Stably Calabi-Yau algebras and skew group algebras

Let G be a finite group and A be a finite-dimensional selfinjective algebra over an algebraically closed field. Suppose A is a left G-module algebra. Some suficient conditions for the skew group algebra AG to be stably Calabi-Yau are provided, and some new examples of stably Calabi-Yau algebras are given as well.     

Quadratic equations over matrix skew series

We introduce notions of skew and dual-skew series of a set of square matrices, and in these terms we formulate deflation conditions for general matrix quadratic equations.     

Quadratic forms in skew normal variates

In this paper first a characterization of the multivariate skew normal distribution is given. Then the joint moment generating functions of two quadratic forms, and a linear compound and a quadratic form in skew normal variates, have been derived and conditions for their independence are given. Distribution of the ratios of quadratic forms in skew normal variates has also been studied.     

Quadratic spaces and quaternion algebras

It is shown that the existence of a nontrivial quaternion algebra over a field F of characteristic not equal to 2 implies the existence of quadratic F[X₁...,X _n ]-spaces (n2) which are not extensions of F.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 110–120, 1978.     

A representation of skew effect algebras

It was shown by the author and R. Hala? that every effect algebra can be organized into a conditionally residuated structure. Skew effect algebras were introduced as a non-associative modification of effect algebras. Hence, there is natural question if a similar characterization by a certain residuated structure is possible. For this we use the so-called skew residuated structure introduced recently by the author and J. Krňávek. It is shown that this is really a suitable tool for the representation.     

Unitary units and skew elements in group algebras

 Let FG be the group algebra of a group G over a field F and let * denote the canonical involution of FG induced by the map g→g ⁻¹ ,gG. Let Un(FG)={uFG|uu ^* =1} be the group of unitary units of FG. In case char F=0, we classify the torsion groups G for which Un(FG) satisfies a group identity not vanishing on 2-elements. Along the way we actually prove that, in characteristic 0, the unitary group Un(FG) does not contain a free group of rank 2 if FG ⁻ , the Lie algebra of skew elements of FG, is Lie nilpotent. Motivated by this connection we characterize most groups G for which FG ⁻ is Lie nilpotent and char F≠2. Received: 15 July 2002 / Revised version: 28 December 2002 Published online: 24 April 2003 Research partially supported by MURST (Italy) and FAPESP and CNPq (Brazil). Mathematics Subject Classification (2000): Primary 16U60; Secondary 16W10, 20C07     

Quadratic algebras with Ext algebras generated in two degrees

Green and Marcos (2005) [2] call a graded k-algebra δ-Koszul if the corresponding Yoneda algebra is finitely generated and there exists a function δ:N→N such that is zero if j≠δ(i). For any integer m≥3 we exhibit a noncommutative quadratic δ-Koszul algebra for which the Yoneda algebra is generated in degrees (1,1) and (m,m+1). These examples answer a question of Green and Marcos. These algebras are not Koszul but m-Koszul (in the sense of Backelin).     

Solution of the twisting problem for skew group algebras

LetK be any field of characteristicp>0 and letG be a finite group acting onK via a map τ. The skew group algebraK _τG may be nonsemisimple (precisely whenP|(H), H=Kert). In [1] necessary conditions were given for the existence of a class α∈H ²(G,K^*) which “twists” the skew group algebraK _τG into a semisimple crossed productK _τ ^αG . The “twisting problem” asks whether these conditions are sufficient. In [1] we showed that this is indeed so in many cases. In this paper we prove it in general. During the period of this research the second author was an Associate at the Center for Advanced Study, Urbana, Illinois.     

On clones, transformation monoids, and finite Boolean algebras

We prove that, for a finite Boolean algebra, there exist only finitely many clones which consist of polynomial functions of the algebra and contain the monoid of its unary polynomial functions. Received January 7, 2000; accepted in final form December 13, 2000.     

Endomorphism monoids of algebras whose subalgebra lattices are chains

A monoidM and a latticeL arealgebraic if there is an algebraA with endomorphism monoid EndA M and subalgebra lattice SuA L. For each chainC we characterize those monoidsM for whichM and C are algebraic. In particular we show that a finite monoidM is algebraic with the three-chain iff the equalizers ofM form a chainE 3. The same assertion however fails for infinite monoids. This generalizes the corresponding result for two-chains and solves a problem posed by B. Jónsson ([2], p. 147). We settle the same question for all longer chainsK. Presented by Ivo Rosenberg.     

Quadratic algebras related to elliptic curves

We construct quadratic finite-dimensional Poisson algebras corresponding to a rank-N degree-one vector bundle over an elliptic curve with n marked points and also construct the quantum version of the algebras. The algebras are parameterized by the moduli of curves. For N = 2 and n = 1, they coincide with Sklyanin algebras. We prove that the Poisson structure is compatible with the Lie-Poisson structure defined on the direct sum of n copies of sl(N). The origin of the algebras is related to the Poisson reduction of canonical brackets on an affine space over the bundle cotangent to automorphism groups of vector bundles. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 163–183, August, 2008.     

Presentations for rook partition monoids and algebras and their singular ideals

We obtain several presentations by generators and relations for the rook partition monoids and algebras, as well as their singular ideals. Among other results, we also calculate the minimal sizes of generating sets (some of our presentations use such minimal-size generating sets), and show that the singular part of the rook partition monoid is generated by its idempotents.