Schur-Weyl duality and the free Lie algebra (original) (raw)
Related papers
2002
We introduce and study a special class of infinite-dimensional Lie algebras that are finite-dimensional modules over a ring of polynomials. The Lie algebras of this class are said to be polynomial. Some classification results are obtained. An associative co-algebra structure on the rings k[x 1 , . . . , x n ]/(f 1 , . . . , f n ) is introduced and, on its basis, an explicit expression for convolution matrices of invariants for isolated singularities of functions is found. The structure polynomials of moving frames defined by convolution matrices are constructed for simple singularities of the types A, B , C , D, and E 6 .
A New Proof of the Existence of Free Lie Algebras and an Application
The existence of free Lie algebras is usually derived as a consequence of the Poincaré-Birkhoff-Witt theorem. Moreover, in order to prove that given a set X and a field à of characteristic zero the Lie algebra L à X of the Lie polynomials in the letters of X over the field à is a free Lie algebra generated by X, all available proofs use the embedding of a Lie algebra into its enveloping algebra U. The aim of this paper is to give a much simpler proof of the latter fact without the aid of the cited embedding nor of the Poincaré-Birkhoff-Witt theorem. As an application of our result and of a theorem due to Cartier 1956 , we show the relationships existing between the theorem of Poincaré-Birkhoff-Witt, the theorem of Campbell-Baker-Hausdorff, and the existence of free Lie algebras.
The present paper introduces a method of basis transformation of a vector space that is specifically applicable to polynomials space and differential equations with certain polynomials solutions such as Hermite, Laguerre and Legendre polynomials. The method based on separated transformations of vector space basis by a set of operators that are equivalent to the formal basis transformation and connected to it by linear combination with projection operators. Applying the Forbenius covariants yields a general method that incorporates the Rodrigues formula as a special case in polynomial space. Using the Lie algebra modules, specifically sl(2,R) , on polynomial space results in isomorphic algebras whose Cartan sub-algebras are Hermite, Laguerre and Legendre differential operators. Commutation relations of these algebras and Baker-Campbell-Hausdorff formula gives new formulas for special polynomials.
Some properties of the space ofn-dimensional Lie algebras
Sbornik: Mathematics, 2009
Some general properties of the space Ln of n-dimensional Lie algebras are studied. This space is defined by the system of Jacobi's quadratic equations. It is proved that these equations are linearly independent and equivalent to each other (more precisely, the quadratic forms defining these equations are affinely equivalent). Moreover, the problem on the closures of some orbits of the natural action of the group GLn on Ln is considered. Two Lie algebras are indicated whose orbits are closed in the projectivization of the space Ln. The intersection of all irreducible components of the space Ln is also treated. It is proved that this intersection is nontrivial and consists of nilpotent Lie algebras. Two Lie algebras belonging to this intersection are indicated. Some other results concerning arbitrary Lie algebras and the space Ln formed by these algebras are presented. Bibliography: 17 titles.
Integral Schur-Weyl duality for partition algebras
2019
Let VVV be a free module of rank nnn over a (not necessarily commutative) unital ring kkk. We prove that tensor space VotimesrV^{\otimes r}Votimesr satisfies Schur-Weyl duality, regarded as a bimodule for the action of the group algebra of the Weyl group of rmGL(V){\rm GL}(V)rmGL(V) and the partition algebra Pr(n)P_r(n)Pr(n) over kkk. We also prove a similar result for the half partition algebra.
2023
The present paper introduces a method of basis transformation of vector fields that is specifically applicable to polynomials space and differential equations with certain polynomials solutions such as Hermite, Laguerre and Legendre polynomials. The method based on separated transformation of vector space basis by a set of operators that are equivalent to the formal basis transformation and connected to it by linear combination with projection operators. Applying the Forbenius covariants yields a general method that incorporates the Rodrigues formula as a special case in polynomial space. Using the Lie algebra modules, specifically (2,) , on polynomial space results in isomorphic algebras whose Cartan sub-algebras are Hermite, Laguerre and Legendre differential operators. Commutation relations of these algebras and Baker-Campbell-Hausdorff formula gives new formulas for special polynomials.
The configuration basis of a Lie algebra and its dual
2010
We use the Lie coalgebra and configuration pairing framework presented previously in [11] to derive a new, left-normed monomial basis for free Lie algebras (built from associative Lyndon-Shirshov words), as well as a dual monomial basis for Lie coalgebras. Our focus is on computational dexterity gained by using the configuration framework and basis. We include several explicit examples using the dual coalgebra basis and configuration pairing to perform Lie algebra computations. As a corollary of our work, we get a new multiplicative basis for the shuffle algebra.
Equations in simple Lie algebras
Journal of Algebra, 2012
Given an element P (X 1 ,. .. , X d) of the finitely generated free Lie algebra L d , for any Lie algebra g we can consider the induced polynomial map P : g d → g. Assuming that K is an arbitrary field of characteristic = 2, we prove that if P is not an identity in sl(2, K), then this map is dominant for any Chevalley algebra g. This result can be viewed as a weak infinitesimal counterpart of Borel's theorem on the dominancy of the word map on connected semisimple algebraic groups. We prove that for the Engel monomials [[[X, Y ], Y ],. .. , Y ] and, more generally, for their linear combinations, this map is, moreover, surjective onto the set of noncentral elements of g provided that the ground field K is big enough, and show that for monomials of large degree the image of this map contains no nonzero central elements. We also discuss consequences of these results for polynomial maps of associative matrix algebras.