Let n be a positive integer and let k be a field (of arbitrary characteristic) accessible to symbolic computation. We describe an algorithmic test for determining whether or not a finitely presented k-algebra R has infinitely many equivalence classes of semisimple representations R → Mn(k′), where k is the algebraic closure of k. The test reduces the problem to computational commutative algebra...

We show that if T is an isometry (as metric spaces) from an open subgroup of the invertible group A of a unital Banach algebra A onto an open subgroup of the invertible group B of a unital Banach algebra B, then T is extended to a real-linear isometry up to translation between these Banach algebras. We consider multiplicativity or unti-multiplicativity of the isometry. Note that a unital linear...

The full Kostant–Toda hierarchy on a semisimple Lie algebra is system of Lax equations, in which the flows are determined by gradients Chevalley invariants. This paper concerned with even orthogonal algebra. By using Pfaffian matrix as one invariants, we construct an explicit form flow associated to this invariant. As main result, introduce extension Schur’s Q-functions time variables, and use ...

In this topic proposal, we discuss the representation theory of semisimple Lie algebras, including results on the structure of these algebras and their modules. In particular, the theory of root systems and Verma modules are very important in this context. We classify all simple Lie algebras, and obtain a presentation for any semisimple Lie algebra. We also encounter several striking formulas t...

We establish a bijective correspondence between gauge equivalence classes of dynamical twists in a finite-dimensional Hopf algebra H based on a finite abelian group A and equivalence classes of pairs (K, {Vλ}λ∈Â), where K is an H-simple left H-comodule semisimple algebra and {Vλ}λ∈Â is a family of irreducible representations satisfying certain conditions. Our results generalize the results obta...

Let Λ be a preprojective algebra of Dynkin type, and let G be the corresponding complex semisimple simply connected algebraic group. We study rigid modules in subcategories Sub Q for Q an injective Λ-module, and we introduce a mutation operation between complete rigid modules in Sub Q. This yields cluster algebra structures on the coordinate rings of the partial flag varieties attached to G.

Different versions of Wiener’s Tauberian theorem are discussed for the generalized group algebra LI(G,A) (of integrable functions on a locally compact abelian group G taking values in a commutative semisimple regular Banach algebra A) using A-valued Fourier transforms. A weak form of Wiener’s Tauberian property is introduced and it is proved that LI(G,A) is weakly Tauberian if and only if A is....

We establish a bijective correspondence between gauge equivalence classes of dynamical twists in a finite-dimensional Hopf algebra H based on a finite abelian group A and equivalence classes of pairs (K, {V λ } λ∈ b A), where K is an H-simple left H-comodule semisimple algebra and {V λ } λ∈ b A is a family of irreducible representations satisfying certain conditions. Our results generalize the ...