Augmented Algebra Research Articles

Parafree augmented algebras and Gröbner-Shirshov bases for complete augmented algebras

We develop a theory of parafree augmented algebras similar to the theory of parafree groups and explore some questions related to the Parafree Conjecture. We provide an example of finitely generated parafree augmented algebra of infinite cohomological dimension. Motivated by this example, we prove a version of the Composition-Diamond lemma for complete augmented algebras and provide a sufficient condition for augmented algebra to be residually nilpotent on the language of its relations.

Rota-Baxter Leibniz Algebras and Their Constructions

In this paper, we introduce the concept of Rota-Baxter Leibniz algebras and explore two characterizations of Rota-Baxter Leibniz algebras. And we construct a number of Rota-Baxter Leibniz algebras from Leibniz algebras and associative algebras and discover some Rota-Baxter Leibniz algebras from augmented algebra, bialgebra, and weak Hopf algebra. In the end, we give all Rota-Baxter operators of weight 0 and -1 on solvable and nilpotent Leibniz algebras of dimension ≤3, respectively.

Cyclic homology of cleft extensions of algebras

Let [Formula: see text] be a commutative algebra with [Formula: see text] and let [Formula: see text] be a cleft extension of [Formula: see text]. We obtain a new mixed complex, simpler than the canonical one, giving the Hochschild and cyclic homologies of [Formula: see text] relative to [Formula: see text]. This complex resembles the canonical reduced mixed complex of an augmented algebra. We begin the study of our complex showing that it has a harmonic decomposition like the one considered by Cuntz and Quillen for the normalized mixed complex of an algebra.

Operad bimodules and composition products on André–Quillen filtrations of algebras

If O is a reduced operad in symmetric spectra, an O-algebra I can be viewed as analogous to the augmentation ideal of an augmented algebra. Implicit in the literature on Topological Andre-Quillen homology is that such an I admits a canonical (and homotopically meaningful) decreasing O-algebra filtration I > I^2 > I^3 > ... satisfying various nice properties analogous to powers of an ideal in a ring. In this paper, we are explicit about these constructions. With R a commutative S-algebra, we study derived versions of the circle product M o_O I, where M is an O-bimodule, and I is an O-algebra in R-modules. Letting M run through a decreasing O-bimodule filtration of O itself then yields the augmentation ideal filtration as above. The composition structure of the operad induces algebra maps from (I^i)^j to I^{ij}, fitting nicely with previously studied structure. As a formal consequence, an O-algebra map from I to J^d induces compatible maps from I^n to J^{dn}, for all n. This is an essential tool in the first author's study of Hurewicz maps for infinite loop spaces, and its utility is illustrated here with a lifting theorem.

Koszul–Morita duality

We construct a generalization of Koszul duality in the sense of Keller–Lefèvre for not necessarily augmented algebras. This duality is closely related to classical Morita duality and specializes to it in certain cases.

