Narrow Results

If $E$ is a directed graph and $K$ is a field, the Leavitt path algebra $L_K(E)$ of $E$ over $K$ is naturally graded by the group of integers $\mathbb{Z}.$ We formulate properties of the graph $E$ which are equivalent with $L_K(E)$ being a crossed product, a skew group ring, or a group ring with respect to this natural grading. We state this main result so that the algebra properties of $L_K(E)$ are also characterized in terms of the pre-ordered group properties of the Grothendieck $\mathbb{Z}$-group of $L_K(E)$. If $E$ has finitely many vertices, we characterize when $L_K(E)$ is strongly graded in terms of the properties of $K_0^{\mathrm{\Gamma }}(L_K(E)).$ Our proof also provides an alternative to the known proof of the equivalence $L_K(E)$ is strongly graded if and only if $E$ has no sinks for a finite graph $E.$ We also show that, if unital, the algebra $L_K(E)$ is strongly graded and graded unit-regular if and only if $L_K(E)$ is a crossed product.

In the process of showing the main result, we obtain conditions on a group $\mathrm{\Gamma }$ and a $\mathrm{\Gamma }$-graded division ring $K$ equivalent with the requirements that a $\mathrm{\Gamma }$-graded matrix ring $R$ over $K$ is strongly graded, a crossed product, a skew group ring, or a group ring. We characterize these properties also in terms of the action of the group $\mathrm{\Gamma }$ on the Grothendieck $\mathrm{\Gamma }$-group $K_0^{\mathrm{\Gamma }}(R).$

The aim of this paper is to take benefit of the foam nature of the Khovanov–Kuperberg algebras to compute the Grothendieck groups of their categories of finite-dimensional projective modules. The computation relies on the Hattori–Stallings trace and some geometrical properties of foams in the solid torus.

It is known that there exist polynomial solutions $\mathrm{\Gamma }$ with infinite support base $P$, of certain functional equations arising from quantum arithmetics, which cannot be constructed from quantum integers. A description of the necessary and sufficient conditions on a set of primes $P$ for the existence of a polynomial solution, with field of coefficients of characteristic zero and support base $P$, which cannot be constructed from quantum integers is also known, leading to the classification of the set of polynomial solutions. In his papers on quantum arithmetics, Melvyn Nathanson raises a question concerning the classification of the possibly non-trivially broader set of solutions, namely the set of rational function solutions. It is not known at the time that the set of rational function solutions is more than just the set of ratio of polynomial solutions. However, it is now known that there are infinitely many rational function solutions $\mathrm{\Gamma }$, with support base $P$ and field of coefficients of characteristic zero, which are not ratios of polynomial solutions with the same support base, even in the purely cyclotomic case. Thus, a natural question that should be asked in order to classify the set of rational function solutions, is: If polynomial solutions are replaced by merely rational function solutions, what would the necessary and sufficient conditions be on the support base $P$? In this paper, we give a complete description of the necessary and sufficient conditions on the set of primes $P$ for the existence of a rational function solution, with field of coefficients of characteristic zero and support base $P$, which cannot be constructed from quantum integers.

Let $G$ be a finite group acting on a fusion category ${𝒞}$ and let $H$ be a subgroup of $G$. Let ${ℳ}$ be a semisimple indecomposable module category over ${𝒞}$. Considering $N$ a simple object in ${ℳ}$ and $H_N$ the stabilizer (or inertia) subgroup of $N$, we establish a bijective correspondence between isomorphism classes of simple objects in equivariantizations ${{ℳ}}^H$ and ${{ℳ}}^{H_N}$, where such simple objects contain $N$ as a direct summand. Also, as an application to projective representations of $H_N$, we relate isomorphism classes of simple objects in ${{ℳ}}^{H_N}$ with irreducible projective representations of $H_N$.

We provide a diagrammatic approach to compute the algebraic structures on the Grothendieck group of a tower of the Temperley–Lieb algebras.

In [S. Das, M. K. Sen and S. K. Maity, Leavitt path algebras for power graphs of finite groups, J. Algebra Appl., doi:10.1142/S0219498822502097], we studied several algebraic properties of Leavitt path algebra of the directed power graph $\overrightarrow{\mathcal{𝒫}}(G)$ and also of the directed punctured power graph ${\overrightarrow{\mathcal{𝒫}}}^{}*(G)$ of a finite group $G$. In this paper, we compute the Grothendieck group of purely infinite simple Leavitt path algebra of the directed punctured power graph of a finite group.

In this paper, we define directed enhanced power graphs for finite groups and study some graph-theoretic properties. Moreover, we investigate several algebraic properties of Leavitt path algebras of the directed enhanced power graphs and also of the directed punctured enhanced power graphs. We also compute Grothendieck groups of the Leavitt path algebras of directed punctured enhanced power graphs for cyclic groups and give partial answers to “the algebraic Kirchberg–Phillips question”.

