with a runtime of a millisecond. But Monico's attack breaks these public keys' sizes in minutes. [ABSTRACT FROM AUTHOR]

We show that the hermitian K-theory space of a commutative ring R can be identified, up to 퐀 1 -homotopy, with the group completion of the groupoid of oriented finite Gorenstein R-algebras, i.e., finite locally free R-algebras with trivialized dualizing sheaf. We deduce that hermitian K-theory is universal among generalized motivic cohomology theories with transfers along oriented finite Gorenstein morphisms. As an application, we obtain a Hilbert scheme model for hermitian K-theory as a motivic space. We also give an application to computational complexity: we prove that 1-generic minimal border rank tensors degenerate to the big Coppersmith–Winograd tensor. [ABSTRACT FROM AUTHOR]

Digital circuits are widely utilized in computers, because they provide models for various digital components and arithmetic operations. Arithmetic circuits are a subclass of digital circuits that are used to execute Boolean algebra. To avoid problems like the infamous Pentium FDIV bug, it is critical to ensure that arithmetic circuits are correct. Formal verification can be used to determine the correctness of a circuit with respect to a certain specification. However, arithmetic circuits, particularly integer multipliers, represent a challenge to current verification methodologies and, in reality, still necessitate a significant amount of manual labor. In my dissertation we examine and develop automated reasoning approaches based on computer algebra, where the word-level specification, modeled as a polynomial, is reduced by a Gröbner basis inferred by the gate-level representation of the circuit. We provide a precise formalization of this reasoning process, which includes soundness and completeness arguments and adds to the mathematical background in this field. On the practical side we present an unique incremental column-wise verification algorithm and preprocessing approaches based on variable elimination that simplify the inferred Gröbner basis. Furthermore, we provide an algebraic proof calculus in this thesis that allows obtaining certificates as a by-product of circuit verification in order to boost confidence in the outcomes of automated reasoning tools. These certificates can be efficiently verified with independent proof checking tools. [ABSTRACT FROM AUTHOR]

We construct an equivalence of graded Abelian categories from a category of representations of the quiver-Hecke algebra of type A 1 (1) to the category of equivariant perverse coherent sheaves on the nilpotent cone of type A. We prove that this equivalence is weakly monoidal. This gives a representation-theoretic categorification of the preprojective K-theoretic Hall algebra considered by Schiffmann and Vasserot. Using this categorification, we compare the monoidal categorification of the quantum open unipotent cells of type A 1 (1) given by Kang, Kashiwara, Kim, Oh and Park in terms of quiver-Hecke algebras with the one given by Cautis and Williams in terms of equivariant perverse coherent sheaves on the affine Grassmannians. [ABSTRACT FROM AUTHOR]

We show that, given an almost-source algebra 𝐴 of a 𝑝-block of a finite group 𝐺, then the unit group of 𝐴 contains a basis stabilized by the left and right multiplicative action of the defect group if and only if, in a sense to be made precise, certain relative multiplicities of local pointed groups are invariant with respect to the fusion system. We also show that, when 𝐺 is 𝑝-solvable, those two equivalent conditions hold for some almost-source algebra of the given 𝑝-block. One motive lies in the fact that, by a theorem of Linckelmann, if the two equivalent conditions hold for 𝐴, then any stable basis for 𝐴 is semicharacteristic for the fusion system. [ABSTRACT FROM AUTHOR]

Thus HT ht as required. Let U be an open neighbourhood of e in G, and choose an open symmetric neighbourhood V of e with HT ht . Suppose there is a continuous map HT ht extending . We claim that HT ht . [Extracted from the article]

Let (K , 풪 , k) {(K,\mathcal{O},k)} be a p-modular system with k algebraically closed and 풪 {\mathcal{O}} unramified, and let Λ be an 풪 {\mathcal{O}} -order in a separable K-algebra. We call a Λ-lattice L rigid if Ext Λ 1 ⁡ (L , L) = 0 {{\operatorname{Ext}}^{1}_{\Lambda}(L,L)=0} , in analogy with the definition of rigid modules over a finite-dimensional algebra. By partitioning the Λ-lattices of a given dimension into "varieties of lattices", we show that there are only finitely many rigid Λ-lattices L of any given dimension. As a consequence we show that if the first Hochschild cohomology of Λ vanishes, then the Picard group and the outer automorphism group of Λ are finite. In particular, the Picard groups of blocks of finite groups defined over 풪 {\mathcal{O}} are always finite. [ABSTRACT FROM AUTHOR]

We construct a categorification of parabolic Verma modules for symmetrizable Kac–Moody algebras using KLR-like diagrammatic algebras. We show that our construction arises naturally from a dg-enhancement of the cyclotomic quotients of the KLR-algebras. As a consequence, we are able to recover the usual categorification of integrable modules. We also introduce a notion of dg-2-representation for quantum Kac–Moody algebras, and in particular of parabolic 2-Verma modules. [ABSTRACT FROM AUTHOR]

Considering prime Leavitt path algebras L K ⁢ (E) {L_{K}(E)} , with E being an arbitrary graph with at least two vertices, and K being any field, we construct a class of maximal commutative subalgebras of L K ⁢ (E) {L_{K}(E)} such that, for every algebra A from this class, A has zero intersection with the commutative core ℳ K ⁢ (E) {\mathcal{M}_{K}(E)} of L K ⁢ (E) {L_{K}(E)} defined and studied in [C. Gil Canto and A. Nasr-Isfahani, The commutative core of a Leavitt path algebra, J. Algebra 511 2018, 227–248]. We also give a new proof of the maximality, as a commutative subalgebra, of the commutative core ℳ R ⁢ (E) {\mathcal{M}_{R}(E)} of an arbitrary Leavitt path algebra L R ⁢ (E) {L_{R}(E)} , where E is an arbitrary graph and R is a commutative unital ring. [ABSTRACT FROM AUTHOR]

We consider smooth representations of the unit group G = A × G=\mathcal{A}^{\times} of a finite-dimensional split basic algebra 풜 over a non-Archimedean local field. In particular, we prove a version of Gutkin's conjecture, namely, we prove that every irreducible smooth representation of 퐺 is compactly induced by a one-dimensional representation of the unit group of some subalgebra of 풜. We also discuss admissibility and unitarisability of smooth representations of 퐺. [ABSTRACT FROM AUTHOR]

We study GL2(F)-invariant periods on representations of GL2(A), where F is a non-archimedean local field and A/F a product of field extensions of total degree 3. For irreducible representations, a theorem of Prasad shows that the space of such periods has dimension ⩽ 1, and is non-zero when a certain ε-factor condition holds. We give an extension of this result to a certain class of reducible representations (of Whittaker type), extending results of Harris–Scholl when A is the split algebra F × F × F. [ABSTRACT FROM AUTHOR]