Stable representation homology and Koszul duality

Abstract This paper is a sequel to [Adv. Math. 245 (2013), 625–689], where we study the derived affine scheme DRep n ( A ) ${{\mathrm {DRep}}_n(A)}$ parametrizing the n-dimensional representations of an associative k-algebra A. In [Adv. Math. 245 (2013), 625–689], we have constructed canonical trace maps Tr n ( A ) • : HC • ( A ) → H • [ DRep n ( A ) ] GL n ${\operatorname{Tr}_n(A)_{\bullet }: \mathrm {HC}_{\bullet }(A) \rightarrow \mathrm {H}_{\bullet }[{\mathrm {DRep}}_n(A)]^{{\mathrm {GL}}_n} }$ extending the usual characters of representations to higher cyclic homology. This raises the natural question whether a well-known theorem of Procesi [Adv. Math. 19 (1976), 306–381.] holds in the derived setting: namely, is the algebra homomorphism Λ Tr n ( A ) • : Λ k [ HC • ( A ) ] → H • [ DRep n ( A ) ] GL n ${\Lambda \mathrm {Tr} _n(A)_{\bullet } : \Lambda _k[\mathrm {HC}_{\bullet }(A)] \rightarrow \mathrm {H}_{\bullet }[{\mathrm {DRep}}_n(A)]^{{\mathrm {GL}}_n}}$ defined by Tr n ( A ) • ${ \operatorname{Tr}_n(A)_{\bullet } }$ surjective? In the present paper, we answer this question for augmented algebras. Given such an algebra, we construct a canonical dense subalgebra DRep ∞ ( A ) Tr ${{\mathrm {DRep}}_{\infty }(A)^{\operatorname{Tr}}}$ of the topological DG algebra lim ← DRep n ( A ) GL n ${\varprojlim {\mathrm {DRep}}_n(A)^{{\mathrm {GL}}_n} }$ . Our main result is that on passing to the inverse limit, the family of maps Λ Tr n ( A ) • ${\Lambda \mathrm {Tr} _n(A)_{\bullet }}$ `stabilizes' to an isomorphism Λ k ( HC ¯ • ( A ) ) ≅ H • [ DRep ∞ ( A ) Tr ] ${\Lambda _k(\overline{\mathrm {HC}}_{\bullet }(A)) \cong \mathrm {H}_{\bullet }[{\mathrm {DRep}}_{\infty }(A)^{\operatorname{Tr}}]}$ . The derived version of Procesi's theorem does therefore hold in the limit as n → ∞. However, for a fixed (finite) n, there exist homological obstructions to the surjectivity of Λ Tr n ( A ) • ${ \Lambda \mathrm {Tr} _n(A)_{\bullet }}$ , and we show on simple examples that these obstructions do not vanish in general. We compare our result with the classical theorem of Loday, Quillen and Tsygan on stable homology of matrix Lie algebras. We show that the Chevalley–Eilenberg complex 𝒞 • ( 𝔤𝔩 ∞ ( A ) , 𝔤𝔩 ∞ ( k ) ; k ) ${{\mathcal {C}}_\bullet ({\mathfrak {gl}}_\infty (A), {\mathfrak {gl}}_\infty (k); k)}$ equipped with a natural coalgebra structure is Koszul dual to the DG algebra DRep ∞ ( A ) Tr ${{\mathrm {DRep}}_{\infty }(A)^{\operatorname{Tr}}}$ . We also extend our main results to bigraded DG algebras, in which case we show the equality DRep ∞ ( A ) Tr = DRep ∞ ( A ) GL ∞ ${ {\mathrm {DRep}}_{\infty }(A)^{\operatorname{Tr}} = {\mathrm {DRep}}_{\infty }(A)^{{\mathrm {GL}}_{\infty }}}$ . As an application, we compute the Euler characteristics of DRep ∞ ( A ) GL ∞ ${{\mathrm {DRep}}_{\infty }(A)^{{\mathrm {GL}}_{\infty }}}$ and HC ¯ • ( A ) ${\overline{\mathrm {HC}}_{\bullet }(A)}$ and derive some interesting combinatorial identities.

The pure virtual braid group is quadratic

If an augmented algebra K over Q is filtered by powers of its augmentation ideal I, the associated graded algebra grK need not in general be quadratic: although it is generated in degree 1, its relations may not be generated by homogeneous relations of degree 2. In this paper we give a sufficient criterion (called the PVH Criterion) for grK to be quadratic. When K is the group algebra of a group G, quadraticity is known to be equivalent to the existence of a (not necessarily homomorphic) universal finite type invariant for G. Thus the PVH Criterion also implies the existence of such a universal finite type invariant for the group G. We apply the PVH Criterion to the group algebra of the pure virtual braid group (also known as the quasi-triangular group), and show that the corresponding associated graded algebra is quadratic, and hence that these groups have a (not necessarily homomorphic) universal finite type invariant.

