A Lie version of Turaev’s G-Frobenius algebras from 2-dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a g-quasi-Frobenius Lie algebra for g a finite dimensional Lie algebra. The latter consists of a quasi-Frobenius Lie algebra (q, β) together with a left g-module structure which acts on q via derivations and for which β is g-inva...

1 Under the microscope 1.1 Osculating spaces of homogeneous varieties Let G be a simply connected complex semisimple Lie group, g its Lie algebra, and g its tensor algebra. The universal envelopping algebra of U(g) is the quotient of this tensor algebra by the ideal generated by the elements xy ? yx ? x; y], x; y 2 g. This quotient algebra inherits a ltration from the natural grading of the ten...

We construct a family of two parameter quantum grou-\ps‎ ‎$U_{r,s}(mathfrak{g})$ associated with a generalized Kac-Moody‎ ‎algebra corresponding to symmetrizable admissible Borcherds Cartan‎ ‎matrix‎. ‎We also construct the $textbf{A}$-form $U_{textbf{A}}$ and‎ ‎the classical limit of $U_{r,s}(mathfrak{g})$‎. ‎Furthermore‎, ‎we‎ ‎display the equitable presentation for a subalgebra‎ ‎$U_{r...

Given an exterior algebra over a finite dimension vector space v, and let , where is graded ideal in . The relation between the regarding to -quadratic - quadratic will be investigated. We show that if only quadratic. Furthermore, it has been shown

Let (R;G) be a pair consisting of an elliptic root system R with a marking G. Assume that the attached elliptic Dynkin diagram (R;G) is simply-laced (see Sect. 2). We associate three Lie algebras, explained in 1), 2) and 3) below, to the elliptic root system, and show that all three are isomorphic. The isomorphism class is called the elliptic algebra. 1) The rst one is the subalgebra e g(R) gen...

We replace the group of group-like elements of the quantized enveloping algebra Uq(g) of a finite dimensional semisimple Lie algebra g by some regular monoid and get the weak Hopf algebra w q (g). It is a new subclass of weak Hopf algebras but not Hopf algebras. Then we devote to constructing a basis of w q (g) and determine the group of weak Hopf algebra automorphisms of w q (g) when q is not ...

We prove that the Drinfeld double of an arbitrary finite group scheme has finitely generated cohomology. That is to say, for G any scheme, and D(G) ring kG, we show self-extension algebra trivial representation a algebra, each D(G)-representation V extensions from form module over aforementioned algebra. As corollary, find all categories \({{\,\mathrm{rep}\,}}(G)^*_\mathscr {M}\) dual \({{\,\ma...

With a nilpotent element in a semisimple Lie algebra g one associates a finitely generated associative algebra W called a W -algebra of finite type. This algebra is obtained from the universal enveloping algebra U(g) by a certain Hamiltonian reduction. We observe that W is the invariant algebra for an action of a reductive group G with Lie algebra g on a quantized symplectic affine variety and ...

In this article, we give necessary and sufficient conditions under which the Leavitt path algebra $$L_K(\mathcal {G})$$ of an ultragraph $$\mathcal {G}$$ over a field K is purely infinite simple that it von Neumann regular. Consequently, obtain every graded either locally matricial algebra, or full matrix ring $$K[x, x^{-1}]$$ , algebra.

For a locally compact group G and p ∈ (1,∞), we define Bp(G) to be the space of all coefficient functions of isometric representations of G on quotients of subspaces of Lp spaces. For p = 2, this is the usual Fourier–Stieltjes algebra. We show that Bp(G) is a commutative Banach algebra that contractively (isometrically, if G is amenable) contains the Figà-Talamanca–Herz algebra Ap(G). If 2 ≤ q ...