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Let D be a finite-dimensional division algebra over its center and R = D[t;σ,δ] a skew polynomial ring. Under certain assumptions on δ and σ, the ring of central quotients D(t;σ,δ) = {f/g|f ∈ D[t;σ,δ],g ∈ C(D[t;σ,δ])} of D[t;σ,δ] is a central simple algebra with reduced norm N. We calculate the norm N(f) for some skew polynomials f ∈ R and investigate when and how the reducibility of N(f) reflects the reducibility of f.

Naloga se ukvarja s tenzorskimi produkti in njihovo uporabo pri študiju algeber. Predvsem se osredotočimo na centralne enostavne algebre. Iz lastnosti tenzorskega produkta na takih algebrah sledijo mnogi zanimivi izreki. The thesis deals with tensor products and their applications in studying algebras. We mainly focus on central simple algebras. Many interesting theorems follow from the properties of tensor products of such algebras.

We show that any adjoint absolutely simple linear algebraic group over a field of characteristic zero is the automorphism group of some projector on a central simple algebra. Projective homogeneous varieties can be described in these terms; in particular, we reproduce quadratic equations by Lichtenstein defining them.

For any central simple algebra over a field F which contains a maximal subfield M with non-trivial automorphism group G = AutF(M), G is solvable if and only if the algebra contains a finite chain of subalgebras which are generalized cyclic algebras over their centers (field extensions of F) satisfying certain conditions. These subalgebras are related to a normal subseries of G. A crossed product algebra F is hence solvable if and only if it can be constructed out of such a finite chain of subalgebras. This result was stated for division crossed product algebras by Petit and overlaps with a similar result by Albert which, however, was not explicitly stated in these terms. In particular, every solvable crossed product division algebra is a generalized cyclic algebra over F.

We study possible decompositions of totally decomposable algebras with involution, that is, tensor products of quaternion algebras with involution. In particular, we are interested in decompositions in which one or several factors are the split quaternion algebra $M_2(F)$, endowed with an orthogonal involution. Using the theory of gauges, developed by Tignol-Wadsworth, we construct examples of algebras isomorphic to a tensor product of quaternion algebras with $k$ split factors, endowed with an involution which is totally decomposable, but does not admit any decomposition with $k$ factors $M_2(F)$ with involution. This extends an earlier result of Sivatski where the algebra considered is of degree $8$ and index $4$, and endowed with some orthogonal involution., Comment: 16 pages

Research partially supported by grant PID2019-110525GB-I00 from Agencia Estatal de Investigacion (AEI) and from Fondo Europeo de Desarrollo Regional (FEDER)., We consider the algorithmic problem of computing a primitive idempotent of a central simple algebra over the field of rational functions over a finite field. The algebra is given by a set of structure constants. The problem is reduced to the computation of a division algebra Brauer equivalent to the central simple algebra. This division algebra is constructed as a cyclic algebra, once the Hasse invariants have been computed. We give an application to skew constacyclic convolutional codes., Agencia Estatal de Investigacion (AEI) PID2019-110525GB-I00, European Commission

A theorem of Pfister asserts that every $12$-dimensional quadratic form with trivial discriminant and trivial Clifford invariant over a field of characteristic different from $2$ decomposes as a tensor product of a binary quadratic form and a $6$-dimensional quadratic form with trivial discriminant. The main result of the paper extends Pfister's result to orthogonal involutions: every central simple algebra of degree $12$ with orthogonal involution of trivial discriminant and trivial Clifford invariant decomposes into a tensor product of a quaternion algebra and a central simple algebra of degree $6$ with orthogonal involutions. This decomposition is used to establish a criterion for the existence of orthogonal involutions with trivial invariants on algebras of degree $12$, and to calculate the $f_3$-invariant of the involution if the algebra has index $2$.

The glider Brauer-Severi variety GBS ( A ) of a central simple algebra A over a field K is introduced as the set of all irreducible left glider ideals in A for some filtration FA. For fields we deduce that GBS ( K ) equals R ( K ) × Z , the product of the Riemann surface of K and Z . For a csa A over K it turns out that GBS ( A ) = BS ( A ) × GBS ( K ) , where BS ( A ) denotes the classical Brauer-Severi variety of A.

