A finite W-algebra U (g, e) is a certain finitely generated algebra that can be viewed as the enveloping algebra of the Slodowy slice to the adjoint orbit of a nilpotent element e of a complex reductive Lie algebra g. It is possible to give the tensor product of a U (g, e)-module with a finite dimensional U (g)-module the structure of a U (g, e)-module; we refer to such tensor products as trans...

Let G be a group, S a subgroup of G, and F a field of characteristic p. We denote the augmentation ideal of the group algebra FG by ω(G). The Zassenhaus-Jennings-Lazard series of G is defined by Dn(G) = G ∩ (1 + ω(G)). We give a constructive proof of a Theorem of Quillen stating that the graded algebra associated to FG is isomorphic as an algebra to the enveloping algebra of the restricted Lie ...

Let Uε(g) be the simply connected quantized enveloping algebra associated to an exceptional simple Lie algebra g at the roots of unity. The De Concini-Kac-Procesi conjecture on the dimension of the irreducible representations of Uε(g) is proved for representations corresponding to the spherical conjugacy classes of the simply connected group with Lie algebra g.

Let U(g) denote the universal enveloping algebra of a Lie algebra g. We show the existence of a ribbon algebra structure in a particular deformation of U(g) which leads to a symmetric pre-monoidal category of U(g)-modules.

A finite W -algebra U(g, e) is a certain finitely generated algebra that can be viewed as the enveloping algebra of the Slodowy slice to the adjoint orbit of a nilpotent element e of a complex reductive Lie algebra g. It is possible to give the tensor product of a U(g, e)-module with a finite dimensional U(g)module the structure of a U(g, e)-module; we refer to such tensor products as translati...

that g* has the same derived algebra as g itself and that every ideal in g is also an ideal in t*. Let g be any algebraic Lie algebra. Denote by b the radical of g (i.e., the largest solvable ideal in g) and by n the largest ideal of g composed only of nilpotent matrices. By Levi's theorem, g is the direct sum of t and of a semi-simple subalgebra J. It can be proved that f is the direct sum of ...

We show that to cubic order double field theory is encoded in Yang-Mills theory. To this end we use algebraic structures from string as follows: The ${L}_{\ensuremath{\infty}}$-algebra of the tensor product $\mathcal{K}\ensuremath{\bigotimes}\mathfrak{g}$ Lie algebra $\mathfrak{g}$ gauge group and a ``kinematic algebra'' $\mathcal{K}$ ${C}_{\ensuremath{\infty}}$-algebra. This structure induces ...

Abstra t Is there an alternative mathemati s? In parti ular, does intuitionism yield an essentially new approa h that annot be spe i ed within lassi al mathemati s? The intended informal meaning of intuitionisti logi Int was given in the 1930s by the Brouwer-HeytingKolmogorov semanti s whi h understands intuitionisti truth as provability. Moreover, Kolmogorov (and later Godel) suggested interp...

Let $\mathfrak{g}$ be a complex simple Lie algebra and let $\mathfrak{g}_0$ the sub-algebra fixed by diagram automorphism of $\mathfrak{g}$. $G$ complex, simply-connected, algebraic group with $\mathfrak{g}$, $G_0$ connected subgroup $\mathfrak{g}_0$. $\rho$ half sum positive roots In this article, we give necessary sufficient condition for highest weight $\mathfrak{g}_0$-representation $V_0(d\...

For each simple Lie algebra g consider the corresponding affine vertex algebra Vcrit(g) at the critical level. The center of this vertex algebra is a commutative associative algebra whose structure was described by a remarkable theorem of Feigin and Frenkel about two decades ago. However, only recently simple formulas for the generators of the center were found for the Lie algebras of type A fo...