Twisted product and cohomology

LetH be a Hopf algebra,H ₁ be a sub-Hopf algebra ofH, H ₂ be the quotient Hopt algebra ofH modularH ₁. This paper gives a simplified complex by defining a new base for the cobar complex and proves that the cobar complex ofH has the same cohomology algebra with a twisted product of the cobar complexes ofH ₁ andH ₂. Supported by National Natural Science Foundation of China     

Poisson identities of enveloping algebras

Let L be a restricted Lie algebra. The symmetric algebra S_p(L) of the restricted enveloping algebra u(L) has the structure of a Poisson algebra. We give necessary and sufficient conditions on L in order for the symmetric algebra S_p(L) to satisfy a multilinear Poisson identity. We also settle the same problem for the symmetric algebra S(L) of a Lie algebra L over an arbitrary field. The first author was partially supported by MIUR of Italy. The second author was partially supported by Grant RFBR-04-01- 00739. Received: 31 October 2005     

Brauer algebras of simply laced type

The diagram algebra introduced by Brauer that describes the centralizer algebra of the n-fold tensor product of the natural representation of an orthogonal Lie group has a presentation by generators and relations that only depends on the path graph A _{n − 1} on n − 1 nodes. Here we describe an algebra depending on an arbitrary graph Q, called the Brauer algebra of type Q, and study its structure in the cases where Q is a Coxeter graph of simply laced spherical type (so its connected components are of type A _{n − 1}, D _n , E₆, E₇, E₈). We find its irreducible representations and its dimension, and show that the algebra is cellular. The algebra is generically semisimple and contains the group algebra of the Coxeter group of type Q as a subalgebra. It is a ring homomorphic image of the Birman-Murakami-Wenzl algebra of type Q; this fact will be used in later work determining the structure of the Birman-Murakami-Wenzl algebras of simply laced spherical type.     

Multiplicative Semiprimeness of Skew Lie Algebras

Abstract

Let A be a semiprime associative algebra with an involution over a field of characteristic not 2, let K be the Lie algebra of all skew elements of A, and let Z _{[K, K]} denote the annihilator of the Lie algebra [K, K]. We will prove that the multiplication algebra of the semiprime Lie algebra [K, K]/Z _{[K, K]} is also semiprime. As a consequence, the multiplication algebra of [K, K]/Z _{[K, K]} is prime, whenever [K, K]/Z _{[K, K]} is prime. We will obtain similar results for the Lie algebra K/Z _K whenever the base field has characteristic zero.     

The Birman–Murakami–Wenzl Algebras of Type D n

The Birman–Murakami–Wenzl algebra (BMW algebra) of type D _n is shown to be semisimple and free of rank (2 ⁿ  + 1)n!! ? (2 ^n?1 + 1)n! over a specified commutative ring R, where n!! =1·3…(2n ? 1). We also show it is a cellular algebra over suitable ring extensions of R. The Brauer algebra of type D _n is the image of an R-equivariant homomorphism and is also semisimple and free of the same rank, but over the ring ?[δ^±1]. A rewrite system for the Brauer algebra is used in bounding the rank of the BMW algebra above. As a consequence of our results, the generalized Temperley–Lieb algebra of type D _n is a subalgebra of the BMW algebra of the same type.     

The Hopf Algebra of Fliess Operators and Its Dual Pre-lie Algebra

We study the Hopf algebra H of Fliess operators coming from Control Theory in the one-dimensional case. We prove that it admits a graded, finite-dimensional, connected grading. Dually, the vector space ? ? x ₀, x ₁ ? is both a pre-Lie algebra for the pre-Lie product dual of the coproduct of H, and an associative, commutative algebra for the shuffle product. These two structures admit a compatibility which makes ? ? x ₀, x ₁ ? a Com-Pre-Lie algebra. We give a presentation of this object as a pre-Lie algebra.     