Research partially supported by grant PID2019-110525GB-I00 from Agencia Estatal de Investigacion (AEI) and from Fondo Europeo de Desarrollo Regional (FEDER).

An orthogonal involution σ on a central simple algebra A, after scalar extension to the function field F ( A ) of the Severi–Brauer variety of A, is adjoint to a quadratic form q σ over F ( A ) , which is uniquely defined up to a scalar factor. Some properties of the involution, such as hyperbolicity, and isotropy up to an odd-degree extension of the base field, are encoded in this quadratic form, meaning that they hold for the involution σ if and only if they hold for q σ . As opposed to this, we prove that there exists non-totally decomposable orthogonal involutions that become totally decomposable over F ( A ) , so that the associated form q σ is a Pfister form. We also provide examples of nonisomorphic involutions on an index 2 algebra that yield similar quadratic forms, thus proving that the form q σ does not determine the isomorphism class of σ, even when the underlying algebra has index 2. As a consequence, we show that the e 3 invariant for orthogonal involutions is not classifying in degree 12, and does not detect totally decomposable involutions in degree 16, as opposed to what happens for quadratic forms.

Positivstellensatz is a fundamental result in real algebraic geometry providing algebraic certificates for positivity of polynomials on semialgebraic sets. In this article Positivstellensatze for trace polynomials positive on semialgebraic sets of $n\times n$ matrices are provided. A Krivine-Stengle-type Positivstellensatz is proved characterizing trace polynomials nonnegative on a general semialgebraic set $K$ using weighted sums of hermitian squares with denominators. The weights in these certificates are obtained from generators of $K$ and traces of hermitian squares. For compact semialgebraic sets $K$ Schmudgen- and Putinar-type Positivstellensatze are obtained: every trace polynomial positive on $K$ has a sum of hermitian squares decomposition with weights and without denominators. The methods employed are inspired by invariant theory, classical real algebraic geometry and functional analysis. Procesi and Schacher in 1976 developed a theory of orderings and positivity on central simple algebras with involution and posed a Hilbert's 17th problem for a universal central simple algebra of degree $n$: is every totally positive element a sum of hermitian squares? They gave an affirmative answer for $n=2$. In this paper a negative answer for $n=3$ is presented. Consequently, including traces of hermitian squares as weights in the Positivstellensatze is indispensable.

Let b be a block ideal of the group algebra of a finite group G over an algebraically closed field k of prime characteristic p with a defect group P. Some direct summands, as k [ P × P ] -module, of a source algebra of the block ideal b outside of the inertial group of a maximal b-Brauer pair will be given; their multiplicities are congruent to 1 modulo p.

The descent algebra of a Coxeter group is a subalgebra of the group algebra with interesting representation theoretic properties. For instance, the natural map from the descent algebra of the symmetric group to the character ring is a surjective algebra homomorphism, so the descent algebra implicitly encodes information about the representations of the symmetric group. However, this property does not hold for other Coxeter groups. Moreover, a complete set of primitive idempotents in the descent algebra of the symmetric group leads to a decomposition of the group algebra as a direct sum of induced linear characters of centralizers of conjugacy class representatives. In this dissertation, I consider the hyperoctahedral group. When the descent algebra of a hyperoctahedral group is replaced with a generalization called the Mantaci-Reutenauer algebra, the natural map to the character ring is surjective. In 2008, Bonnafe asked whether a complete set of idempotents in the Mantaci-Reutenauer algebra could lead to a decomposition of the group algebra of the hyperoctahedral group as a direct sum of induced linear characters of centralizers. In this dissertation, I will answer this question positively and go through the construction of the idempotents, conjugacy class representatives, and linear characters required to do so.

Let A be a central simple algebra central over a number field K whose ring of integers is R. An outlier is an element r of R so that: r is a reduced norm of an element of A, but not the norm of an algebraic integer in A. We study properties and distribution of outliers. We end with a structure theorem about products of super singular elliptic curves over GF(p).

