Lambda Algebra Research Articles

Reflexive combinatory algebras

Abstract We introduce the notion of reflexivity for combinatory algebras. Reflexivity can be thought of as an equational counterpart of the Meyer–Scott axiom of combinatory models, which indeed allows us to characterize an equationally definable counterpart of combinatory models. This new structure, called strongly reflexive combinatory algebra, admits a finite axiomatization with seven closed equations, and the structure is shown to be exactly the retract of combinatory models. Lambda algebras can be characterized as strongly reflexive combinatory algebras that are stable. Moreover, there is a canonical construction of a lambda algebra from a strongly reflexive combinatory algebra. The resulting axiomatization of lambda algebras by the seven axioms for strong reflexivity together with those for stability is shown to correspond to the axiomatization of lambda algebras due to Selinger (2002, J. Funct. Program., 12, 549–566).

A NOTE ON THE NON-TRIVIAL ELEMENTS IN THE COHOMOLOGY GROUPS OF THE STEENROD ALGEBRA

Let $F_2$ be the prime field of two elements and let $GL_s:= GL(s, F_2)$ be the general linear group of rank $s.$ Denote by $\mathscr A$ the Steenrod algebra over $F_2.$ The (mod-2) Lambda algebra, $\Lambda,$ is one of the tools to describe those mysterious "Ext-groups". In addition, the $s$-th algebraic transfer of William Singer \cite{Singer} is also expected to be a useful tool in the study of them. This transfer is a homomorphism $Tr_s: F_2 \otimes_{GL_s}P_{\mathscr A}(H_{*}(B\mathbb V_s, F_2))\to {\rm Ext}_{\mathscr {A}}^{s,s+*}(F_2, F_2),$ where $\mathbb V_s$ denotes the elementary abelian $2$-group of rank $s$, and $H_*(B\mathbb V_s)$ is the homology group of the classifying space of $\mathbb V_s,$ while $P_{\mathscr A}(H_{*}(B\mathbb V_s, F_2))$ means the primitive part of $H_{*}(B\mathbb V_s, F_2)$ under the action of $\mathscr A.$ It has been shown that $Tr_s$ is highly non-trivial and, more precisely, that $Tr_s$ is an isomorphism for $s\leq 3.$ In addition, Singer proved that $Tr_4$ is an isomorphism in some internal degrees. He was also investigated the image of the fifth transfer by using invariant theory. In this note, we use another method to study the image of $Tr_5.$ More precisely, by direct computations using a representation of $Tr_5$ over the algebra $\Lambda,$ we show that $Tr_5$ detects the non-zero elements $h_0d_0\in {\rm Ext}_{\mathscr A}^{5, 5+14}(F_2, F_2),\ h_2e_0\in {\rm Ext}_{\mathscr A}^{5, 5+20}(F_2, F_2)$ and $h_1h_4c_0\in {\rm Ext}_{\mathscr A}^{5, 5+24}(F_2, F_2).$ The same argument can be used for homological degrees $s\geq 6$ under certain conditions.

Length-preserving monomorphisms for some algebras of operations

Let p be an odd prime. Investigating the existence of a fractal structure for the universal Steenrod algebra \(\mathcal Q(p)\) and the Lambda algebra leads to determine the group of their length-preserving automorhisms. Contrarily to the \(p=2\) case, no length-preserving strict monomorphism turns out to exist, and this makes reasonable to conjecture that the algebras above do not contain proper subalgebras isomorphic to themselves.

Chasing non-diagonal cycles in a certain system of algebras of operations

The mod 2 universal Steenrod algebra Q is a non-locally finite homoge- neous quadratic algebra closely related to the ordinary mod 2 Steenrod algebra and the Lambda algebra. The algebra Q provides an example of a Koszul algebra which is a direct limit of a family of certain non-Koszul algebras Rk's. In this paper we see how far the several Rk's are to be Koszul by chasing in their cohomology non-trivial cocycles of minimal homological degree.

Lambda algebra and the Singer transfer

We modify Singer's idea to give a direct description of the lambda algebra using modular invariant theory. As an application, we describe the algebraic transfer in purely invariant-theoretic framework, thus, provides an effective computational tool for the algebraic transfer. The induced action of the Steenrod algebra on lambda algebra is also investigated and clarified.