Class Partition Algebras as Centralizer Algebras of Wreath Products

In this article, we study an important subalgebra of the tensor product partition algebra P _k (x)? P _k (y), denoted by P _k (x, y) and called “Class Partition Algebra.” We show that the algebra P _k (n, m) is the centralizer algebra of the wreath product S _m ? S _n . Furthermore, the algebra P _k (x, y) and the tensor product partition algebra P _k (x)? P _k (y) are subalgebras of the G-colored partition algebra P _k (x;G) and G-vertex colored partition algebra P _k (x, G) respectively, for every group G with |G|=y ≥ 2k.     

D4型量子包络代数的Gelfand-Kirillov维数

本文研究了D₄ 型量子包络代数的Gelfand-Kirillov 维数的计算问题. 利用文献[1] 中给出的Gelfand-Kirillov 维数的计算方法和文献[2] 中给出的D₄ 型量子包络代数的Groebner-Shirshov 基计算了D₄型量子包络代数的Gelfand-Kirillov 维数, 得到的主要结果是D₄ 型量子包络代数的Gelfand-Kirillov 维数为28. 希望此结果为计算D_n型量子包络代数的Gelfand-Kirillov 维数提供一些思路.     

Application of a semi-direct sum of Lie algebras to the TD hierarchy

We construct a Lie algebra G by using a semi-direct sum of Lie algebra G₁ with Lie algebra G₂. A direct application to the TD hierarchy leads to a novel hierarchy of integrable couplings of the TD hierarchy. Furthermore, the generalized variational identity is applied to Lie algebra G to obtain quasi-Hamiltonian structures of the associated integrable couplings.     

The Yoneda Algebra of a 𝒦2 Algebra Need not be Another 𝒦2 Algebra

The Yoneda algebra of a Koszul algebra or a D-Koszul algebra is Koszul. 𝒦₂ algebras are a natural generalization of Koszul algebras, and one would hope that the Yoneda algebra of a 𝒦₂ algebra would be another 𝒦₂ algebra. We show that this is not necessarily the case by constructing a monomial 𝒦₂ algebra for which the corresponding Yoneda algebra is not 𝒦₂.     

Polynomial Identities of RA and RA2 Loop Algebras

Let F be an algebraically closed field of characteristic zero and L an RA loop. We prove that the loop algebra FL is in the variety generated by the split Cayley–Dickson algebra Z _F over F. For RA2 loops of type M(Dih(A), ^?1,g ₀), we prove that the loop algebra is in the variety generated by the algebra  ₃ which is a noncommutative simple component of the loop algebra of a certain RA2 loop of order 16. The same does not hold for the RA2 loops of type M(G, ^?1,g ₀), where G is a non-Abelian group of exponent 4 having exactly 2 squares.     

Gröbner–Shirshov bases for brace algebras

Let A be a brace algebra. This structure implies that A is also a pre-Lie algebra. In this paper, we establish Composition-Diamond lemma for brace algebras. For each pre-Lie algebra L, we find a Gröbner–Shirshov basis for its universal brace algebra U_b(L). As applications, we determine an explicit linear basis for U_b(L) and prove that L is a pre-Lie subalgebra of U_b(L).     

Construction of vertex operator algebras from commutative associative algebras

Given a commutative associative algebra A with an associative form (’), we construct a vertex operator algebra V with the weight two space V₂;? A If in addition the form (’) is nondegenerate, we show that there is a simple vertex operator algebra with V₂;? A We also show that if A is semisimple, then the vertex operator algebra constructed is the tensor products of a certain number of Virasoro vertex operator algebras.     

A new kind of graded Lie algebra and parastatistical supersymmetry

In this paper the usualZ ₂ graded Lie algebra is generalized to a new form, which may be calledZ _2,2 graded Lie algebra. It is shown that there exist close connections between theZ _2,2 graded Lie algebra and parastatistics, so theZ _2,2 can be used to study and analyse various symmetries and supersymmetries of the paraparticle systems     