A subset $S$ of an integral domain is called a semidomain if the pairs $(S,+)$ and $(S\backslash \{0\},\cdot )$ are commutative and cancellative semigroups with identities. The multiplication of $S$ extends to the group of differences $\mathcal{𝒢}(S)$, turning $\mathcal{𝒢}(S)$ into an integral domain. In this paper, we study the arithmetic of semisubtractive semidomains (i.e. semidomains $S$ for which either $s\in S$ or $-s\in S$ for every $s\in \mathcal{𝒢}(S)$). Specifically, we provide necessary and sufficient conditions for a semisubtractive semidomain to be atomic, to satisfy the ascending chain condition on principals ideals, to be a bounded factorization semidomain, and to be a finite factorization semidomain, which are subsequent relaxations of the property of having unique factorizations. In addition, we present a characterization of factorial and half-factorial semisubtractive semidomains. Throughout the paper, we present examples to provide insight into the arithmetic aspects of semisubtractive semidomains.

Let k be a field and A_n(ω) be the Taft n²-dimensional Hopf algebra. When n is odd, the Drinfeld quantum double D(A_n(ω)) of A_n(ω) is a ribbon Hopf algebra. We have constructed an n⁴-dimensional Hopf algebra H_n(p,q) which is isomorphic to D(A_n(ω)) if p ≠ 0 and q = ω^-1, and studied the irreducible and finite-dimensional representations of H_n(1,q). In this paper, we continue our study of H_n(1,q), examine the Grothendieck group G₀(H_n(1,q)) ≅ G₀(D(A_n(ω)), and describe its ring structure. We also give the Loewy length of the tensor product of two simple modules over H_n(1,q).

In [7] we have introduced the Frobenius categories over a finite p-group P, and we have associated to — suitably endowed with some central k*-extensions — a “Grothendieck group” as an inverse limit of Grothendieck groups of categories of modules in characteristic p obtained from , determining its rank. Our purpose here is to introduce an analogous inverse limit of Grothendieck groups of categories of modules in characteristic zero obtained from , determining its rank and proving that its extension to a field is canonically isomorphic to the direct sum of the corresponding extensions of the “Grothendieck groups” above associated with the centralizers in of a suitable set of representatives of the -classes of elements of P.

In [Frobenius Categories Versus Brauer Blocks, Progress in Math., 274] and in [Ordinary Grothendieck groups of a Frobenius $P$-category, Algebra Colloq. 18 (2011) 1–76], we consider suitable inverse limits of Grothendieck groups of categories of modules in characteristics $p$ and zero, obtained from a folded Frobenius$P$-category$(\mathcal{F},{\widehat{\mathfrak{a}\mathfrak{u}\mathfrak{t}}}_{{\mathcal{F}}^{\mathrm{sc}}})$, which covers the case of the Frobenius$P$-categories associated with blocks; moreover, in [Beyond a question of Markus Linckelmann, arxiv.org/abs/1507.04278] we show that a folded Frobenius$P$-categoryis actually equivalent to the choice of a regular central$k^{\ast }$-extension${\hat{\mathcal{F}}}^{\mathrm{sc}}$ of ${\mathcal{F}}^{\mathrm{sc}}$. Here, taking advantage of the existence of a perfect${\mathcal{F}}^{\mathrm{sc}}$-locality${\mathcal{P}}^{\mathrm{sc}}$, we exhibit those inverse limits as the true Grothendieck groups of the categories of ${\mathcal{K}}_{\ast }\hat{\mathcal{G}}$-and $k_{\ast }\hat{\mathcal{G}}$-modules for a suitable $k^{\ast }$-group $\hat{\mathcal{G}}$ associated to the $k^{\ast }$-category ${\hat{\mathcal{P}}}^{\mathrm{sc}}$ obtained from ${\mathcal{P}}^{\mathrm{sc}}$ and ${\hat{\mathcal{F}}}^{\mathrm{sc}}$. It depends on a vanishing cohomology result, given with more generality in the Appendix.

In this paper we study some new theories of characteristic homology classes of singular complex algebraic (or compactifiable analytic) spaces.

We introduce a motivic Chern class transformationmC_y: K₀(var/X) → G₀(X) ⊗ ℤ[y], which generalizes the total λ-class λ_y(T*X) of the cotangent bundle to singular spaces. Here K₀(var/X) is the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration, and G₀(X) is the Grothendieck group of coherent sheaves of -modules. A first construction of mC_y is based on resolution of singularities and a suitable "blow-up" relation, following the work of Du Bois, Guillén, Navarro Aznar, Looijenga and Bittner. A second more functorial construction of mC_y is based on some results from the theory of algebraic mixed Hodge modules due to M. Saito.

We define a natural transformation T_y* : K₀(var/X) → H_*(X) ⊗ ℚ[y] commuting with proper pushdown, which generalizes the corresponding Hirzebruch characteristic. T_y* is a homology class version of the motivic measure corresponding to a suitable specialization of the well-known Hodge polynomial. This transformation unifies the Chern class transformation of MacPherson and Schwartz (for y = -1), the Todd class transformation in the singular Riemann-Roch theorem of Baum–Fulton–MacPherson (for y = 0) and the L-class transformation of Cappell-Shaneson (for y = 1).

We also explain the relation among the "stringy version" of our characteristic classes, the elliptic class of Borisov–Libgober and the stringy Chern classes of Aluffi and De Fernex–Lupercio–Nevins–Uribe.

All our results can be extended to varieties over a base field k of characteristic 0.