Eckhard Meinrenken
This article summarizes joint work with A. Alekseev (Geneva) on the Duflo isomorphism for quadratic Lie algebras. We describe a certain quantization map for Weil algebras, generalizing both the Duflo map and the quantization map for Clifford algebras. In this context, Duflo's theorem generalizes to a statement in equivariant cohomology.
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In the present note we show that the Duflo isomorphism extends to an isomorphism of associative algebras of tangential cohomologies. This result confirms the B.Shoikhet's conjecture.
We prove that when Kontsevich's deformation quantization is applied on weight homogeneous Poisson structures, the operators in the $\ast-$ product formula are weight homogeneous. We then consider the linear Poisson case $X=\mathfrak{g}^\ast$ for a semi simple Lie algebra $\mathfrak{g}$. As an application we provide an isomorphism between the Cattaneo-Felder-Torossian reduction algebra $H^0(\mathfrak{g},\mathfrak{m},χ)$ and the $W-$ algebra $(U(\mathfrak{g})/U(\mathfrak{g})\mathfrak{m}_χ)^\mathfrak{m}$. We also show that in the $W-$ algebra setting, $(S(\mathfrak{g})/S(\mathfrak{g})\mathfrak{m}…
Let $G$ be a finite-dimensional Poisson algebraic, Lie or formal group. We show that the center of the quantization of $G$ provided by an Etingof-Kazhdan functor is isomorphic as an algebra to the Poisson center of the algebra of functions on $G$. This recovers and generalizes Duflo's theorem which gives an isomorphism between the center of the enveloping algebra of a finite-dimensional Lie algebra $\mathfrak{a}$ and the subalgebra of ad-invariant in the symmetric algebra of $\mathfrak{a}$. As our proof relies on Etingof-Kazhdan construction it ultimately depends on the existence of Drinfeld a…
We study the analogue of the Harish-Chandra homomorphism where the universal enveloping algebra is replaced by the Clifford algebra, $Cl(g)$, of a semisimple Lie algebra $g$. Two main goals are achieved. First, we prove that there is a Harish-Chandra type isomorphism between the subalgebra of $g$-invariants in $Cl(g)$ and the Clifford algebra of the Cartan subalgebra of $g$. Second, the Cartan subalgebra is identified, via this isomorphism, with a graded space of the so-called primitive skew-symmetric invariants of $g$. This leads to a distinguished orthogonal basis of the Cartan subalgebra, w…
Keller admissible triples and Duflo theorem
本文研究Keller容许三元组，证明其可诱导一对关联微分分次代数的直和型Hochschild上同调的Gerstenhaber代数同构，并给出有限维李代数Duflo–Kontsevich定理的新具体证明。
In 1979, M. Kashiwara and M. Vergne formulated a conjecture on a Lie group G which implies that the Duflo isomorphism of Z(g) and S(g)^g extends to a natural module isomorphism between the spaces of germs of invariant distributions on G and g=Lie(G), respectively. They also proved their conjecture for G solvable. Using Kontsevich's deformation quantization we establish directly this result for distributions on any real Lie group G. In turn this gives a new proof of Duflo's result on the local solvability of bi-invariant differential operators on G.
Let $G$ be a connected Lie group, with Lie algebra $g$. In 1977, Duflo constructed a homomorphism of $g$-modules $Duf: S(g) -> U(g)$, which restricts to an algebra isomorphism on invariants. Kashiwara and Vergne (1978) proposed a conjecture on the Campbell-Hausdorff series, which (among other things) extends the Duflo theorem to germs of bi-invariant distributions on the Lie group $G$. The main results of the present paper are as follows. (1) Using a recent result of Torossian (2002), we establish the Kashiwara-Vergne conjecture for any Lie group $G$. (2) We give a reformulation of the Kashiwa…
The notion of trigonometric spin double affine Hecke algebras (tsDaHa) and trigonometric double affine Hecke-Clifford algebras (tDaHCa) associated to classical Weyl groups are introduced. The PBW basis property is established. An algebra isomorphism relating tDaHCa to tsDaHa is obtained.
This paper is a survey of work done on $\mathbb{N}$-graded Clifford algebras (GCAs) and $\mathbb{N}$-graded \textit{skew} Clifford algebras (GSCAs) \cite{VVW, SV, CaV, NVZ, VVe1, VVe2}. In particular, we discuss the hypotheses necessary for these algebras to be Artin Schelter-regular \cite{AS, ATV1} and show how certain `points' called, point modules, can be associated to them. We may view an AS-regular algebra as a noncommutative analog of the polynomial ring. We begin our survey with a fundamental result in \cite{VVW} that is essential to subsequent results discussed here: the connection bet…
The notion of rational spin double affine Hecke algebras (sDaHa) and rational double affine Hecke-Clifford algebras (DaHCa) associated to classical Weyl groups are introduced. The basic properties of these algebras such as the PBW basis and Dunkl operator representations are established. An algebra isomorphism relating the rational DaHCa to the rational sDaHa is obtained. We further develop a link between the usual rational Cherednik algebra and the rational sDaHa by introducing a notion of rational covering double affine Hecke algebras.
Clifford algebra analogue of Cartan's theorem for symmetric pairs
该研究将Kostant关于复半单李代数克利福德代数不变量的结果，推广到经典对称对对应克利福德代数的相对情形，证明了克利福德代数的嘉当定理等相关结果。
Let g be a complex simple Lie algebra, and h be a Cartan subalgebra. In the end of 1990s, B. Kostant defined two filtrations on h, one using the Clifford algebras and the odd analogue of the Harish-Chandra projection $hc: Cl(g) \to Cl(h)$, and the other one using the canonical isomorphism $\check{h} = h^*$ (here $\check{h}$ is the Cartan subalgebra in the simple Lie algebra corresponding to the dual root system) and the adjoint action of the principal sl2-triple. Kostant conjectured that the two filtrations coincide. The two filtrations arise in very different contexts, and comparing them prov…
Clifford algebras are naturally associated with quadratic forms. These algebras are Z_2-graded by construction. However, only a Z_n-gradation induced by a choice of a basis, or even better, by a Chevalley vector space isomorphism Cl(V) <-> \bigwedge V and an ordering, guarantees a multi-vector decomposition into scalars, vectors, tensors, and so on, mandatory in physics. We show that the Chevalley isomorphism theorem cannot be generalized to algebras if the Z_n-grading or other structures are added, e.g., a linear form. We work with pairs consisting of a Clifford algebra and a linear form or a…
Work of Clifford, Munn and Ponizovski{\uı} parameterized the irreducible representations of a finite semigroup in terms of the irreducible representations of its maximal subgroups. Explicit constructions of the irreducible representations were later obtained independently by Rhodes and Zalcstein and by Lallement and Petrich. All of these approaches make use of Rees's theorem characterizing 0-simple semigroups up to isomorphism. Here we provide a short modern proof of the Clifford-Munn-Ponizovski{\uı} result based on a lemma of J. A. Green, which allows us to circumvent the theory of 0-simple s…
Let $(V,g)$ be a $2n$-dimensional hyperbolic space and $C(V,g)$ its Clifford algebra. $C(V,g)$ has a $\mathbb Z$-grading, $C^k $, and an algebra isomorphism $C(V,g)\cong End(S)$, $S$ the space of spinors. É. Cartan defined operators $L_k: End(S) \to C^k$ which are involved in the definition of pure spinors. We shall give a more refined study of the operator $L_2$, in fact, obtain explicit formulae for it in terms of spinor inner products and combinatorics, as well as the matrix of it in a basis of pure spinors. Using this information we give a construction of the exceptional Lie algebras $\mat…
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