Action Of Lie Algebra Research Articles

Atiyah sequences of braided Lie algebras and their splittings

Given an equivariant noncommutative principal bundle, we construct an Atiyah sequence of braided derivations whose splittings give connections on the bundle. Vertical braided derivations act as infinitesimal gauge transformations on connections. In the case of the principal \mathrm{SU}(2) -bundle over the sphere S^{4}_{\theta} an equivariant splitting of the Atiyah sequence recovers the instanton connection. An infinitesimal action of the braided conformal Lie algebra \mathrm{so}_{\theta}(5,1) yields a five parameter family of splittings. On the principal \mathrm{SO}_\theta(2n,\mathbb{R}) -bundle of orthonormal frames over the sphere S^{2n}_\theta , a splitting of the sequence leads to the Levi-Civita connection for the ‘round’ metric on S^{2n}_\theta . The corresponding Riemannian geometry of S^{2n}_{\theta} is worked out.

Reduction of $L_\infty$-Algebras of Observables on Multisymplectic Manifolds

We develop a reduction scheme for the $L_\infty$-algebra of observables on a premultisymplectic manifold $(M,\omega)$ in the presence of a compatible Lie algebra action $\mathfrak{g}\curvearrowright M$ and subset $N\subset M$. This reproduces in the symplectic setting the Poisson algebra of observables on the Marsden-Weinstein-Meyer symplectic reduced space, whenever the reduced space exists, but is otherwise distinct from the Dirac, Śniatycki-Weinstein, and Arms-Cushman-Gotay observable reduction schemes. We examine various examples, including multicotangent bundles and multiphase spaces, and we conclude with a discussion of applications to classical field theories and quantization.

Hamiltonian Lie algebroids

In previous work with M.C. Fernandes, we found a Lie algebroid symmetry for the Einstein evolution equations. The present work was motivated by the effort to combine this symmetry with the hamiltonian structure of the equations to explain the coisotropic structure of the constraint subset for the initial value problem. In this paper, we extend the notion of hamiltonian structure from Lie algebra actions to general Lie algebroids over presymplectic manifolds. Application of this construction to the problem in general relativity is still work in progress. After comparing a number of possible compatibility conditions between an anchor map A → T M A\to TM on a vector bundle A A and a presymplectic structure on the base M M , we choose the most natural of them, best formulated in terms of a suitably chosen connection on A A . We define a notion of momentum section of A ∗ A^* , and, when A A is a Lie algebroid, we specify a condition for compatibility with the Lie algebroid bracket. Compatibility conditions on an anchor, a Lie algebroid bracket, a momentum section, a connection, and a presymplectic structure are then the defining properties of a hamiltonian Lie algebroid. For an action Lie algebroid with the trivial connection, the conditions reduce to those for a hamiltonian action. We show that the clean zero locus of the momentum section of a hamiltonian Lie algebroid is a coisotropic submanifold. To define morphisms of hamiltonian Lie algebroids, we express the structure in terms of a bigraded algebra generated by Lie algebroid forms and de Rham forms on its base. We give an Atiyah-Bott type characterization of a bracket-compatible momentum map; it is equivalent to a closed basic extension of the presymplectic form, within the generalization of the BRST model of equivariant cohomology to Lie algebroids. We show how to construct a groupoid by reduction of an action Lie groupoid G × M G\times M by a subgroup H H of G G which is not necessarily normal, and we find conditions which imply that a hamiltonian structure descends to such a reduced Lie algebroid. We consider many examples and, in particular, find that the tangent Lie algebroid over a symplectic manifold is hamiltonian with respect to some connection if and only if the symplectic structure has a nowhere vanishing primitive. Recent results of Stratmann and Tang show that this is the case whenever the symplectic structure is exact.

Lie algebra actions on module categories for truncated shifted yangians

Abstract We develop a theory of parabolic induction and restriction functors relating modules over Coulomb branch algebras, in the sense of Braverman-Finkelberg-Nakajima. Our functors generalize Bezrukavnikov-Etingof’s induction and restriction functors for Cherednik algebras, but their definition uses different tools. After this general definition, we focus on quiver gauge theories attached to a quiver $\Gamma $ . The induction and restriction functors allow us to define a categorical action of the corresponding symmetric Kac-Moody algebra $\mathfrak {g}_{\Gamma }$ on category $ \mathcal {O}$ for these Coulomb branch algebras. When $ \Gamma $ is of Dynkin type, the Coulomb branch algebras are truncated shifted Yangians and quantize generalized affine Grassmannian slices. Thus, we regard our action as a categorification of the geometric Satake correspondence. To establish this categorical action, we define a new class of ‘flavoured’ KLRW algebras, which are similar to the diagrammatic algebras originally constructed by the second author for the purpose of tensor product categorification. We prove an equivalence between the category of Gelfand-Tsetlin modules over a Coulomb branch algebra and the modules over a flavoured KLRW algebra. This equivalence relates the categorical action by induction and restriction functors to the usual categorical action on modules over a KLRW algebra.