The non-commutative Weil algebra

For any compact Lie group G, together with an invariant inner product on its Lie algebra ?, we define the non-commutative Weil algebra ? _G as a tensor product of the universal enveloping algebra U(?) and the Clifford algebra Cl(?). Just like the usual Weil algebra W _G =S(?^*)⊗∧?^*, ? _G carries the structure of an acyclic, locally free G-differential algebra and can be used to define equivariant cohomology ℋ _G (B) for any G-differential algebra B. We construct an explicit isomorphism ?: W _G →? _G of the two Weil algebras as G-differential spaces, and prove that their multiplication maps are G-chain homotopic. This implies that the map in cohomology H _G (B)→ℋ _G (B) induced by ? is a ring isomorphism. For the trivial G-differential algebra B=ℝ, this reduces to the Duflo isomorphism S(?) ^G ≅U(?) ^G between the ring of invariant polynomials and the ring of Casimir elements. Oblatum 13-III-1999 & 27-V-1999 / Published online: 22 September 1999     

Involutions for upper triangular matrix algebras

In this paper we describe completely the involutions of the first kind of the algebra UT_n(F) of n×n upper triangular matrices. Every such involution can be extended uniquely to an involution on the full matrix algebra. We describe the equivalence classes of involutions on the upper triangular matrices. There are two distinct classes for UT_n(F) when n is even and a single class in the odd case.Furthermore we consider the algebra UT₂(F) of the 2×2 upper triangular matrices over an infinite field F of characteristic different from 2. For every involution *, we describe the *-polynomial identities for this algebra. We exhibit bases of the corresponding ideals of identities with involution, and compute the Hilbert (or Poincaré) series and the codimension sequences of the respective relatively free algebras.Then we consider the *-polynomial identities for the algebra UT₃(F) over a field of characteristic zero. We describe a finite generating set of the ideal of *-identities for this algebra. These generators are quite a few, and their degrees are relatively large. It seems to us that the problem of describing the *-identities for the algebra UT_n(F) of the n×n upper triangular matrices may be much more complicated than in the case of ordinary polynomial identities.     

Presenting Schur Algebras as Quotients of the Universal Enveloping Algebra of gl2

We give a presentation of the Schur algebras S _Q (2,d) by generators and relations, in fact a presentation which is compatible with Serre's presentation of the universal enveloping algebra of a simple Lie algebra. In the process we find a new basis for S _Q (2,d), a truncated form of the usual PBW basis. We also locate the integral Schur algebra within the presented algebra as the analogue of Kostant's Z-form, and show that it has an integral basis which is a truncated version of Kostant's basis.     

Noncommutative algebras related with Schubert calculus on Coxeter groups

For any finite Coxeter system (W,S) we construct a certain noncommutative algebra, the so-called bracket algebra, together with a family of commuting elements, the so-called Dunkl elements. The Dunkl elements conjecturally generate an algebra which is canonically isomorphic to the coinvariant algebra of the Coxeter group W. We prove this conjecture for classical Coxeter groups and I₂(m). We define a “quantization” and a multiparameter deformation of our construction and show that for Lie groups of classical type and G₂, the algebra generated by Dunkl’s elements in the quantized bracket algebra is canonically isomorphic to the small quantum cohomology ring of the corresponding flag variety, as described by B. Kim. For crystallographic Coxeter systems we define the so-called quantum Bruhat representation of the corresponding bracket algebra. We study in more detail the structure of the relations in B_n-, D_n- and G₂-bracket algebras, and as an application, discover a Pieri-type formula in the B_n-bracket algebra. As a corollary, we obtain a Pieri-type formula for multiplication of an arbitrary B_n-Schubert class by some special ones. Our Pieri-type formula is a generalization of Pieri’s formulas obtained by A. Lascoux and M.-P. Schützenberger for flag varieties of type A. We also introduce a super-version of the bracket algebra together with a family of pairwise anticommutative elements, the so-called flat connections with constant coefficients, which describes “a noncommutative differential geometry on a finite Coxeter group” in the sense of S. Majid.     

On v-closed manis valution rings

ABSTRACT

The graded Lie algebra L associated to the Nottingham group with respect to its natural filtration is known to be a loop algebra of the first Witt algebra W ₁ . The fact that the Schur multiplier of W ₁ , in characteristic p > 3, is one-dimensional implies that L is not finitely presented. Consider the universal covering ? ₁ of W ₁ and the corresponding loop algebra M of ? ₁ . In this paper we prove that M itself is finitely presented for p > 3. In characteristic p >  11 the algebra M turns out to be presented by two relations.