It is a well-known consequence of the elementary theory of vector spaces that if A and B are n-by-n matrices over a field (or even a skew field) such that AB = 1, then BA = 1. This result remains true for matrices over a commutative ring, however, it is not, in general, true for matrices over noncommutatives rings. In this paper we show that if A and B are n-by-n matrices over a commutative sem...

Miriam Cohen raised the question whether the smash product of a semisimple Hopf algebra and a semiprime module algebra is semiprime. In this paper we show that the smash product of a commutative semiprime module algebra over a semisimple cosemisimple Hopf algebra is semiprime. In particular we show that the central H-invariant elements of the Martindale ring of quotients of a module algebra for...

The Birman–Murakami–Wenzl algebra (BMW algebra) of type Dn is shown to be semisimple and free over Z[δ±1, l±1]/(m(1− δ)− (l− l−1)) of rank (2n + 1)n!! − (2n−1 + 1)n!, where n!! = 1 · 3 · · · (2n − 1). We also show it is a cellular algebra over suitable rings. The Brauer algebra of type Dn is a homomorphic ring image and is also semisimple and free of the same rank, but over the ring Z[δ±1]. A r...

It is well known that over an infinite field the ring of symmetric functions in a finite number of variables is isomorphic to the one of polynomial functions on a single matrix that are invariants by the action of conjugation by general linear group. We generalize this result showing that the abelianization of the algebra of the symmetric tensors of fixed order over a free associative algebra i...

We identify the ring of odd symmetric functions introduced by Ellis and Khovanov as the space of skew polynomials fixed by a natural action of the Hecke algebra at q = −1. This allows us to define graded modules over the Hecke algebra at q = −1 that are ‘odd’ analogs of the cohomology of type A Springer varieties. The graded module associated to the full flag variety corresponds to the quotient...

Let G be a connected reductive group acting on a finite dimensional vector space V . Assume that V is equipped with a G-invariant symplectic form. Then the ring O(V ) of polynomial functions becomes a Poisson algebra. The ring O(V ) of invariants is a sub-Poisson algebra. We call V multiplicity free if O(V ) is Poisson commutative, i.e., if {f, g} = 0 for all invariants f and g. Alternatively, ...

In this paper, we show the second part of Schur-Weyl duality for mixed tensor space. The quantum group U = U(gln) of the general linear group and a q-deformation Br,s(q) of the walled Brauer algebra act on V ⊗r ⊗V ∗⊗s where V = R is the natural U-module. We show that EndBnr,s(q)(V ⊗r ⊗ V ∗) is the image of the representation of U, which we call the rational q-Schur algebra. As a byproduct, we o...

In an earlier paper we established that every second countable, completely normal spectral space is homeomorphic to the ℓ-spectrum of some Abelian ℓ-group. We extend result ℓ-spectra vector lattices over any countable totally ordered division ring k. Combining those methods with Baro's Normal Triangulation Theorem, obtain following result: TheoremFor formally real field k, spectrum commutative ...

We give a structure theorem for a ring homomorphism of a commutative regular Banach algebra into another commutative Banach algebra. In particular, it is shown that: (i) A ring homomorphism of a commutative C∗-algebra onto another commutative C∗-algebra with connected infinite Gelfand space is either linear or anti-linear. (ii) A ring automorphism of L1(R ) is either linear or anti-linear. (iii...

We describe presentations of the Roger-Yang generalized skein algebras for punctured spheres with an arbitrary number punctures. This algebra is a quantization decorated Teichmuller space and generalizes construction Kauffman bracket algebra. In this paper, we also obtain new interpretation homogeneous coordinate ring Grassmannian planes in terms theory.