New coefficients for Hopf Cyclic Cohomology

In this note, the categories of coefficients for Hopf cyclic cohomology of comodule algebras and comodule coalgebras are extended. We show that these new categories have two proper different subcategories where the smallest one is the known category of stable anti Yetter–Drinfeld modules. We prove that components of Hopf cyclic cohomology such as cup products work well with these new coefficients.     

Conjugacy Classes of P-Cycles of Type D in Alternating Groups

We classify the conjugacy classes of p-cycles of type D in alternating groups. This finishes the open cases in [3 Andruskiewitsch , N. , Fantino , F. , Graña , M. , Vendramin , L. ( 2011 ). Finite-dimensional pointed Hopf algebras with alternating groups are trivial . Ann. Mat. Pura Appl 190 : 225 – 245 .[Web of Science ®] , [Google Scholar]]. Also we determine all the subracks of those conjugacy classes which are not of type D.     

Involutions of a Clifford Algebra Induced by Involutions of Orthogonal Group in Characteristic 2

Among the involutions of a Clifford algebra, those induced by the involutions of the orthogonal group are the most natural ones. In this work, several basic properties of these involutions, such as the relations between their invariants, their occurrences, and their decompositions, are investigated.     

Symmetric Elements,Hermitian Forms,and Cyclic Involutions

Let k be a field, K/k be a quadratic separable field extension, and 𝒜 a finite dimensional central simple algebra over K. If k is global or the field of fractions of a two-dimensional excellent henselian local domain with an algebraically closed residue field of characteristic zero and the degree of 𝒜 is odd, we prove that all K/k-involutions on 𝒜 are cyclic.     

Some Results about Projective Modules over Monoid Algebras

(1) Let R be a 1-dimensional commutative Noetherian anodal ring with finite seminormalization and M a commutative cancellative torsion-free monoid. Let P be a projective R[M]-module of rank r. Then P ? ∧^rP ⊕ R[M]^r?1.

(2) Murthy and Pedrini [11 Murthy, M. P., Pedrini, C. (1973). K₀ and K₁ of polynomial rings. In: Algebraic K-Theory, II: “Classical” Algebraic K-Theory and Connections with Arithmetic (Proc. Conf., Seattle Res. Center, Battelle Memorial Inst., Seattle, Wash., 1972). Lecture Notes in Math., Vol. 342. Berlin: Springer, pp. 109–121.[Crossref] , [Google Scholar]] proved K₀ homotopy invariance of polynomial extension of some affine normal surfaces. We extend this result to a monoid extension (see 1.5).     

Free Structures in Division Rings

Makar–Limanov's conjecture states that, if a division ring D is finitely generated and infinite dimensional over its center k, then D contains a free k-subalgebra of rank 2. In this work, we will investigate the existence of such structures in D, the division ring of fractions of the skew polynomial ring L[t; σ], where t is a variable and σ is a k-automorphism of L. For instance, we prove Makar-Limanov's conjecture when either L is the function field of an abelian variety or the function field of the n-dimensional projective space.     

Existence of Gradings on Associative Algebras

In this article, we study the existence of gradings on finite dimensional associative algebras. We prove that a connected algebra A does not have a nontrivial grading if and only if A is basic, its quiver has one vertex, and its group of outer automorphisms is unipotent. We apply this result to prove that up to graded Morita equivalence there do not exist nontrivial gradings on the blocks of group algebras with quaternion defect groups and one isomorphism class of simple modules.     

Lie and Jordan Products in Interchange Algebras

We study Lie brackets and Jordan products derived from associative operations ○, ? satisfying the interchange identity (w?x) ○ (y?z) ≡ (w ○ y)?(x ○ z). We use computational linear algebra, based on the representation theory of the symmetric group, to determine all polynomial identities of degree ≤7 relating (i) the two Lie brackets, (ii) one Lie bracket and one Jordan product, and (iii) the two Jordan products. For the Lie–Lie case, there are two new identities in degree 6 and another two in degree 7. For the Lie–Jordan case, there are no new identities in degree ≤6 and a complex set of new identities in degree 7. For the Jordan–Jordan case, there is one new identity in degree 4, two in degree 5, and complex sets of new identities in degrees 6 and 7.