Action Of Lie Algebra Research Articles

Asymptotics of H-identities for associative algebras with an H-invariant radical

We prove the existence of the Hopf PI-exponent for finite dimensional associative algebras A with a generalized Hopf action of an associative algebra H with 1 over an algebraically closed field of characteristic 0 assuming only the invariance of the Jacobson radical J(A) under the H-action and the existence of the decomposition of A/J(A) into the sum of H-simple algebras. As a consequence, we show that the analog of Amitsurʼs conjecture holds for G-codimensions of finite dimensional associative algebras over a field of characteristic 0 with an action of an arbitrary group G by automorphisms and anti-automorphisms and for differential codimensions of finite dimensional associative algebras with an action of an arbitrary Lie algebra by derivations.

Higher Lie algebra actions on Lie algebroids

We consider a simple instance of action up to homotopy. More precisely, we consider strict actions of DGLAs in degrees −1 and 0 on degree 1 NQ-manifolds. In a more conventional language this means: strict actions of Lie algebra crossed modules on Lie algebroids.When the action is strict, we show that it integrates to group actions in the categories of Lie algebroids and Lie groupoids (i.e. actions of LA-groups and 2-groups). We perform the integration in the framework of Mackenzie’s doubles.

Formula omitted]-algebra actions

We define the notion of action of an L∞-algebra g on a graded manifold M, and show that such an action corresponds to a homological vector field on g[1]×M of a specific form. This generalizes the correspondence between Lie algebra actions on manifolds and transformation Lie algebroids. In particular, we consider actions of g on a second L∞-algebra, leading to a notion of “semidirect product” of L∞-algebras more general than those we found in the literature.

Local structure of 2 dimensional solvable lie algebra actions on the plane

In this paper the local classification of 2-dimensional solvable Lie algebra actions on the plane is given. Normal forms of such actions are found.

Degenerate two-boundary centralizer algebras

Diagram algebras (e.g. graded braid groups, Hecke algebras, Brauer algebras) arise as tensor power centralizer algebras, algebras of commuting operators for a Lie algebra action on a tensor space. This work explores centralizers of the action of a complex reductive Lie algebra $\mathfrak{g}$ on tensor space of the form $M \otimes N \otimes V^{\otimes k}$. We define the degenerate two-boundary braid algebra $\mathcal{G}_k$ and show that centralizer algebras contain quotients of this algebra in a general setting. As an example, we study in detail the combinatorics of special cases corresponding to Lie algebras $\mathfrak{gl}_n$ and $\mathfrak{sl}_n$ and modules $M$ and $N$ indexed by rectangular partitions. For this setting, we define the degenerate extended two-boundary Hecke algebra $\mathcal{H}_k^{\mathrm{ext}}$ as a quotient of $\mathcal{G}_k$, and show that a quotient of $\mathcal{H}_k^{\mathrm{ext}}$ is isomorphic to a large subalgebra of the centralizer. We further study the representation theory of $\mathcal{H}_k^{\mathrm{ext}}$ to find that the seminormal representations are indexed by a known family of partitions. The bases for the resulting modules are given by paths in a lattice of partitions, and the action of $\mathcal{H}_k^{\mathrm{ext}}$ is given by combinatorial formulas.

Polynomial Representations and Categorifications of Fock Space

The rings of symmetric polynomials form an inverse system whose limit, the ring of symmetric functions, is the model for the bosonic Fock space representation of the affine Lie algebra. We show that this limit naturally carries an action of the affine Lie algebra (in the sense of Rouquier), thereby obtaining a family of categorifications of the bosonic Fock space representation.

On deformations of quasi-Miura transformations and the Dubrovin–Zhang bracket

In our recent paper, we proved the polynomiality of a Poisson bracket for a class of infinite-dimensional Hamiltonian systems of partial differential equations (PDEs) associated to semi-simple Frobenius structures. In the conformal (homogeneous) case, these systems are exactly the hierarchies of Dubrovin and Zhang, and the bracket is the first Poisson structure of their hierarchy.Our approach was based on a very involved computation of a deformation formula for the bracket with respect to the Givental–Lee Lie algebra action. In this paper, we discuss the structure of that deformation formula. In particular, we give an alternative derivation using a deformation formula for the weak quasi-Miura transformation that relates our hierarchy of PDEs with its dispersionless limit.

Braiding via geometric Lie algebra actions

AbstractWe introduce the idea of a geometric categorical Lie algebra action on derived categories of coherent sheaves. The main result is that such an action induces an action of the braid group associated to the Lie algebra. The same proof shows that strong categorical actions in the sense of Khovanov–Lauda and Rouquier also lead to braid group actions. As an example, we construct an action of Artin’s braid group on derived categories of coherent sheaves on cotangent bundles to partial flag varieties.

