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We furnish any category of a universal (co)homology theory. Universal (co)homologies and universal relative (co)homologies are obtained by showing representability of certain functors and take values in $R$-linear abelian categories of motivic nature, where $R$ is any commutative unitary ring. Universal homology theory on the one point category yields "hieratic" $R$-modules, i.e. the indization of Freyd's free abelian category on $R$. Grothendieck $\partial$-functors and satellite functors are recovered as certain additive relative homologies on an abelian category for which we also show the existence of universal ones., Comment: References added

We address the study of multiparameter quamtum groups (=MpQG's) at roots of unity, namely quantum universal enveloping algebras $ U_{\boldsymbol{\rm q}}(\mathfrak{g}) $ depending on a matrix of parameters $ \boldsymbol{\rm q} = {\big( q_{ij} \big)}_{i, j \in I} \, $. This is performed via the construction of quantum root vectors and suitable "integral forms" of $ U_{\boldsymbol{\rm q}}(\mathfrak{g}) \, $, a \textsl{restricted one} - generated by quantum divided powers and quantum binomial coefficients - and an \textsl{unrestricted\/} one - where quantum root vectors are suitably renormalized. The specializations at roots of unity of either forms are the "MpQG's at roots of unity" we look for. In particular, we study special subalgebras and quotients of our MpQG's at roots of unity - namely, the multiparameter version of small quantum groups - and suitable associated quantum Frobenius morphisms, that link the MpQG's at roots of 1 with MpQG's at 1, the latter being classical Hopf algebras bearing a well precise Poisson-geometrical content. A key point in the discussion - often at the core of our strategy - is that every MpQG is actually a 2-cocycle deformation of the algebra structure of (a lift of) the "canonical" one-parameter quantum group by Jimbo-Lusztig, so that we can often rely on already established results available for the latter. On the other hand, depending on the chosen multiparameter $ \boldsymbol{\rm q} $ our quantum groups yield (through the choice of integral forms and their specialization) different semiclassical structures, namely different Lie coalgebra structures and Poisson structures on the Lie algebra and algebraic group underlying the canonical one-parameter quantum group., 71 pages. Final version, shortened and polished after the referee's comments - the mathematical content, though, is unchanged with respect to the previous version. To appear in "Journal of Noncommutative Geometry"

In 2012 Ayesha Asloob Qureshi connected collections of cells to Commutative Algebra assigning to every collection $\mathcal{P}$ of cells the ideal of inner 2-minors, denoted by $I_{\mathcal{P}}$, in the polynomial ring $S_{\mathcal{P}}=K[x_v:v\text{ is a vertex of }\mathcal{P}]$. Investigating the main algebraic properties of $K[\mathcal{P}]=S_{\mathcal{P}}/I_{\mathcal{P}}$ depending on the shape of $\mathcal{P}$ is the purpose of this research. Many problems are still open and they seem to be fascinating and exciting challenges.\\ In this thesis we prove several results about the primality of $I_{\mathcal{P}}$ and the algebraic properties of $K[\mathcal{P}]$ like Cohen-Macaulyness, normality and Gorensteiness, for some classes of non-simple polyominoes. The study of the Hilbert-Poincar\'e series and the related invariants as Krull dimension and Castelnuovo-Mumford regularity are given. Finally we provide the code of the package \texttt{PolyominoIdeals} developed for \texttt{Macaulay2}.

Let [Formula: see text] be the free *-superalgebra over a field [Formula: see text] of characteristic zero and let [Formula: see text] be the [Formula: see text]-ideal generated by the set of the *-graded Capelli polynomials [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] alternating on [Formula: see text] symmetric variables of homogeneous degree zero, on [Formula: see text] skew variables of homogeneous degree zero, on [Formula: see text] symmetric variables of homogeneous degree one and on [Formula: see text] skew variables of homogeneous degree one, respectively. We study the asymptotic behavior of the sequence of *-graded codimensions of [Formula: see text] In particular, we prove that the *-graded codimensions of the finite dimensional simple *-superalgebras are asymptotically equal to the *-graded codimensions of [Formula: see text], for some fixed natural numbers [Formula: see text] and [Formula: see text].

We define the product of admissible abstract kernels of the form [Formula: see text], where [Formula: see text] is a monoid, [Formula: see text] is a group and [Formula: see text] is a monoid homomorphism. Identifying [Formula: see text]-equivalent abstract kernels, where [Formula: see text] is the center of [Formula: see text], we obtain that the set [Formula: see text] of [Formula: see text]-equivalence classes of admissible abstract kernels inducing the same action of [Formula: see text] on [Formula: see text] is a commutative monoid. Considering the submonoid [Formula: see text] of abstract kernels that are induced by special Schreier extensions, we prove that the factor monoid [Formula: see text] is an abelian group. Moreover, we show that this abelian group is isomorphic to the third cohomology group [Formula: see text].

