Algebras and Representation Theory最新文献

We consider the group algebra of the symmetric group as a superalgebra, and describe its Lie subsuperalgebra generated by the transpositions.

We realise the Bott-Samelson resolutions of type A Schubert varieties as quiver Grassmannians. In order to explicitly describe this isomorphism, we introduce the notion of a geometrically compatible decomposition for any permutation in the symmetric group (S_n). For smooth type A Schubert varieties, we identify a suitable dimension vector such that the corresponding quiver Grassmannian is isomorphic to the Schubert variety. To obtain these isomorphisms, we construct a special quiver with relations and investigate two classes of quiver Grassmannians for this quiver.

We investigate cases where the finite dual coalgebra of a twisted tensor product of two algebras is a cotwisted tensor product of their respective finite dual coalgebras. This is achieved by interpreting the finite dual as a topological dual; in order to prove this, we show that the continuous dual is a strong monoidal functor on linearly topologized vector spaces whose open subspaces have finite codimension. We describe a sufficient condition for the result on finite dual coalgebras to be applied, and we specialize this condition to particular constructions including Ore extensions, smash product algebras, and bitwisted tensor products of bialgebras.

The finitistic dimension conjecture is closely connected to the symmetry of the finitistic dimension. Recent work indicates that such connection extends to one of its upper bounds, the delooping level. In this paper, we show that the same holds for the derived delooping level, which is an improvement of the delooping level. This reduces the finitistic dimension conjecture to considering algebras whose opposite algebra has (derived) delooping level zero. We thereby demonstrate ways to utilize the new concept of derived delooping level to obtain new results and present additional work involving tensor product of algebras.

In this paper, we show that there are infinitely many semisimple monoidal categories of rank two over an algebraically closed field (mathbb {F}).

We prove a version of Gabriel’s theorem for (possibly infinite dimensional) representations of infinite quivers. More precisely, we show that the representation theory of a quiver (varvec{Omega }) is of unique type (each dimension vector has at most one associated indecomposable) and infinite Krull-Schmidt (every, possibly infinite dimensional, representation is a direct sum of indecomposables) if and only if (varvec{Omega }) is eventually outward and of generalized ADE Dynkin type ((varvec{A_n}), (varvec{D_n}), (varvec{E_6}), (varvec{E_7}), (varvec{E_8}), (varvec{A_infty }), (varvec{A_{infty , infty }}), or (varvec{D_infty })). Furthermore we define an analog of the Euler-Tits form on the space of eventually constant infinite roots and show that a quiver is of generalized ADE Dynkin type if and only if this form is positive definite. In this case the indecomposables are all locally finite-dimensional and eventually constant and correspond bijectively to the positive roots (i.e. those of length (varvec{1})).

This article continues our study of P- and Q-key polynomials, which are (non-symmetric) “partial” Schur P- and Q-functions as well as “shifted” versions of key polynomials. Our main results provide a crystal interpretation of P- and Q-key polynomials, namely, as the characters of certain connected subcrystals of normal crystals associated to the queer Lie superalgebra (mathfrak {q}_n). In the P-key case, the ambient normal crystals are the (mathfrak {q}_n)-crystals studied by Grantcharov et al., while in the Q-key case, these are replaced by the extended (mathfrak {q}_n)-crystals recently introduced by the first author and Tong. Using these constructions, we propose a crystal-theoretic lift of several conjectures about the decomposition of involution Schubert polynomials into P- and Q-key polynomials. We verify these generalized conjectures in a few special cases. Along the way, we establish some miscellaneous results about normal (mathfrak {q}_n)-crystals and Demazure (mathfrak {gl}_n)-crystals.

In this paper, we calculate explicitly automorphism group of the Suzuki’s Hopf algebra (A_{Nn}^{mu lambda }) by viewing Yetter-Drinfeld modules as invariants of Hopf algebra automorphisms.

In this paper we characterize the relative Gorenstein weak global dimension of the Gorenstein (mathcal {B})-flat R-modules and projectively coresolved Gorenstein (mathcal {B})-flat R-modules recently studied by S. Estrada, A. Iacob, and M. A. Pérez, which are a relativisation of the ones introduced by J. Šaroch and J. Št’ovíchěk. As application we prove that the weak global dimension with respect to the Gorenstein (textrm{FP}_n)-flat R-modules is finite over a Gorenstein n-coherent ring R and in this case coincides with the flat dimension of the right (textrm{FP}_n)-injective R-modules. This result extends the known for Gorenstein flat modules over Iwanaga-Gorenstein and Ding-Chen rings. We also show that there is a close relationship between the relative global dimension of the Gorenstein (textrm{FP}_n)-projectives and the Gorenstein weak global dimension respect to the class of Gorenstein (textrm{FP}_n)-flat R-modules. We also get an hereditary and complete cotorsion triple and consequently a balanced pair.

We study a relation between the Drinfeld modules and the even dimensional noncommutative tori. A non-abelian class field theory is developed based on this relation. Explicit generators of the Galois extensions are constructed.