Generalised nice sets

For a finite-dimensional semisimple Lie algebra and a root of unity, Lusztig defined an infinite-dimensional quantum group of divided powers. Under certain restrictions on the order of the root of unity, he constructed a Frobenius homomorphism with finite-dimensional Hopf kernel and with the image of the universal enveloping algebra. In this article, we define and completely describe the Frobenius homomorphism for arbitrary roots of unity by systematically using the theory of Nichols algebras. In several new exceptional cases, the Frobenius–Lusztig kernel is associated to a different Lie algebra than the initial Lie algebra. Moreover, the Frobenius homomorphism often switches short and long roots and/or maps to a braided category.

In this communication I would like to give a progress report on a program to construct graded geometric structures on graded differential manifolds. Most of the results so far are on the formal algebraic level and many important problems of analysis remain. Kostant I has given a definition of a graded differential manifold of dimension (n,m) and this is discussed in these proceedings by M. Batchelor. I only mention here that it consists of a pair (M,A) where M is an ordinary C ~ manifold of dimension n and A is a sheaf of Z2-graded commutative algebras whose underlying graded vector space V = V 0 + V I has dim V 0 = n, dim V 1 = m. A graded Lie group (G,A) then is a pair where G is a Lie group and the restricted dual A ° (vanishes on some ideal of finite codimension) has the structure of a graded Hopf algebra and can be represented as A ° = R(G) ^ E(G) where R(G) is the group ring, E(G) is the universal enveloping algebra of the graded Lie algebra G = G O + G I with G O the Lie algebra of G, and ^ denotes semidirect product. In analogy with the usual situation, graded frames can be introduced and the graded frame bundle constructed. This consists of a principal fibre bundle L(M,G) with group G = GL(n)×GL(m) such that the graded tangent bundle T(M,A) defined by Kostant is an associated vector bundle for L(M,G). The graded frame bundle (not really a bundle) is then given by the graded manifold (L(M,G), A8 H) where H ° = R(G) ^ E(G) , G = GL(n)× GL(m), G = gl(n,m) = End V. Now if G'C G is a closed Lie subgroup of G with a subHopf algebra H' C H , then (L(M,G'), A8 H') is called a graded G-struc ture if L(M,G') is reduced subbundle of L(M,G). There is a natural right action R a of H' defined on (L(M,G'), ASH'). Associated with the graded tangent bundle of a G-structure, there is the tangent sheaf derA + derH (der A = derivations of A), and this is a free A@ H module. A complement H to derH is called a horizontal subspace, and as usual H defines a connection if R * H = H. As underlying graded vector spaa ces H is isomorphic to V. In analogy with the ordinary ease, we study graded G-structures by studying graded Spencer cohomology. Let GC gl(n,m) = V8 V* be a graded Lie algebra. We define G (1) i i I I the first prolongation of G by all T E Hom(V,G) which satisfy T(u)v =(-i) luILvi T(V)U where u,v are homogeneous elements in V of degree lul, Ivl respectively. The k th prolongation is defined inductively by G (k) = (G(k-l)) (I) • We construct Z 2 gravector spaces C k+l'l = G (k) O Al(V *) where AI(V *) = AI(V~)0 Sl(V~) is the graded ded exterior algebra over V*. The cochain map ~:C k+l'/ >?,/+i is defined ~ by

We present several embedding results for 3 3 -graded Lie algebras and KKT algebras that are generated by two homogeneous elements of degrees 1 1 and − 1 -1 . We also propose the canonical kernel function for a “universal Bergman kernel” which extends the usual Bergman kernel on a bounded symmetric domain to a group-valued function or, in terms of formal series, to an element in the formal completion of the universal enveloping algebra of the free 3 3 -graded Lie algebra in a pair of generators.

A nonassociative algebra endowed with a Lie bracket, called a torsion algebra, is viewed as an algebraic analog of a manifold with an affine connection. Its elements are interpreted as vector fields and its multiplication is interpreted as a connection. This provides a framework for differential geometry on a formal manifold with a formal connection. A torsion algebra is a natural generalization of pre‐Lie algebras which appear as the “torsionless” case. The starting point is the observation that the associator of a nonassociative algebra is essentially the curvature of the corresponding Hochschild quasicomplex. It is a cocycle, and the corresponding equation is interpreted as Bianchi identity. The curvature‐associator‐monoidal structure relationships are discussed. Conditions on torsion algebras allowing to construct an algebra of functions, whose algebra of derivations is the initial Lie algebra, are considered. The main example of a torsion algebra is provided by the pre‐Lie algebra of Hochschild cochains of a K‐module, with Lie bracket induced by Gerstenhaber composition.

Lie color algebras are generalizations of Lie superalgebras and graded Lie algebras. The properties of a Lie color algebra can often be related directly to the ring structure of its universal enveloping algebra. We study the effects of torsion elements and torsion subspaces. Let [Formula: see text] denote a Lie color algebra. If [Formula: see text] is homogeneous and torsion then x2 = 0 in [Formula: see text]. If no homogeneous element of [Formula: see text] is torsion, then [Formula: see text] so [Formula: see text] is semiprime. In this case we can give a test which uses Gröbner basis methods to determine when [Formula: see text] is a domain. This is applied in an example to show [Formula: see text] may be a domain even if [Formula: see text] contains torsion elements and torsion subspaces.

Let 𝔤 be a finite-dimensional simple Lie algebra of rank l over an algebraically closed field 𝕜 of characteristic zero. We identify 𝔤 with 𝔤∗ through the Killing form of 𝔤. Let (e, h, f) be an 𝔰𝔩2-triple of 𝔤. Denote by 𝔤 e the centralizer of e in 𝔤 and by $\mathrm {S}(\mathfrak {g}^{e})^{\mathfrak {g}^{e}}$ the algebra of symmetric invariants of 𝔤 e . We say that e is good if the nullvariety of some l homogeneous elements of $\mathrm {S}(\mathfrak {g}^{e})^{\mathfrak {g}^{e}}$ in (𝔤 e )∗ has codimension l. In our previous work (Charbonnel and Moreau. Math. Zeitsch. 282, n° 1-2, 273–339 2016), we showed that if e is good then $\mathrm {S}(\mathfrak {g}^{e})^{\mathfrak {g}^{e}}$ is a polynomial algebra. In this paper, we prove that the converse of the main result of Charbonnel and Moreau (Math. Zeitsch. 282, n° 1-2, 273–339 2016) is true. Namely, we prove that e is good if and only if for some homogeneous generating sequence q 1, … , q l of S(𝔤) 𝔤 , the initial homogeneous components of their restrictions to e + 𝔤 f are algebraically independent over 𝕜.