A variety V has an alg-universal n-expansion if the addition of n nullary operations to algebras from V produces an alg-universal category. It is proved that any semigroup variety V containing a semigroup that is neither an inflation of a completely simple semigroup nor an inflation of a semilattice of groups has an alg-universal 3-expansion. We say that a variety V is var-relatively alg-univer...

To any Hopf algebra H we associate two commutative Hopf algebras Hl1(H) and Hl2(H), which we call the lazy homology Hopf algebras of H . These Hopf algebras are built in such a way that they are “predual” to the lazy cohomology groups based on the so-called lazy cocycles. We establish two universal coefficient theorems relating the lazy cohomology to the lazy homology and we compute the lazy ho...

We construct a new efficient algorithm for finding Gröbner–Shirshov bases for noncommutative algebras and their representations. This algorithm uses the Macaulay matrix [F.S. Macaulay, On some formula in elimination, Proc. London Math. Soc. 33 (1) (1902) 3–27], and can be viewed as a representation theoretic analogue of the F4 algorithm developed by J.C. Faugère. We work out some examples of un...

In this paper, deformations of L∞-algebras are defined in such a way that the bases of deformations are L∞-algebras, as well. A universal and a semiuniversal deformation is constructed for L∞-algebras, whose cotangent complex admits a splitting. The paper also contains an explicit construction of a minimal L∞-structure on the homology H of a differential graded Lie algebra L and of an L∞-quasi-...

A category is said to be alg-universal, if every category of universal algebras can be fully embedded into it. We prove here that the category of finitary endofunctors of the category Set is alg-universal. We also present an example of a proper class of accessible set functors with no natural transformations between them (except the obvious identities).

We apply the construction of universal lower-bounded generalized twisted modules by author to construct and grading-restricted for affine vertex (operator) algebras. prove that these algebras are equivalent suitable induced corresponding Lie algebra or quotients such explicitly given submodules .

We show that for a class of operator algebras satisfying a natural condition the C∗-envelope of the universal free product of operator algebras Ai is given by the free product of the C ∗-envelopes of the Ai. We apply this theorem to, in special cases, the C∗-envelope of the semicrossed products for multivariable dynamics in terms of the single variable semicrossed products of Peters. An importa...

These notes are intended as a fairly self contained explanation of Fock space and various algebras that act on it, including a Clifford algebras, a Weyl algebra, and an affine Kac-Moody algebra. We also discuss how the various algebras are related, and in particular describe the celebrated boson-fermion correspondence. We finish by briefly discussing a deformation of Fock space, which is a repr...

Let A, B be separable simple unital tracially AF C*-algebras. Assuming that A is exact and satisfies the Universal Coefficient Theorem (UCT) in KK-theory, we prove the existence, and uniqueness modulo approximately inner automorphisms, of nuclear ∗-homomorphisms from A to B with prescribed K-theory data. This implies the AF-embeddability of separable exact residually finite dimensional C*-algeb...