Let k be an algebraically closed field of characteristic zero and K a finitely generated field over k. Let Σ be a central simple K-algebra, X a normal projective model of K and Λ a sheaf of maximal O X -orders in Σ. There is a ramification Q -divisor Δ on X, which is related to the canonical bimodule ω Λ by an adjunction formula. It only depends on the class of Σ in the Brauer group of K. When the numerical abundance conjecture holds true, or when Σ is a central simple algebra, we show that the Gelfand-Kirillov dimension (or GK dimension) of the canonical ring of Λ is one more than the Iitaka dimension (or D-dimension) of the log pair ( X , Δ ) . In the case that Σ is a division algebra, we further show that this GK dimension is also one more than the transcendence degree of the division algebra of degree zero fractions of the canonical ring of Λ. We prove that these dimensions are birationally invariant when the b-log pair determined by the ramification divisor has b-canonical singularities. In that case we refer to the Iitaka (or D-dimension) of ( X , Δ ) as the Kodaira dimension of the order Λ. For this, we establish birational invariance of the Kodaira dimension of b-log pairs with b-canonical singularities. We also show that the Kodaira dimension can not decrease for an embedding of central simple algebras, finite dimensional over their centres, which induces a Galois extension of their centres, and satisfies a condition on the ramification which we call an effective embedding. For example, this condition holds if the target central simple algebra has the property that its period equals its index. In proving our main result, we establish existence of equivariant b-terminal resolutions of G-b-log pairs and we also find two variants of the Riemann-Hurwitz formula. The first variant applies to effective embeddings of central simple algebras with fixed centres while the second applies to the pullback of a central simple algebra by a Galois extension of its centre. We also give two different local characterizations of effective embeddings. The first is in terms of complete local invariants, while the second uses Galois cohomology.

Let p p be an odd prime. Let F F be the function field of a p p -adic curve. Let A A be a central simple algebra of period 2 over F F with an involution σ \sigma . There are known upper bounds for the u u -invariant of hermitian forms over ( A , σ ) (A, \sigma ) . In this article we compute the exact values of the u u -invariant of hermitian forms over ( A , σ ) (A, \sigma ) .

If the enveloping central simple algebra of an Okubo algebra comes with a valuation, then we can compute the residue of that Okubo algebra, at least as a vector subspace of the central simple algebra. We give a criterion for the residue of an Okubo algebra without nonzero idempotents to be an Okubo algebra. We also prove that Okubo algebras without nonzero idempotents over a field of characteristic 3 are always the residue of an Okubo algebra over a field of characteristic 0.

In characteristic two, it is shown that a central simple algebra of degree equal to a power of two with anisotropic orthogonal involution is totally decomposable, if it becomes either anisotropic or metabolic over all extensions of the ground field. A similar result is obtained for the case where this algebra with involution is Brauer-equivalent to a quaternion algebra and it becomes adjoint to a bilinear Pfister form over all splitting fields of the algebra.

For a maximal separable subfield K of a central simple algebra A, we provide a semiring isomorphism between K–K-sub-bimodules of A and H–H-sub-bisets of G = Gal ( L / F ) , where F = Cent ( A ) , L is the Galois closure of K / F , and H = Gal ( L / K ) . This leads to a combinatorial interpretation of the growth of dim K ⁡ ( ( K a K ) i ) , for fixed a ∈ A , especially in terms of Kummer subspaces.

We study systems of quadratic forms over fields and their isotropy over [Formula: see text]-extensions. We apply this to obtain particular splitting fields for quaternion algebras defined over a finite field extension. As a consequence we obtain that every central simple algebra of degree [Formula: see text] is split by a [Formula: see text]-extension of degree at most [Formula: see text].

We prove that the essential dimension of central simple algebras of degree $p^{\ell m}$ and exponent $p^{m}$ over fields $F$ containing a base-field $k$ of characteristic $p$ is at least $\ell +1$ when $k$ is perfect. We do this by observing that the $p$-rank of $F$ bounds the symbol length in $\text{Br}_{p^{m}}(F)$ and that there exist indecomposable $p$-algebras of degree $p^{\ell m}$ and exponent $p^{m}$. We also prove that the symbol length of the Kato-Milne cohomology group $\text{H}_{p^{m}}^{n+1}(F)$ is bounded from above by $\binom{r}{n}$ where $r$ is the $p$-rank of the field, and provide upper and lower bounds for the essential dimension of Brauer classes of a given symbol length.