Symmetry Reduction of States I

We develop a general theory of symmetry reduction of states on (possibly non-commutative) *-algebras that are equipped with a Poisson bracket and a Hamiltonian action of a commutative Lie algebra g . The key idea advocated for in this article is that the "correct" notion of positivity on a *-algebra A is not necessarily the algebraic one whose positive elements are the sums of Hermitian squares a^* a with a \in A , but can be a more general one that depends on the example at hand, like pointwise positivity on *-algebras of functions or positivity in a representation as operators. The notion of states (normalized positive Hermitian linear functionals) on A thus depends on this choice of positivity on A , and the notion of positivity on the reduced algebra A_{red} should be such that states on A_{red} are obtained as reductions of certain states on A . We discuss three examples in detail: Reduction of the *-algebra of smooth functions on a Poisson manifold M , which reproduces the coisotropic reduction of M ; reduction of the Weyl algebra with respect to translation symmetry; and reduction of the polynomial algebra with respect to a U(1)-action.

Manin–Drinfeld cycles and derivatives of $L$-functions

We study algebraic cycles in the moduli space of \operatorname{PGL}_2 -shtukas, arising from the diagonal torus. Our main result shows that their intersection pairing with the Heegner–Drinfeld cycle is the product of the r -th central derivative of an automorphic L -function L(\pi,s) and Waldspurger's toric period integral. When L(\pi,1/2) \neq 0 , this gives a new geometric interpretation for the Taylor series expansion. When L(\pi,1/2) = 0 , the pairing vanishes, suggesting higher order analogues of the vanishing of cusps in the modular Jacobian, as well as other new phenomena. Our proof sheds new light on the algebraic correspondence introduced by Yun and Zhang, which is the geometric incarnation of “differentiating the L -function”. We realize it as the Lie algebra action of e+f \in \mathfrak{sl}_2 on (\mathbb{Q}_\ell^2)^{\otimes 2d} . The comparison of relative trace formulas needed to prove our formula is then a consequence of Schur–Weyl duality.

On symmetries of singular foliations

This paper shows that a weak symmetry action of a Lie algebra g on a singular foliation F induces a unique up to homotopy Lie ∞-morphism from g to the DGLA of vector fields on a universal Lie ∞-algebroid of F. Such a morphism is known as L∞-algebra action in [1]. We deduce from this general result several geometrical consequences. For instance, we give an example of a Lie algebra action on an affine sub-variety which cannot be extended on the ambient space. Last, we introduce the notion of bi-submersion towers over a singular foliation and lift symmetries to those.

3D geometric moment invariants from the point of view of the classical invariant theory

The aim of this paper is to clear up the problem of the connection between the 3D geometric moments invariants and the invariant theory, considering a problem of describing of the 3D geometric moments invariants as a problem of the classical invariant theory.Using the remarkable fact that the complex groups $SO(3,\mathbb{C})$ and $SL(2,\mathbb{C})$ are locally isomorphic, we reduced the problem of deriving 3D geometric moments invariants to the well-known problem of the classical invariant theory.
 We give a precise statement of the 3D geometric invariant moments computation, intro\-ducing the notions of the algebras of simultaneous 3D geometric moment invariants, and prove that they are isomorphic to the algebras of joint $SL(2,\mathbb{C})$-invariants of several binary forms. To simplify the calculating of the invariants we proceed from an action of Lie group $SO(3,\mathbb{C})$ to equivalent action of the complex Lie algebra $\mathfrak{sl}_2$. The author hopes that the results will be useful to the researchers in thefields of image analysis and pattern recognition.