Heisenberg algebras and rational double affine Hecke algebras

We relate the filtration by the support on the Grothendieck group[O][\mathcal {O}]of the categoryO\mathcal {O}of cyclotomic rational double affine Hecke algebras to a representation-theoretic grading on[O][\mathcal {O}], defined using the action of an affine Lie algebra and of a Heisenberg algebra on the Fock space. This implies a recent conjecture of Etingof. The proof uses a categorification of the Heisenberg action, which is new, and a categorification of the affine Lie algebra action, which was introduced by the first author in an earlier paper.

Morita theory in deformation quantization

Various aspects of Morita theory of deformed algebras and in particular of star product algebras on general Poisson manifolds are discussed. We relate the three flavours ring-theoretic Morita equivalence, *-Morita equivalence, and strong Morita equivalence and exemplify their properties for star product algebras. The complete classification of Morita equivalent star products on general Poisson manifolds is discussed as well as the complete classification of covariantly Morita equivalent star products on a symplectic manifold with respect to some Lie algebra action preserving a connection.

Smooth actions of Lie groups and Lie algebras on manifolds

Necessary or sufficient conditions are presented for the existence of various types of actions of Lie groups and Lie algebras on manifolds.

Courant Algebroids and Poisson Geometry

Given a manifold M with an action of a quadratic Lie algebra d, such that all stabilizer algebras are co-isotropic in d, we show that the product M\times d becomes a Courant algebroid over M. If the bilinear form on d is split, the choice of transverse Lagrangian subspaces g_1, g_2 of d defines a bivector field on M, which is Poisson if (d,g_1,g_2) is a Manin triple. In this way, we recover the Poisson structures of Lu-Yakimov, and in particular the Evens-Lu Poisson structures on the variety of Lagrangian Grassmannians and on the de Concini-Procesi compactifications. Various Poisson maps between such examples are interpreted in terms of the behaviour of Lagrangian splittings under Courant morphisms.

Formula omitted]-invariant ideals in Iwasawa algebras

Let k G be the completed group algebra of a uniform pro- p group G with coefficients in a field k of characteristic p . We study right ideals I in k G that are invariant under the action of another uniform pro- p group Γ . We prove that if I is non-zero then an irreducible component of the characteristic support of k G / I must be contained in a certain finite union of rational linear subspaces of Spec gr k G . The minimal codimension of these subspaces gives a lower bound on the homological height of I in terms of the action of a certain Lie algebra on G / G p . If we take Γ to be G acting on itself by conjugation, then Γ -invariant right ideals of k G are precisely the two-sided ideals of k G , and we obtain a non-trivial lower bound on the homological height of a possible non-zero two-sided ideal. For example, when G is open in SL n ( Z p ) this lower bound equals 2 n − 2 . This gives a significant improvement of the results of [K. Ardakov, F. Wei, J.J. Zhang, Reflexive ideals in Iwasawa algebras, Adv. Math. 218 (2008) 865–901].

A superalgebraic interpretation of the quantization maps of Weil algebras

Let G be a Lie group whose Lie algebra $$ \mathfrak{g} $$ is quadratic. In the paper “the non-commutative Weil algebra”, Alekseev and Meinrenken constructed an explicit G-differential space homomorphism , called the quantization map, between the Weil algebra $$ W_\mathfrak{g} = S(\mathfrak{g}*) \otimes \wedge (\mathfrak{g}*) $$ and (which they call the noncommutative Weil algebra) for $$ \mathfrak{g} $$ . They showed that induces an algebra isomorphism between the basic cohomology rings H bas * ( $$ W_\mathfrak{g} $$ ) and H bas * ( ). In this paper, we will interpret the quantization map as the super Duflo map between the symmetric algebra $$ S\left( {\widetilde{T\mathfrak{g}\left[ 1 \right]}} \right) $$ and the universal enveloping algebra $$ U\left( {\widetilde{T\mathfrak{g}\left[ 1 \right]}} \right) $$ of a super Lie algebra $$ \widetilde{T\mathfrak{g}\left[ 1 \right]} $$ which is canonically associated with the quadratic Lie algebra $$ \mathfrak{g} $$ . The basic cohomology rings H bas * ( $$ W_\mathfrak{g} $$ ) and H bas * ( ) correspond exactly to $$ S\left( {\widetilde{T\mathfrak{g}\left[ 1 \right]}} \right)^{inv} $$ and $$ U\left( {\widetilde{T\mathfrak{g}\left[ 1 \right]}} \right)^{inv} $$ , respectively. So what they proved is equivalent to the fact that the super Duflo map commutes with the adjoint action of the super Lie algebra, and that the super Duflo map is an algebra homomorphism when restricted to the space of invariants.

