# Groups and Algebras

Organizers:

- Pavel Zalesski (University of Brasilia)
- Ivan Shestakov (State University of Sao Paulo)
- Olga Kharlampovich (Hunter College CUNY)
- Alexei Miasnikov (Stevens Institute of Technology)
- Yuri Bahturin (Memorial University)

Schedule:

- Tuesday, Jul 25 [McGill U., Trottier Building (Engineering), Room 60]
- 11:45 Yuri Bahturin (Memorial University of Newfoundland, Canada), Real graded division algebras
- 14:15 Inna Bumagina (Carleton University, Canada), Untitled
- 14:45 Liudmila Sabinina (Universidad Autónoma del Estado de Morelos), On Malcev algebras with the identity $J(x_1x_2...x_n, y, z) =0.$
- 15:45 Jeremy MacDonald (Concordia University), Untitled
- 16:15 Svetla Vassileva (Champlain College Saint-Lambert), Untitled
- 17:00 Victor Petrogradsky (University of Brasilia, Brazil), Lie identities of symmetric Poisson algebras
- Wednesday, Jul 26 [McGill U., Trottier Building (Engineering), Room 60]
- 11:15 Pavel Zalesskii (University of Brasilia), The profinite completion of 3-manifold groups
- 11:45 Ivan Shestakov (State University of Sao Paulo), A finite representation of Jordan algebras
- 13:45 Olga Kharlampovich (Hunter College, CUNY), Tarski-type problems for free Lie algebras
- 14:15 Dessislava Kochloukova (Brazil State University of Campinas), Subdirect sums and fibre sums of Lie algebras
- 14:45 Alexei Krassilnikov (University of Brasilia, Brazil), Lie nilpotent asssociative algebras
- 15:15 Evegeny Plotkin (Bar-Ilan University, Israel), Word equations and word equations with constants
- 16:15 Alexei Myasnikov (CUNY, USA), Untitled
- 16:45 Jacob Mostovoy (Instituto Politécnico Nacional, Mexico), Multiplicative graphs and related algebras
- Unscheduled [McGill U., Trottier Building (Engineering), Room 60]
- Nikolay Romanovski (Syberian Branch of the Russian Academy of Sciences), Untitled

- Yuri Bahturin

Memorial University of Newfoundland, CanadaReal graded division algebrasIn this joint work with Professor Mikhail Zaicev of Moscow State University, we classify, up to equivalence, all finite-dimensional real graded division algebras. The grading is by any abelian group. The classification in the case where the algebras are simple was previously obtained by us and independently by Adrián Rodrigo-Escudero of University of Zaragoza. - Liudmila Sabinina

Universidad Autónoma del Estado de MorelosOn Malcev algebras with the identity $J(x_1x_2...x_n, y, z) =0.${\it 2010 Mathematics Subject Classification}: 20N05, 17D10\\ Let $ \mathfrak MA_n$ be a variety of Malcev algebras with the identity $J(x_1x_2...x_n, y, z) =0,\,n\in \mathbb N.$ We study a structure of Malcev algebras from this variety. In particular, a Malcev algebra $A\in \mathfrak MA_2$ is a tangent algebra of some smooth left automorphic Moufang loop $M$. Let us call a variety of smooth Moufang loops with the identity $([...[x_1,x_2],x_3]...,x_k], y, z)=1$ the variety of $k$-generalized left automorphic Moufang loops.\\ We study the correspondence between smooth $k$-generalized left automorphic Moufang loops and the variety $\mathfrak MA_k$ of their tangent algebras. We show that a local smooth $k$- generalized left automorphic Moufang loop defines a global smooth $k$-generalized left automorphic Moufang loop. This talk is based on the joint research with M. Rasskazova and R. Carrillo Catalan. - Victor Petrogradsky