On the algebraic K-theory of formal power series

AbstractIn this paper we extend the computation of the the typical curves of algebraic K-theory done by Lars Hesselholt and Ib Madsen to general tensor algebras. The models used allow us to determine the stages of the Taylor tower of algebraic K-theory as a functor of augmented algebras, as defined by Tom Goodwillie, when evaluated on derived tensor algebras.For R a discrete ring, and M a simplicial R-bimodule, we let R(M) denote the (derived) tensor algebra of M over R, and πR denote the ring of formal (derived) power series in M over R. We define a natural transformation of functors of simplicial R-bimodules Φ: which is closely related to Waldhausen's equivalence We show that Φ induces an equivalence on any finite stage of Goodwillie's Taylor towers of the functors at any simplicial bimodule. This is used to show that there is an equivalence of functors , and for connected bimodules, also an equivalence

Cohomology of Products and Coproducts of Augmented Algebras

We show that the ordinary cohomology functor $\Lambda \mapsto {\operatorname{Ext}} ^* _\Lambda (k,k)$ from the category of augmented k-algebras to itself exchanges coproducts and products, then that Hochschild cohomology is close to sending coproducts to products if the factors are self-injective. We identify the multiplicative structure of the Hochschild cohomology of a product, modulo a certain ideal, in terms of the cohomology of the factors.

Algebra Structures on the Comparison of the Reduced Bar Construction and the Reduced W-Construction

For a simplicial augmented algebra K, Eilenberg–Mac Lane constructed a chain map . They proved that g is a reduction (homology isomorphism) and conjectured that it is also the injection of a contraction (special homotopy equivalence). The contraction is followed at once by using homological perturbation techniques. If K is commutative, Eilenberg–Mac Lane proved that g is a morphism of DGA-algebras. The present article is devoted to proving that f and φ satisfy certain multiplicative properties (weaker than g) and showing how they can be used for computing in an economical way the homology of twisted cartesian products of two Eilenberg–Mac Lane spaces.

The plus-construction, Bousfield localization, and derived completion

We define a plus-construction on connective augmented algebras over operads in symmetric spectra using Quillen homology. For associative and commutative algebras, we show that this plus-construction is related to both Bousfield localization and Carlsson’s derived completion.

Generalized Koszul properties for augmented algebras

Under certain conditions, a filtration on an augmented algebra A admits a related filtration on the Yoneda algebra E ( A ) : = Ext A ( K , K ) . We show that there exists a bigraded algebra monomorphism gr E ( A ) ↪ E Gr ( gr A ) , where E Gr ( gr A ) is the graded Yoneda algebra of gr A. This monomorphism can be applied in the case where A is connected graded to determine that A has the K 2 property recently introduced by Cassidy and Shelton.

The [formula omitted] deformation theory of a point and the derived categories of local Calabi–Yaus

Let A be an augmented algebra over a semi-simple algebra S. We show that the Ext algebra of S as an A-module, enriched with its natural A-infinity structure, can be used to reconstruct the completion of A at the augmentation ideal. We use this technical result to justify a calculation in the physics literature describing algebras that are derived equivalent to certain non-compact Calabi–Yau three-folds. Since the calculation produces superpotentials for these algebras we also include some discussion of superpotential algebras and their invariants.

Relative cyclic homology of square zero extensions

Let k be a characteristic zero field, C a k-algebra and M a square zero two sided ideal of C. We obtain a new mixed complex, simpler than the canonical one, giving the Hochschild and cyclic homologies of C relative to M. This complex resembles the canonical reduced mixed complex of an augmented algebra. We begin the study of our complex showing that it has a harmonic decomposition like to the one considered by Cuntz and Quillen for the normalized mixed complex of an algebra. We also give new proofs of two theorems of Goodwillie, obtaining a light improvement of one of them.

On the Schur multiplier of augmented algebras

Letk be a field, andA a finitely generatedk-algebra, with augmentation. Suppose there is a presentation ofA 0→I→R→A→0 whereR is a finitely generated freek-algebra andI is non-zero. IfA is infinite dimensional overk, Lewin proved thatR/I2 is not finitely presented. A stronger statement would be that the ‘Schur multiplier’ ofR/I2 is not finite dimensional. In the case thatA is an augmented domain, we prove this stronger statement, and some related statements.