Singer's formulas for the Steenrod algebra

We prove Singer's formulas and unstable acyclic relations for the mod $2$ Steenrod algebra, which are important tools for the computations of products. These are given by analogy of the corresponding dual results for the lambda algebra.

Combinatorial proofs of the Lambda algebra basis and EHP sequence

Combinatorial proofs of the Lambda algebra basis and EHP sequence

The Cohomology of the Universal Steenrod Algebra

The mod 2 universal Steenrod algebra Q is a homogeneous quadratic algebra closely related to the ordinary mod 2 Steenrod algebra and the Lambda algebra introduced in [1]. In this paper we show that Q is Koszul. It follows by [7] that its cohomology, being purely diagonal, is isomorphic to a completion of Q itself with respect to a suitable chain of two-sided ideals.

Some acyclic relations in the lambda algebra

We consider the relations og ¼ 0 A L, and show that if oa ¼ 0 then a ¼ gb for some b. These relations give the acyclic chain complex L ! g L ! o L. We consider

Spherical classes and the Lambda algebra

Spherical classes and the Lambda algebra

On Singer's invariant-theoretic description of the lambda algebra: A mod p analogue

Let p be an odd prime. The purpose of the paper is to give a mod p analogue for the Singer invariant-theoretic description of the lambda algebra. In other words, we give an invariant-theoretic interpretation for the homology of the mod p Steenrod algebra A. More precisely, we associate to any left A-module M a chain complex Γ + M whose homology is isomorphic to Tor ∗ A( Z/p, M) . This chain complex is built from the Dickson-Mùi invariants of the general linear group GL(n, Z/p) for n > 0. Particularly, in the case M = Z/p , the complex Γ + = Γ + Z/p is dual to the lambda algebra, which is the E 1 term of the Adams spectral sequence for spheres. Notably, we naturalize a little bit Singer's way to define Γ + M. In fact, we replace Singer's Γ + M by its image under the so-called total power. Consequently, the action of A on the new Γ + M is diagonal, meanwhile that on Singer's Γ + M is not. Also, the differential of the new Γ + M becomes simpler. This naturalization is valid for p = 2 as well as for p an odd prime.

The Poincare limit in 2+1 dimensional quantum de Sitter gravity

The quantum traces algebra for the 2+1 Poincare gravity in a first order formalism is explicitly constructed by contracting the corresponding traces algebra of the de Sitter gravity. Unbounded representations of the latter, in the case Lambda (0, are constructed in terms of an underlying SU(1, 1) algebra. Unfortunately, these representations do not possess a well defined Poincare limit. Nevertheless an explicit realization of the Poincare traces algebra is constructed in terms of two pairs of canonical variables.

Level Number Sequences of Trees and the Lambda Algebra

We enumerate ordered partitions of positive integers, restricted by the condition that the (j + 1)th entry is not greater than d times the jth entry for all j. We also consider restrictions on the first and/or last entries, and on congruence classes modulo 2d - 2 of all the entries. Such sequences arise as level number sequences of d-ary trees, but we are motivated by constructions arising in algebraic topology.

Computing the homology of the lambda algebra

Computing the homology of the lambda algebra

Invariant theory and the lambda algebra

Let A A be the Steenrod algebra over the field F 2 {F_2} . In this paper we construct for any left A A -module M M a chain complex whose homology groups are isomorphic to the groups Tor s A ⁡ ( F 2 , M ) \operatorname {Tor}_s^A({F_2},M) . This chain complex in homological degree s s is built from a ring of invariants associated with the action of the linear group G L s ( F 2 ) G{L_s}({F_2}) on a certain algebra of Laurent series. Thus, the homology of the Steenrod algebra (and so the Adams spectral sequence for spheres) is seen to have a close relationship to invariant theory. A key observation in our work is that the Adem relations can be described in terms of the invariant theory of G L 2 ( F 2 ) G{L_2}({F_2}) . Our chain complex is not new: it turns out to be isomorphic to the one constructed by Kan and his coworkers from the dual of the lambda algebra. Thus, one effect of our work is to give an invariant-theoretic interpretation of the lambda algebra. As a consequence we find that the dual of lambda supports an action of the Steenrod algebra that commutes with the differential. The differential itself appears as a kind of "residue map". We are also able to describe the coalgebra structure of the dual of lambda using our invariant-theoretic language.