In this paper we give a thoughtful exposition of the Clifford algebra Cl(HV ) of multivecfors which is naturally associated with a hyperbolic space HV , whose elements are called vecfors. Geometrical interpretation of vecfors and multivecfors are given. Poincaré automorphism (Hodge dual operator) is introduced and several useful formulas derived. The role of a particular ideal in Cl(HV ) whose ...

Let g be a complex semisimple Lie algebra, and let G be a complex semisimple group with trivial center whose root system is dual to that of g. We establish a graded algebra isomorphism H q (Xλ,C) ∼= Sg e/Iλ, where Xλ is an arbitrary spherical Schubert variety in the loop Grassmannian for G, and Iλ is an appropriate ideal in the symmetric algebra of g, the centralizer of a principal nilpotent in...

Let H be a finitely-generated free abelian group and L(H) the graded free Lie algebra on H . There is a natural homomorphism H ⊗ L(H)→ L(H) defined by bracketing, whose kernel is denoted D(H). If H supports a non-singular symplectic form, eg H = H1(Σ), where Σ is a closed orientable surface, with symplectic basis {xi, yi}, then D(H) is, in fact, a Lie algebra. It can be identified with the Lie ...

this article presents a unified approach to the abstract notions of partial convolution and involution in $l^p$-function spaces over semi-direct product of locally compact groups. let $h$ and $k$ be locally compact groups and $tau:hto aut(k)$ be a continuous homomorphism.  let $g_tau=hltimes_tau k$ be the semi-direct product of $h$ and $k$ with respect to $tau$. we define left and right $tau$-c...

We prove that the reflection equation (RE) algebra LR associated with a finite dimensional representation of a quasitriangular Hopf algebra H is twist-equivalent to the corresponding Faddeev-Reshetikhin-Takhtajan (FRT) algebra. We show that LR is a module algebra over the twisted tensor squareH R ⊗H and the double D(H). We define FRTand RE-type algebras and apply them to the problem of equivari...

Let A be a finite dimensional Hopf algebra, D(A) = A ⊗A its Drinfel’d double and H(A) = A#A its Heisenberg double. The relation between D(A) and H(A) has been found by J.-H. Lu in [24] (see also [33], p. 196): the multiplication of H(A) may be obtained by twisting the multiplication of D(A) by a certain left 2-cocycle which in turn is obtained from the R-matrix of D(A). It was also obtained in ...

The main goal of this paper is to investigate the structure of Hopf algebras with the property that either its Jacobson radical is a Hopf ideal or its coradical is a subalgebra. In order to do that we define the Hochschild cohomology of an algebra in an abelian monoidal category. Then we characterize those algebras which have dimension less than or equal to 1 with respect to Hochschild cohomolo...

Let L be a Lie algebra with universal enveloping algebra U(L). We prove that if H is another Lie algebra with the property that U(L) ∼= U(H) then certain invariants of L are inherited by H. For example, we prove that if L is nilpotent then H is nilpotent with the same class as L. We also prove that if L is nilpotent of class at most two then L is isomorphic to H.

We introduce an equivariant version of cyclic cohomology for Hopf module algebras. For any H-module algebra A, where H is a Hopf algebra with S2 = idH we define the cocyclic module C ♮ H(A) and we find its relation with cyclic cohomology of crossed product algebra A ⋊ H. We define K 0 (A), the equivariant K-theory group of A, and its pairing with cyclic and periodic cyclic cohomology of C H(A).

In 2013 Benkart, Lopes and Ondrus introduced studied in a series of papers the infinite-dimensional unital associative algebra Ah generated by elements x,y, which satisfy relation yx?xy=h for some 0?h?F[x]. We generalize this construction to Ah(B) working over fixed F-algebra B instead F. describe polynomial identities infinite field F case h?B[x] satisfies certain restrictions.