University of Brasilia, BrazilLie identities of symmetric Poisson algebrasWe recall results on existence of identical relations in group rings and universal (restricted) enveloping algebras of (restricted) Lie (super)algebras in case of different characteristics. Now we consider related results on identities in Poisson algebras. Let $L$ be a Lie algebra over a field of characteristic $p>0$. Consider its symmetric algebra $S(L)=\oplus_{n=0}^\infty U_n/U_{n-1}$, which is isomorphic to a polynomial ring. It also has a structure of a Poisson algebra, where the Lie product is traditionally denoted by $\{\ ,\ \}$. This bracket naturally induces the structure of a Poisson algebra on the truncated polynomial ring $\mathbf{s}(L)=S(L)/(x^p\,|\, x\in L)$, which we call a {\it truncated symmetric Poisson algebra}. We study Lie identical relations of $\mathbf{s}(L)$. Namely, we determine necessary and sufficient conditions for $L$ under which $\mathbf{s}(L)$ is Lie nilpotent, strongly Lie nilpotent, solvable and strongly solvable, where we assume that $p>2$ to specify the solvability. We compute the strong Lie nilpotency class of $\mathbf{s}(L)$. Also, we prove that the Lie nilpotency class coincides with the strong Lie nilpotency class in case $p>3$. Shestakov proved that the symmetric algebra $S(L)$ of an arbitrary Lie algebra $L$ satisfies the identity $\{x,\{y,z\}\}\equiv 0$ if, and only if, $L$ is abelian. We extend this result for the (strong) Lie nilpotency and the (strong) solvability of $S(L)$. We show that the solvability of $\mathbf{s}(L)$ and $S(L)$ in case $\mathrm{char} K=2$ is different to other characteristics, namely, we construct examples of such algebras which are solvable but not strongly solvable. - Ivan Shestakov

State University of Sao PauloA finite representation of Jordan algebrasLet $R$ be an associative algebra. Let $\star : R\rightarrow R$ be an involution. We study the following question: when are the Jordan algebras $R^{(+)}$ and $H(R,\star) = \{a \in R\,|\, a^{\star} = a\}$ finitely presented? It is a joint work with Efim Zelmanov. - Dessislava Kochloukova

Brazil State University of CampinasSubdirect sums and fibre sums of Lie algebrasWe will discuss subdirect sums and fibre sums of Lie algebras and show that some group theoretic results on subdirect products and fibre products due to Baumslag,Bridson, Roseblade, Howie , Miller and Short have versions for Lie algebras. In particular we will discuss subdirect sums of finitely generated free Lie algebras and will give some sufficient conditions for the the fibre sum of two Lie algebras to be finitely presented in terms of generators and relations (this is a Lie algebra version of the 1-2-3 Theorem from group theory). This is a joint work with Conchita Martínez-Perez (Zaragoza). - Alexei Krassilnikov

University of Brasilia, BrazilLie nilpotent asssociative algebrasAn associative algebra $A$ is called Lie nilpotent if its associated Lie algebra $A^{(-)}$ (with the Lie bracket defined by $[a,b] = ab - ba$) is nilpotent. Recent interest in Lie nilpotent associative algebras has been motivated by the study of the quotients of the lower central series of $A^{(-)}$ for various associative algebras A. This study was initiated in 2007 in the pioneering work of Feigin and Shoikhet and developed in a number of articles by various authors. In my talk I will present and discuss some new results about Lie nilpotent associative algebras. - Evegeny Plotkin

Bar-Ilan University, IsraelWord equations and word equations with constantsI will survey some old and recent results on word maps with the focus on word maps with constants. It turns out that a sort of Borel’s dominancy theorem for word maps evaluated on simple algebraic groups remains valid for word maps with constants. Other results deal with Borel-type theorem for linear perfect groups. Joint work with N.Gordeev and B.Kunyavskii - Jacob Mostovoy

Instituto Politécnico Nacional, MexicoMultiplicative graphs and related algebrasA multiplicative graph $Q$ is a graph together with an associative morphism of its cartesian square $Q\,\square\, Q$ to $Q$. There are several examples of multiplicative graphs that arise in knot theory; the category of multiplicative graphs whose vertices form a group is equivalent to the category of augmented racks. We show how to construct rack homology via multiplicative graphs and define several Hopf algebras associated with a multiplicative graph.