We present recent results about Capelli polynomials with involution or graded involution and their asymptotics. In the associative case, the asymptotic equality between the codimensions of the T -ideal generated by the Capelli polynomial of rank k2 + 1 and the codimensions of the matrix algebra Mk(F) was proved. This result was extended to superalgebras. Recently, similar results have been determined by the authors in the case of algebras with involution and superalgebras with graded involution.

Let A be an associative algebra endowed with an automorphism or an antiautomorphism phi of order

We construct ( 2 , 3 ) (2,3) -generators for the finite 8-dimensional orthogonal groups, proving the following results: the groups Ω 8 + ⁢ ( q ) \Omega_{8}^{+}(q) and P ⁢ Ω 8 + ⁢ ( q ) \mathrm{P}\Omega_{8}^{+}(q) are ( 2 , 3 ) (2,3) -generated if and only if q ≥ 4 q\geq 4 ; the groups Ω 8 - ⁢ ( q ) \Omega_{8}^{-}(q) and P ⁢ Ω 8 - ⁢ ( q ) \mathrm{P}\Omega_{8}^{-}(q) are ( 2 , 3 ) (2,3) -generated for all q ≥ 2 q\geq 2 .

In this paper we introduce a special class of partially filled arrays. A magic partially filled array $\mathrm{MPF}_\Omega(m,n; s,k)$ on a subset $\Omega$ of an abelian group $(\Gamma,+)$ is a partially filled array of size $m\times n$ with entries in $\Omega$ such that $(i)$ every $\omega \in \Omega$ appears once in the array; $(ii)$ each row contains $s$ filled cells and each column contains $k$ filled cells; $(iii)$ there exist (not necessarily distinct) elements $x,y\in \Gamma$ such that the sum of the elements in each row is $x$ and the sum of the elements in each column is $y$. In particular, if $x=y=0_\Gamma$, we have a zero-sum magic partially filled array ${}^0\mathrm{MPF}_\Omega(m,n; s,k)$. Examples of these objects are magic rectangles, $\Gamma$-magic rectangles, signed magic arrays, (integer or non integer) Heffter arrays. Here, we give necessary and sufficient conditions for the existence of a magic rectangle with empty cells, i.e., of an $\mathrm{MPF}_\Omega(m,n;s,k)$ where $\Omega=\{1,2,\ldots,nk\}\subset\mathbb{Z}$. We also construct zero-sum magic partially filled arrays when $\Omega$ is the abelian group $\Gamma$ or the set of its nonzero elements.

For a category B with finite products, we first characterize pseudofunctors from B to Cat whose corresponding opfibration is cartesian monoidal. Among those, we then characterize the ones which extend to pseudofunctors from internal groups to 2-groups. If B is additive, this is the case precisely when the corresponding opfibration has groupoidal fibres.

In this paper, we set η ⁢ ( G ) \eta(G) to be the number of conjugacy classes of maximal cyclic subgroups of a finite group 𝐺. We compute η ⁢ ( G ) \eta(G) for all metacyclic 𝑝-groups. We show that if 𝐺 is a metacyclic 𝑝-group of order p n p^{n} that is not dihedral, generalized quaternion, or semi-dihedral, then η ⁢ ( G ) ≥ n - 2 \eta(G)\geq n-2 , and we determine when equality holds.

Let $ G $ be a periodic group, and let $ \omega(G)\subseteq{𝕅} $ be the spectrum of $ G $ that is the set of orders of elements in $ G $ . We prove that the alternating group $ A_{7} $ is uniquely recognized by its spectrum in the class of all groups.

Let G be a finite group of even order that has no 2-rank 1. We will prove, using only elementary methods, that there is an involution $$t \in G$$ such that $$|G| < |C_{G}(t)|^{6}$$ .

We focus on the transfer of some known orthogonal factorization systems from$$\mathsf {Cat}$$Catto the 2-category$${\mathsf {Fib}}(B)$$Fib(B)of fibrations over a fixed base categoryB: the internal version of thecomprehensive factorization, and the factorization systems given by (sequence of coidentifiers, discrete morphism) and (sequence of coinverters, conservative morphism) respectively. For the class of fibrewise opfibrations in$${\mathsf {Fib}}(B)$$Fib(B), the construction of the latter two simplify to a single coidentifier (respectively coinverter) followed by an internal discrete opfibration (resp. fibrewise opfibration in groupoids). We show how these results follow from their analogues in$$\mathsf {Cat}$$Cat, providing suitable conditions on a 2-category$${\mathcal {C}}$$C, that allow the transfer of the construction of coinverters and coidentifiers from$${\mathcal {C}}$$C to$${\mathsf {Fib}}_{{\mathcal {C}}}(B)$$FibC(B).