Continuous and Discrete Clebsch Variational Principles

The Clebsch method provides a unifying approach for deriving variational principles for continuous and discrete dynamical systems where elements of a vector space are used to control dynamics on the cotangent bundle of a Lie group \emph{via} a velocity map. This paper proves a reduction theorem which states that the canonical variables on the Lie group can be eliminated, if and only if the velocity map is a Lie algebra action, thereby producing the Euler-Poincar\'e (EP) equation for the vector space variables. In this case, the map from the canonical variables on the Lie group to the vector space is the standard momentum map defined using the diamond operator. We apply the Clebsch method in examples of the rotating rigid body and the incompressible Euler equations. Along the way, we explain how singular solutions of the EP equation for the diffeomorphism group (EPDiff) arise as momentum maps in the Clebsch approach. In the case of finite dimensional Lie groups, the Clebsch variational principle is discretised to produce a variational integrator for the dynamical system. We obtain a discrete map from which the variables on the cotangent bundle of a Lie group may be eliminated to produce a discrete EP equation for elements of the vector space. We give an integrator for the rotating rigid body as an example. We also briefly discuss how to discretise infinite-dimensional Clebsch systems, so as to produce conservative numerical methods for fluid dynamics.

GEOMETRIC QUANTIZATION OF HAMILTONIAN ACTIONS OF LIE ALGEBROIDS AND LIE GROUPOIDS

We construct Hermitian representations of Lie algebroids and associated unitary representations of Lie groupoids by a geometric quantization procedure. For this purpose, we introduce a new notion of Hamiltonian Lie algebroid actions. The first step of our procedure consists of the construction of a prequantization line bundle. Next, we discuss a version of Kähler quantization suitable for this setting. We proceed by defining a Marsden–Weinstein quotient for our setting and prove a "quantization commutes with reduction" theorem. We explain how our geometric quantization procedure relates to a possible orbit method for Lie groupoids. Our theory encompasses the geometric quantization of symplectic manifolds, Hamiltonian Lie algebra actions, actions of bundles of Lie groups, and foliations, as well as some general constructions from differential geometry.

THE COVARIANT PICARD GROUPOID IN DIFFERENTIAL GEOMETRY

In this paper we discuss some general results on the covariant Picard groupoid in the context of differential geometry and interpret the problem of lifting Lie algebra actions to line bundles in the Picard groupoid approach.

Laguerre functions on symmetric cones and recursion relations in the real case

In this article we derive differential recursion relations for the Laguerre functions on the cone Ω of positive definite real matrices. The highest weight representations of the group Sp ( n , R ) play a fundamental role. Each such representation acts on a Hilbert space of holomorphic functions on the tube domain Ω + i Sym ( n , R ) . We then use the Laplace transform to carry the Lie algebra action over to L 2 ( Ω , d μ ν ) . The differential recursion relations result by restricting to a distinguished three-dimensional subalgebra, which is isomorphic to sl ( 2 , R ) .

Symmetries of WDVV equations

We say that a function F(τ) obeys WDVV equations, if for a given invertible symmetric matrix ηαβ and all τ∈T⊂Rn, the expressions cβαγ(τ)=ηαλcλβγ(τ)=ηαλ∂λ∂β∂γF can be considered as structure constants of commutative associative algebra; the matrix ηαβ inverse to ηαβ determines an invariant scalar product on this algebra. A function xα(z,τ) obeying ∂α∂βxγ(z,τ)=z−1cαɛβ∂ɛxγ(z,τ) is called a calibration of a solution of WDVV equations. We show that there exists an infinite-dimensional group acting on the space of calibrated solutions of WDVV equations (in different form such a group was constructed in [A. Givental, math.AG/0305409]). We describe the action of Lie algebra of this group.

Completing Lie algebra actions to Lie group actions

For a finite-dimensional Lie algebra g \mathfrak {g} of vector fields on a manifold M M we show that M M can be completed to a G G -space in a universal way, which however is neither Hausdorff nor T 1 T_1 in general. Here G G is a connected Lie group with Lie-algebra g \mathfrak {g} . For a transitive g \mathfrak {g} -action the completion is of the form G / H G/H for a Lie subgroup H H which need not be closed. In general the completion can be constructed by completing each g \mathfrak {g} -orbit.