Algebras and Representation Theory最新文献

A ghor algebra is the path algebra of a dimer quiver on a surface, modulo relations that come from the perfect matchings of its quiver. Such algebras arise from abelian quiver gauge theories in physics. We show that a ghor algebra (Lambda ) on a torus is a dimer algebra (a quiver with potential) if and only if it is noetherian, and otherwise (Lambda ) is the quotient of a dimer algebra by homotopy relations. Furthermore, we classify the simple (Lambda )-modules of maximal dimension and give an explicit description of the center of (Lambda ) using a special subset of perfect matchings. In our proofs we introduce formalized notions of Higgsing and the mesonic chiral ring from quiver gauge theory.

The semisimple bismash product Hopf algebra (J_n=k^{S_{n-1}}#kC_n) for an algebraically closed field k is constructed using the matched pair actions of (C_n) and (S_{n-1}) on each other. In this work, we reinterpret these actions and use an understanding of the involutions of (S_{n-1}) to derive a new Froebnius-Schur indicator formula for irreps of (J_n) and show that for n odd, all indicators of (J_n) are nonnegative. We also derive a variety of counting formulas including Theorem 6.2.2 which fully describes the indicators of all 2-dimensional irreps of (J_n) and Theorem 6.1.2 which fully describes the indicators of all odd-dimensional irreps of (J_n) and use these formulas to show that nonzero indicators become rare for large n.

We use the Gauss decomposition of the generator matrix in the R-matrix presentation of the Yangian for the orthosymplectic Lie superalgebra (mathfrak {osp}_{N|2m}) to produce its Drinfeld-type presentation. The results rely on a super-version of the embedding theorem which allows one to identify a subalgebra in the R-matrix presentation which is isomorphic to the Yangian associated with (mathfrak {osp}_{N|2m-2}).

Recently, Buan and Marsh showed that if two complete (tau )-exceptional sequences agree in all but at most one term, then they must agree everywhere, provided the algebra is (tau )-tilting finite. They conjectured that the result holds without that assumption. We prove their conjecture. Along the way, we also show that the dimension vectors of the modules in a (tau )-exceptional sequence are linearly independent.

We provide identities of inverse Chevalley type for the graded characters of level-zero Demazure submodules of extremal weight modules over a quantum affine algebra of type C. These identities express the product (e^{mu } text {gch} ~V_{x}^{-}(lambda )) of the (one-dimensional) character (e^{mu }), where (mu ) is a (not necessarily dominant) minuscule weight, with the graded character gch(V_{x}^{-}(lambda )) of the level-zero Demazure submodule (V_{x}^{-}(lambda )) over the quantum affine algebra (U_{textsf{q}}(mathfrak {g}_{textrm{af}})) as an explicit finite linear combination of the graded characters of level-zero Demazure submodules. These identities immediately imply the corresponding inverse Chevalley formulas for the torus-equivariant K-group of the semi-infinite flag manifold (textbf{Q}_{G}) associated to a connected, simply-connected and simple algebraic group G of type C. Also, we derive cancellation-free identities from the identities above of inverse Chevalley type in the case that (mu ) is a standard basis element ({varepsilon }_{k}) in the weight lattice P of G.

Fock and Goncharov described a quantization of cluster (mathcal {X})-varieties (also known as cluster Poisson varieties) in Fock and Goncharov (Ann. Sci. Éc. Norm. Supér. 42(6), 865–930 2009). Meanwhile, families of deformations of cluster (mathcal {X})-varieties were introduced in Bossinger et al. (Compos. Math. 156(10), 2149–2206, 2020). In this paper we show that the two constructions are compatible– we extend the Fock-Goncharov quantization of (mathcal {X})-varieties to the families of Bossinger et al. (Compos. Math. 156(10), 2149–2206, 2020). As a corollary, we obtain that these families and each of their fibers have Poisson structures. We relate this construction to the Berenstein-Zelevinsky quantization of (mathcal {A})-varieties (Berenstein and Zelevinsky, Adv. Math. 195(2), 405–455, 2005). Finally, inspired by the counter-example to quantum positivity of the quantum greedy basis in Lee, et al. (Proc. Natl. Acad. Sci. 111(27), 9712–9716, 2014), we compute a counter-example to quantum positivity of the quantum theta basis.

In this note, we consider the twisted Yangians (text {Y}(mathfrak {g}_N)) associated with the orthogonal and symplectic Lie algebras (mathfrak {g}_N=mathfrak {o}_N,mathfrak {sp}_N). First, we introduce a certain subalgebra (text {A}_c(mathfrak {g}_N)) of the double Yangian for (mathfrak {gl}_N) at the level (cin mathbb {C}), which contains the centrally extended (text {Y}(mathfrak {g}_N)) at the level c as well as its vacuum module (mathcal {M}_c(mathfrak {g}_N)). Next, we employ its structure to construct examples of quasi modules for the quantum affine vertex algebra (mathcal {V}_c(mathfrak {gl}_N)) associated with the Yang R-matrix. Finally, we use the description of the center of (mathcal {V}_c(mathfrak {gl}_N)) to obtain explicit formulae for families of central elements for a certain completion of (text {A}_c(mathfrak {g}_N)) and invariants of (mathcal {M}_c(mathfrak {g}_N)).

This paper aims to initiate a study of the representation theory of representation-finite artin algebras in terms of the nilpotency of the radical of their module category. Firstly, we shall calculate this nilpotency explicitly for hereditary algebras of type (mathbb {A}_n) and for Nakayama algebras. Surprisingly, this nilpotency for a given algebra coincides with its Loewy length if and only if the algebra is a hereditary Nakayama algebra. Secondly, we shall find all artin algebras for which this nilpotency is equal to any given positive integer up to four and describe completely their module category.

We study automorphic Lie algebras using a family of evaluation maps parametrised by the representations of the associative algebra of functions. This provides a descending chain of ideals for the automorphic Lie algebra which is used to prove that it is of wild representation type. We show that the associated quotients of the automorphic Lie algebra are isomorphic to twisted truncated polynomial current algebras. When a simple Lie algebra is used in the construction, this allows us to describe the local Lie structure of the automorphic Lie algebra in terms of affine Kac-Moody algebras.

We study a class of derived discrete Nakayama algebras. All indecomposable compact objects in the derived module category are determined and all recollements generated by the indecomposable compact exceptional objects are classified. It reveals that all such recollements are derived equivalent to stratifying recollements. As a byproduct, this confirms a question due to Xi for these recollements.