Weierstrass sections for parabolic adjoint action in type A

The notion of “Weierstrass Section”, comes from Weierstrass canonical form for elliptic curves. In celebrated work Kostant (1963) [26] constructed such a section for the action of a semisimple Lie algebra on its dual using a principal s-triple. Actually it is enough to have an “adapted pair” and indeed the construction in Joseph and Shafrir (2010) [25] works rather well for the co-adjoint action of an algebraic, but not necessarily reductive Lie algebra.In the present work a Weierstrass section is constructed for the adjoint action of a parabolic subgroup P on the nilradical m of its Lie algebra in type A. The starting point is Richardson's theorem which asserts that m admits a dense P orbit (in all types).For type A an explicit generator of this dense orbit was given in Brüstle et al. (1999) [4]. It involves the joining boxes in the Young tableau associated to P.On the other hand Richardson's theorem implies the polynomiality of the (semi)-invariant subalgebra, where the generators themselves are given through ideals of definition of hypersurface orbital varieties Joseph and Melnikov (2003) [24]. (Again this relationship holds in all types.)In this adapted pairs seldom exist.Thus an entirely new construction is developed and this is mainly combinatorial.The initial step is the construction of Ringel et al. ([24]). In this the joining of boxes is significantly modified to take account of the explicit form of the generators of the invariant algebra suggested in Benlolo and Sanderson (2001) [3].Indications are given as to how one might extend this construction of a Weierstrass section to other types.

Cohomologies and crossed modules for pre-Lie Rinehart algebras

A pre-Lie-Rinehart algebra is an algebraic generalization of the notion of a left-symmetric algebroid. We construct pre-Lie-Rinehart algebras from r-matrices through Lie algebra actions. We study cohomologies of pre-Lie-Rinehart algebras and show that abelian extensions of pre-Lie-Rinehart algebras are classified by the second cohomology groups. We introduce the notion of crossed modules for pre-Lie-Rinehart algebras and show that they are classified by the third cohomology groups of pre-Lie-Rinehart algebras. At last, we use (pre-)Lie-Rinehart 2-algebras to characterize the crossed modules for (pre-)Lie Rinehart algebras.

Momentum section on Courant algebroid and constrained Hamiltonian mechanics

We propose a generalization of the momentum map on a symplectic manifold with a Lie algebra action to a Courant algebroid structure. The theory of a momentum section on a Lie algebroid is generalized to the theory compatible with a Courant algebroid. As an example, we identify the momentum section in a constrained Hamiltonian mechanics with Courant algebroid symmetry. Moreover, we construct cohomological formulations by considering the BFV and BV formalism of this Hamiltonian system. The Weil algebra for this structure is constructed.

Algebras with representable representations

AbstractJust like group actions are represented by group automorphisms, Lie algebra actions are represented by derivations: up to isomorphism, a split extension of a Lie algebra $B$ by a Lie algebra $X$ corresponds to a Lie algebra morphism $B\to {\mathit {Der}}(X)$ from $B$ to the Lie algebra ${\mathit {Der}}(X)$ of derivations on $X$. In this article, we study the question whether the concept of a derivation can be extended to other types of non-associative algebras over a field ${\mathbb {K}}$, in such a way that these generalized derivations characterize the ${\mathbb {K}}$-algebra actions. We prove that the answer is no, as soon as the field ${\mathbb {K}}$ is infinite. In fact, we prove a stronger result: already the representability of all abelian actions – which are usually called representations or Beck modules – suffices for this to be true. Thus, we characterize the variety of Lie algebras over an infinite field of characteristic different from $2$ as the only variety of non-associative algebras which is a non-abelian category with representable representations. This emphasizes the unique role played by the Lie algebra of linear endomorphisms $\mathfrak {gl}(V)$ as a representing object for the representations on a vector space $V$.

Group properties and solutions for the 1D Hall MHD system in the cold plasma approximation

We study the Lie point symmetries and the similarity transformations for the partial differential equations of the nonlinear one-dimensional magnetohydrodynamic system with the Hall term known as HMHD system. For this 1+1 system of partial differential equations we find that is invariant under the action of a seventh dimensional Lie algebra. Furthermore, the one-dimensional optimal system is derived while the Lie invariants are applied for the derivation of similarity transformations. We present different kinds of oscillating solutions.

