Anti-commutative Grbner-Shirshov basis of a free Lie algebra

The concept of Hall words was first introduced by P. Hall in 1933 in his investigation on groups of prime power order. Then M. Hall in 1950 showed that the Hall words form a basis of a free Lie algebra by using direct construction, that is, first he started with a linear space spanned by Hall words, then defined the Lie product of Hall words and finally checked that the product yields the Lie identities. In this paper, we give a Grbner-Shirshov basis for a free Lie algebra. As an application, by using the ...     

Anti-commutative Gröbner-Shirshov basis of a free Lie algebra

The concept of Hall words was first introduced by P. Hall in 1933 in his investigation on groups of prime power order. Then M. Hall in 1950 showed that the Hall words form a basis of a free Lie algebra by using direct construction, that is, first he started with a linear space spanned by Hall words, then defined the Lie product of Hall words and finally checked that the product yields the Lie identities. In this paper, we give a Gröbner-Shirshov basis for a free Lie algebra. As an application, by using the Composition-Diamond lemma established by Shirshov in 1962 for free anti-commutative (non-associative) algebras, we provide another method different from that of M. Hall to construct a basis of a free Lie algebra.     

On multivariate polynomials in Bernstein–Bézier form and tensor algebra

The Bernstein–Bézier representation of polynomials is a very useful tool in computer aided geometric design. In this paper we make use of (multilinear) tensors to describe and manipulate multivariate polynomials in their Bernstein–Bézier form. As an application we consider Hermite interpolation with polynomials and splines.     

Gosset’s figure in a Clifford algebra

This note describes a way to realize a “projective” version of Gosset’s 240-vertex semiregular polytope 421 using the Clifford algebra Cl(8) generated by an 8-dimensional vector space equipped with a non-degenerate quadratic form. The 120 vertices of this projective Gosset figure are also seen to coincide with a particular basis for the Lie algebra      

Takeuchi’s free Hopf algebra construction revisited

Takeuchi’s famous free Hopf algebra construction is analyzed from a categorical point of view, and so is the construction of the Hopf envelope of a bialgebra. Both constructions in fact can be described as compositions of well known and natural constructions. This way certain partially wrong perceptions of these constructions are clarified and their mutual relation is made precise. The construction of Hopf envelopes finally is shown to provide a construction of a Hopf coreflection of bialgebras by simple dualization. The results provided hold for any commutative von Neumann regular ring, not only for fields.     

What does a group algebra of a free group “know” about the group?

We describe solutions to the problem of elementary classification in the class of group algebras of free groups. We will show that unlike free groups, two group algebras of free groups over infinite fields are elementarily equivalent if and only if the groups are isomorphic and the fields are equivalent in the weak second order logic. We will show that the set of all free bases of a free group F is 0-definable in the group algebra $K(F)$ when K is an infinite field, the set of geodesics is definable, and many geometric properties of F are definable in $K(F)$. Therefore $K(F)$ “knows” some very important information about F. We will show that similar results hold for group algebras of limit groups.     

ℤ/2ℤ Invariants of the free fermion algebra

By using quantum vertex operators we study the invariance of the rank n free-fermion vertex algebra under the action of the group ?∕2? and obtain its minimal generating set. When n = 1, it is well known that this subalgebra is isomorphic to the Virasoro vertex algebra with central charge 1∕2. In the n = 2 case we show that invariant subalgebra is isomorphic to a simple quotient of a certain W-algebra, which we explicitly construct. For n≥3, our approach leads to a rediscovery of the spinor representation of the a?ne vertex algebra associated to the Lie algebra 𝔰𝔬(n) of I. Frenkel.     

Strong majorization in a free ✱-algebra

We study, in the spirit of modern real algebra, the interplay between left ideals of the free ∗-algebra with n generators, and their suitably defined zero sets; and similarly between quadratic submodules of and their positivity sets. Partially supported by grants from the National Science Foundation and the Ford Motor Co.     