Cartan-Eilenberg cohomology and triples

In their classic book, Cartan and Eilenberg described a more-or-less general scheme for defining homology and cohomology theories for a number of different kinds of algebraic structure, using a general theory of augmented algebras. Later, in his doctoral dissertation, Beck showed how to use the theory of triples to derive a very different and completely general scheme for doing the same thing. Originally, it was unclear how the two theories were related, but many of these questions were eventually answered in a paper by Barr and Beck. The present paper answers the remaining such questions, most notably in the case of Lie algebras by finding a general result that takes care of all the cases at once. It also shows that it is possible to extend the Cartan-Eilenberg theory of Lie algebras from algebras that are free over the ground ring to ones that are only projective.

Commutative Augmented Algebras with Two Vanishing Homology Modules

Commutative Augmented Algebras with Two Vanishing Homology Modules

Demonstration of the origins of cooperativity within SU2�L 6 mapping over $$\left\{ {I\tilde H_\upsilon } \right\}$$ Liouvillian carrier subspaces characterising the MQ-NMR cluster [A]6(L 6) and its $$\{ |kq\upsilon :[\tilde \lambda ] > > \} $$ tensorial bases

The fundamental mappings over carrier subspace and substructures associated with $$\{ |kq\upsilon > > \} $$ augmented spin algebras of Liouville space, and their mapping onto a subduced symmetry, are derived for [A]6(L 6) spin clusters within the combinatorial context of Rota-Cayley algebra over a field. Use of suitable lexical sets of combinatorialp-tuples (number partitions) over {|IM(M 1−M n )>}M, followed by the subsequent use ofL n inner tensor product (ITP) algebra, allows the substructure of Liouville space to be derived. For SU2×L 6 mapping over the simply-reducible $$\left\{ {I\tilde H_\upsilon } \right\}$$ carrier subspaces, the $$D^k \left( {\tilde U} \right) \times \tilde \Gamma ^{\left[ {\tilde \lambda } \right]} \left( \upsilon \right)$$ (L 6) dual irreps, also arise as a consequence of the Liouville space recoupling termsv≡{k 1−k n } being distinct labels for $$\left\{ {I\tilde H_\upsilon } \right\}$$ which are themselves amenible to combinatorial analysis within the concept of Rota-Cayley algebra. Hence, theL n -induced symmetry aspects of multiquantum NMR density matrix formalisms and their dual $$\{ |kq\upsilon :[\tilde \lambda ] > > \} $$ tensorial bases of spin cluster problems are derived and the nature of the cooperative, aspect between the individual symmetries comprising the duality is demonstrated, i.e. in the context of the operator bases of Liouville space. These practical arguments correlate, well with those based on an augmented boson pattern algebra derived from a Heisenburg algebra for superoperators, ℐ±,ℐ0. An earlier, treatment of conventional Hilbert space SU2×L 6 dualitycould only be realised in terms of standard SU2 boson algebra. Since the recoupling Rota-‘field’v for Liouville space is an explicit aspect of the dual mapping, a direct demonstration of cooperativity exists.

Equivalence of the relational algebra and calculus for nested relations

The relational model is extended to include nested structures. This extension is formalised using the distinction between a tuple scheme and a relation scheme. The algebra and calculus languages are defined for this model. It is shown that the restricted powerset is a derived operation in the algebra and the full powerset is expressible by a safe formula in the calculus. Since the full powerset cannot be derived from the algebra operations, there does not exist a complete equivalence between the calculus and the algebra. In other words, given any algebra expression, there is a safe calculus formula with equivalent expressive power. Conversely, given any safe calculus formula and a bound on the cardinality of the database instance, there is a corresponding equivalent algebra expression. The relational algebra is then augmented with programming constructs and this augmented algebra is shown to be equivalent in expressive power to the relational calculus for nested relations.

An example in the cohomology of augmented algebras

An example in the cohomology of augmented algebras