Let $(A,\sigma)$ be a central simple algebra with an orthogonal involution. It is well-known that $O(A,\sigma)$ contains elements of reduced norm $-1$ if and only if the Brauer class of $A$ is trivial. We generalize this statement to Azumaya algebras with orthogonal involution over semilocal rings, and show that the "if" part fails if one allows the base ring to be arbitrary., Comment: Changes from previous version: Added Example 7. 5 pages. Comments are welcome

Various classes of simple torsion modules are classified over the quantum spatial ageing algebra (this is a Noetherian algebra of Gelfand-Kirillov dimension 4). Explicit constructions of these modules are given and for each module its annihilator is found.\ud \ud

Let D be a central simple algebra of prime degree over a field and let E be an 𝐒𝐋 1 ⁡ ( D ) {\operatorname{\mathbf{SL}}_{1}(D)} -torsor. We determine the complete motivic decomposition of certain compactifications of E. We also compute the Chow ring of E.

Danz computes the depth of certain twisted group algebra extensions in Comm. Alg. (2011), which are less than the values of the depths of the corresponding untwisted group algebra extensions in Burciu et al, I.E.J.A. (2011). In this paper, we show that the closely related h-depth of any group crossed product algebra extension is less than or equal to the h-depth of the corresponding (finite rank) group algebra extension. A convenient theoretical underpinning to do so is provided by the entwining structure of a right $H$-comodule algebra A and a right H-module coalgebra C for a Hopf algebra H. Then A tensor C is an A-coring, where corings have a notion of depth extending h-depth. This coring is Galois in certain cases where C is the quotient module Q of a coideal subalgebra R < H. We note that this applies for the group crossed product algebra extension, so that the depth of this Galois coring is less than the h-depth of H in G. Along the way, we show that subgroup depth behaves exactly like combinatorial depth with respect to the core of a subgroup, and extend results in Kadison J.Pure Appl.Alg. (2014) to coideal subalgebras of finite dimension., 25 pp. A proposition 1.3 is added connecting relative separability with finite depth, and an example 1.6 computing Q for H* in D(H). The introduction and abstract are rewritten

Let R be a central simple algebra finite-dimensional over its center F of characteristic 0. We will show that every element of reduced trace 0 in R can be expressed as [ a , [ c , b ] ] + λ [ c , [ a , b ] ] for some a , b , c ∈ R where λ ≠ 0 , − 1 . In addition, let D be a division algebra satisfying the conditions above. We will also show that the set of values of any nonzero multilinear polynomial of degree at most three, with coefficients from the center F of D , evaluated on M k ( D ) , k ≥ 2 , contains all matrices of reduced trace 0.

We generalize Amitsur's construction of central simple algebras over a field $F$ which are split by field extensions possessing a derivation with field of constants $F$ to nonassociative algebras: for every central division algebra $D$ over a field $F$ of characteristic zero there exists an infinite-dimensional unital nonassociative algebra whose right nucleus is $D$ and whose left and middle nucleus are a field extension $K$ of $F$ splitting $D$, where $F$ is algebraically closed in $K$. We then give a short direct proof that every $p$-algebra of degree $m$, which has a purely inseparable splitting field $K$ of degree $m$ and exponent one, is a differential extension of $K$ and cyclic. We obtain finite-dimensional division algebras over a field $F$ of characteristic $p>0$ whose right nucleus is a division $p$-algebra., Some minor changes to previous version, some definitions added in Section 2

We show that odd order transcendental elements of the Brauer group of a K3 surface can obstruct the Hasse principle. We exhibit a general K3 surface $Y$ of degree 2 over $\mathbb{Q}$ together with a three torsion Brauer class $\alpha$ that is unramified at all primes except for 3, but ramifies at all 3-adic points of $Y$. Motivated by Hodge theory, the pair $(Y, \alpha)$ is constructed from a cubic fourfold $X$ of discriminant 18 birational to a fibration into sextic del Pezzo surfaces over the projective plane. Notably, our construction does not rely on the presence of a central simple algebra representative for $\alpha$. Instead, we prove that a sufficient condition for such a Brauer class to obstruct the Hasse principle is insolubility of the fourfold $X$ (and hence the fibers) over $\mathbb{Q}_3$ and local solubility at all other primes., Comment: 22 pages; Magma scripts included as ancillary files in the arXiv distribution