Little and big q-Jacobi polynomials and the Askey–Wilson algebra

The Ramanujan Journal - The little and big q-Jacobi polynomials are shown to arise as basis vectors for representations of the Askey–Wilson algebra. The operators that these polynomials...     

On the non-normal Padé table in a non-commutative algebra

In the non-commutative algebra the blocks in the table of orthogonal polynomials and therefore in the Padé table are not square, and generally it is impossible to say anything on the structure of these blocks except for infinite blocks. This last case is extensively studied here for the non-normal Padé table, the non-normal table P of orthogonal polynomials, and the non-normal ϵ-table. Some examples of illustration of different situations are given.     

Melin-Hörmander inequality in a Wiener type pseudo-differential algebra

We prove a Melin-Hörmander inequality for a Banach algebra of pseudo-differential operators whose calculus was developed by Sjöstrand. The main new difficulties in the proof are settled by a stationary phase method tailored to the low-regularity of the symbols.     

A Batalin–Vilkovisky algebra structure on the Hochschild cohomology of truncated polynomials

We calculate the Batalin–Vilkovisky, and the induced Gerstenhaber structures of the Hochschild cohomology of the singular R-cochain complex of n-dimensional K-projective spaces, where K=C,H$\mathbf{K}=\mathbb{C},\mathbb{H}$ and R=Z$R=\mathbb{Z}$ and any field. In the special case that K=C$\mathbf{K}=\mathbb{C}$, n=1$n=1$ and R=Z$R=\mathbb{Z}$, we show that this structure cannot be identified with the BV-structure of the integral loop homology of the 2-dimensional sphere computed by L. Menichi, but their induced Gerstenhaber structures can still be identified. Combined with the work of Y. Félix and J. Thomas, the main result of the present paper calculates the BV-structure of the rational loop homology of projective spaces.     

The centralizer of an element in a Lie algebra of type L

In this paper, we investigate the Lie algebra L(A, α, δ) of type L and obtain the respective sufficient conditions for L(A, α,δ) to be semisimple, and for Z(ω) = Fω as well, where 0 ≠ ω ∈ L(A, α, δ) and Z(ω) is the centralizer of ω.     

On a certain ideal of Külshammer in the centre of a group algebra

Let G be a finite group and let F be a splitting field of characteristic $ p > 0 $ p > 0 . We show that I² = E₀, where I is a certain ideal of the centre Z of FG, and E₀ is the span of the block idempotents of defect zero.     

Cohomology structure for a Poisson algebra:Ⅱ

For a Poisson algebra, we prove that the Poisson cohomology theory introduced by Flato et al.(1995)is given by a certain derived functor. We show that the(generalized) deformation quantization is equivalent to the formal deformation for Poisson algebras under certain mild conditions. Finally we construct a long exact sequence, and use it to calculate the Poisson cohomology groups via the Yoneda-extension groups of certain quasi-Poisson modules and the Lie algebra cohomology groups.     

Egoroff’s theorem in measurable operator spaces associated with a Von Neumann algebra

In this article, some kinds of convergence about the τ–measurable operators affiliated with a von Neumann algebra are defined. Moreover, the relationships among these kinds of convergence are considered.     

Bernstein–Sato polynomials in positive characteristic

In characteristic zero, the Bernstein–Sato polynomial of a hypersurface can be described as the minimal polynomial of the action of an Euler operator on a suitable D-module. We consider analogous D-modules in positive characteristic, and use them to define a sequence of Bernstein–Sato polynomials (corresponding to the fact that we need to consider also divided powers Euler operators). We show that the information contained in these polynomials is equivalent to that given by the F-jumping exponents of the hypersurface, in the sense of Hara and Yoshida [N. Hara, K.-i. Yoshida, A generalization of tight closure and multiplier ideals, Trans. Amer. Math. Soc. 355 (2003) 3143–3174].