On quantum adjacency algebras of Doob graphs and their irreducible modules

For fixed integers $$n \ge 1$$ and $$m \ge 0$$ , we consider the Doob graph $$D=D(n,m)$$ formed by taking direct product of n copies of Shrikhande graph and m copies of complete graph $$K_4$$ . Fix a vertex x of D, and let $$T=T(x)$$ denote the Terwilliger algebra of D with respect to x. Let A denote the adjacency matrix of D. There exists a decomposition of A into a sum $$A = L + F + R$$ of elements in T where L, F, and R are the lowering, flat, and raising matrices, respectively. We call $$A = L + F + R$$ the quantum decomposition of A. Hora and Obata (Quantum Probability and Spectral Analysis of Graphs. Theoretical and Mathematical Physics, Springer, Berlin, 2007) introduced a semi-simple matrix algebra based on the quantum decomposition of the adjacency matrix. This algebra is generated by the quantum components of the decomposition and is called the quantum adjacency algebra of the graph. Let $$Q=Q(x)$$ denote the quantum adjacency algebra of D with respect to x. In this paper, we display an action of the special orthogonal Lie algebra $$\mathfrak {so}_4$$ on the standard module for D. We also prove Q is generated by the center and the homomorphic image of the universal enveloping algebra $$U(\mathfrak {so}_4)$$ . To do these, we exploit the work of Tanabe (JAC 6: 173–195, 1997) on irreducible T-modules of D.

A Lie algebra action on the Chow ring of the Hilbert scheme of points of a K3 surface

We construct an action of the Neron–Severi part of the Looijenga–Lunts–Verbitsky Lie algebra on the Chow ring of the Hilbert scheme of points on a K3 surface. This yields a simplification of Maulik and Negut’s proof that the cycle class map is injective on the subring generated by divisor classes as conjectured by Beauville. The key step in the construction is an explicit formula for Lefschetz duals in terms of Nakajima operators. Our results also lead to a formula for the monodromy action on Hilbert schemes in terms of Nakajima operators.

The Global Sections of Chiral de Rham Complexes on Compact Ricci-Flat Kähler Manifolds

The space of the global sections of the chiral de Rham complex on a compact Ricci-flat Kähler manifold is calculated and it is expressed as the subspace of invariant elements in a \(\beta \gamma -bc\) system under the action of certain Lie algebra of Cartan type.

Differential Polynomial Identities of Upper Triangular Matrices Under the Action of the Two-Dimensional Metabelian Lie Algebra

We study the differential polynomial identities of the algebra UTm(F) under the derivation action of the two dimensional metabelian Lie algebra, obtaining a generating set of the TL-ideal they constitute. Then we determine the Sn-structure of their proper multilinear spaces and, for the minimal cases m = 2, 3, their exact differential codimension sequence.

Quaternionic analysis, representation theory and physics II

We develop further quaternionic analysis introducing left and right doubly regular functions. We derive Cauchy-Fueter type formulas for these doubly regular functions that can be regarded as another counterpart of Cauchy's integral formula for the second order pole, in addition to the one studied in the first paper with the same title. We also realize the doubly regular functions as a subspace of the quaternionic-valued functions satisfying a Euclidean version of Maxwell's equations for the electromagnetic field. Then we return to the study of the original quaternionic analogue of Cauchy's second order pole formula and its relation to the polarization of vacuum. We find the decomposition of the space of quaternionic-valued functions into irreducible components that include the spaces of doubly left and right regular functions. Using this decomposition, we show that a regularization of the vacuum polarization diagram is achieved by subtracting the component corresponding to the one-dimensional subrepresentation of the conformal group. After the regularization, the vacuum polarization diagram is identified with a certain second order differential operator which yields a quaternionic version of Maxwell equations. Next, we introduce two types of quaternionic algebras consisting of spaces of scalar-valued and quaternionic-valued functions. We emphasize that these algebra structures are invariant under the action of the conformal Lie algebra. This uses techniques from our study of the vacuum polarization diagram. These algebras are not associative, but we can define an infinite family of n-multiplications, and we conjecture that they have structures of weak cyclic A-infinity algebras. We also conjecture the relation between the multiplication operations of the scalar and non-scalar quaternionic algebras with the n-photon Feynman diagrams in the scalar and ordinary conformal QED.

A Thom isomorphism in foliated de Rham theory

We prove a Thom isomorphism theorem for differential forms in the setting of transverse Lie algebra actions on foliated manifolds and foliated vector bundles.

Cohomological localization for transverse Lie algebra actions on Riemannian foliations

We prove localization and integration formulas for the equivariant basic cohomology of Riemannian foliations. As a corollary we obtain a Duistermaat–Heckman theorem for transversely symplectic foliations.