In this paper, first we show that the invariant bilinear form in a quadratic Lie-Yamaguti algebra induces an isomorphism between adjoint representation and coadjoint representation. Then introduce notions of relative Rota-Baxter operators on algebras pre-Lie-Yamaguti algebras. We prove gives rise to naturally operator algebra. Finally, study symplectic structures algebra, which give as well As ...

We present an application of the program of groupoidification leading up to a sketch of a categorification of the Hecke algebroid—the category of permutation representations of a finite group. As an immediate consequence, we obtain a categorification of the Hecke algebra. We suggest an explicit connection to new higher isomorphisms arising from incidence geometries, which are solutions of the Z...

We have already seen that given any group G and a normal subgroup H, there is a natural homomorphism φ : G −→ G/H, whose kernel is H. In fact we will see that this map is not only natural, it is in some sense the only such map.

Let G denote a connected reductive group over a nonarchimedean local field F . Let K denote a special maximal parahoric subgroup of G(F ). We establish a Satake isomorphism for the Hecke algebra HK of K-bi-invariant compactly supported functions on G(F ). The key ingredient is a Cartan decomposition describing the double coset space K\G(F )/K. We also describe how our results relate to the trea...

Let M be a closed, oriented, n-manifold, and LM its free loop space. In [4] a commutative algebra structure in homology, H∗(LM), and a Lie algebra structure in equivariant homology H 1 ∗ (LM), were defined. In this paper we prove that these structures are homotopy invariants in the following sense. Let f : M1 → M2 be a homotopy equivalence of closed, oriented n-manifolds. Then the induced equiv...

Let M be a closed, oriented, n-manifold, and LM its free loop space. In [3] a commutative algebra structure in homology, H∗(LM), and a Lie algebra structure in equivariant homology H 1 ∗ (LM), were defined. In this paper we prove that these structures are homotopy invariants in the following sense. Let f : M1 → M2 be a homotopy equivalence of closed, oriented n-manifolds. Then the induced equiv...

In this short note we construct two families of examples large stratifying systems in module categories algebras. The first consist on infinite size the category an algebra A. second family show that a finite system dimensional A can be arbitrarily comparison to number isomorphism classes simple A-modules. We both are built using well-established results higher homological algebra.

We define a higher level version of the affine Hecke algebra and prove that, after completion, this is isomorphic to completion Webster's tensor product type A. then introduce Schur establish, again an isomorphism with quiver algebra. An important observation that surjects Dipper-James-Mathas cyclotomic q-Schur Moreover, we give nice diagrammatic presentations for all algebras introduced in pap...

Let $\mathfrak{g}_{0}$ be a simple Lie algebra of type ADE and let $U'_{q}(\mathfrak{g})$ the corresponding untwisted quantum affine algebra. We show that there exists an action braid group $B(\mathfrak{g}_{0})$ on Grothendieck ring $\mathcal{K}_{t}(\mathfrak{g})$ Hernandez-Leclerc’s category $\mathcal{C}_{\mathfrak{g}}^{0}$. Focused case $A_{N-1}$, we construct family monoidal autofunctors $\{...

In this paper we describe the structure of all sequential isomorphisms between the sets of von Neumann algebra effects. It turns out that if the underlying algebras have no commutative direct summands, then every sequential isomorphism between the sets of their effects extends to the direct sum of a *-isomorphism and a *-antiisomorphism between the underlying von Neumann algebras.