In this paper we prove that if A is any algebra with involution * satisfying a non-trivial polynomial identity, then its sequence of *-codimensions is eventually non-decreasing. Furthermore, by making use of the *-exponent we reconstruct the only two *-algebras, up to T*-equivalence, generating varieties of almost polynomial growth. As a third result we characterize the varieties of algebras with involution whose exponential growth is bounded by 2.

In this thesis we work on problems related to tensor decomposition from a geometrical perspective. In the first part of the thesis we focus on the identifiability problem, which amounts to understand in how many ways a tensor can be decomposed as a minimal sum of elementary tensors. In particular we completely classify the identifiability of any tensor up to rank 3. In the second part of the thesis we continue to work with specific elementsand we introduce the notion of r-thTerracini locus of a Segre variety. This is the locus containing all points for which the differential of the map between the r-th abstarct secant variety and the r-th secant variety of a Segre variety drops rank. We completely determine the r-th Terracini locus of any Segre variety in the case of r = 2, 3.

We set up a fibred categorical theory of obstruction and classification of morphisms that specialises to the one of monoidal functors between categorical groups and also to the Schreier-Mac Lane theory of group extensions. Further applications are provided to crossed extensions and crossed bimodule butterflies, with in particular a classification of non-abelian extensions of unital associative algebras in terms of Hochschild cohomology.

In this paper, we prove the existence of a relation between the prime divisors of the order of a Sylow normalizer and the degree of characters having values in some small cyclotomic fields. This relation is stronger when the group is solvable.

White light can be decomposed into different colors, and a complex sound wave can be decomposed into its partials. While the physics behind transverse and longitudinal waves is quite different and several theories have been developed to investigate the complexity of colors and timbres, we can try to model their structural similarities through the language of categories. Then, we consider color mixing and color transition in painting, comparing them with timbre superposition and timbre morphing in orchestration and computer music in light of bicategories and bigroupoids. Colors and timbres can be a probe to investigate some relevant aspects of visual and auditory perception jointly with their connections. Thus, the use of categories proposed here aims to investigate color/timbre perception, influencing the computer science developments in this area.

The aim of this thesis is to investigate the tensor decomposition of structured tensors related to SL(n)-irreducible representations. Structured tensors are multilinear objects satisfying specific symmetry relations and their decompositions are of great interest in the applications. In this thesis we look for the decompositions of tensors belonging to irreducible representations of SL(n) into sum of elementary objects associated to points of SL(n)-rational hoogeneous varieties. This family includes Veronese varieties (symmetric tensors), Grassmann varieties (skew-symmetric tensors), and flag varieties. A classic tool to study the decomposition of symmetric tensors is the apolarity theory, which dates back to Sylvester. An analogous skew-symmetric apolarity theory for skew-symmetric tensors have been developed only few years ago. In this thesis we describe a global apolarity theory called Schur apolarity theory, which is suitable for tensors belonging to any irreducible representation of SL(n). Examples, properties and applications of such apolarity are studied with details and original results both in algebra and geoemtry are provided.

In this paper, we set $\eta (G)$ to be the number of conjugacy classes of maximal cyclic subgroups of $G$. We consider $\eta$ and direct and semi-direct products. We characterize the normal subgroups $N$ so that $\eta (G/N) = \eta (G)$. We set $G^- = \{ g \in G \mid \langle g \rangle {\rm ~is~not ~maximal~cyclic} \}$. We show if $\langle G^- \rangle < G$, then $G/\langle G^- \rangle$ is either (1) an elementary abelian $p$-group for some prime $p$, (2) a Frobenius group whose Frobenius kernel is a $p$-group of exponent $p$ and a Frobenius complement has order $q$ for distinct primes $p$ and $q$, or (3) isomorphic to $A_5$., Comment: 18 pages

We deal with varieties of nonassociative algebras having polynomial growth of codimensions. We describe some results obtained in recent years in the class of left nilpotent algebras of index two. Recently the authors established a correspondence between the growth rates for left nilpotent algebras of index two and the growth rates for commutative or anticommutative metabelian algebras that allows to transfer the results concerning varieties of left nilpotent algebras of index two to varieties of commutative or anticommutative metabelian algebras.

The trace codimensions give a quantitative description of the identities satisfied by an algebra with trace. Here we study the asymptotic behaviour of the sequence of trace codimensions c tr n(A) and of pure trace codimensions c ptr n (A) of a finite-dimensional algebra A over a field of characteristic zero. We find an upper and lower bound of both codimensions and as a consequence we get that the limits limn→∞ctrn(A)√n and limn→∞cptrn(A) √n always exist and are integers. This result gives a positive answer to a conjecture of Amitsur in this setting. Finally we characterize the varieties of algebras whose exponential growth is bounded by 2

Let A be an associative algebra over a field of characteristic zero. A central polynomial is a polynomial of the free associative algebra that takes central values of A. In this survey, we present some recent results about the exponential growth of the central codimension sequence and the proper central codimension sequence in the setting of algebras with involution and algebras graded by a finite group.