Algebras with given properties of subalgebras and elements

1. A question - Lie-algebraic monster (last updated August 10, 2011; partially in Russian).
2. Lie algebras that can be written as the sum of two nilpotent subalgebras (last revised September 2011).
3. (Alexander Gein) Does there exist an infinite-dimensional associative noncommutative algebra all whose proper subalgebras are commutative?
   (Alexei Belov, D.Sci. dissertation) Does there exist an infinite-dimensional associative skew field generated by any pair of noncommuting elements?
4. (A.I. Shirshov, 1958) Does there exist an infinite-dimensional finitely-generated Lie algebra all whose 2-generated subalgebras are finite-dimensional with dimensions bounded by a fixed number? (For an analogous question for groups, see Strunkov).
5. (I.V. Lvov, Dniester notebook, 1.97) Is it true that an associative algebra of dimension > 1 over Q, all whose proper subalgebras are nilpotent, is also nilpotent? This is true over fields satisfying the Brauer condition; particularly, for finite and algebraically closed fields.
6. (B.E. Barbaumov, Dniester notebook, 1.14). Does there exist an associative division algebra, infinite dimensional over its center, all whose proper subalgebras are PI?
   In general, what can be said about minimal-non-PI (associative, Lie, etc.) algebras?
7. (famous question posed by Nathan Jacobson) Whether a Lie p-algebra satisfying the condition x^pn(x) = x, is abelian? (This is known to be true if the Lie algebra is finite-dimensional, due to Alexander Premet).
8. (A.T. Gainov, Dniester notebook, 3.26) Describe all finite-dimensional simple anticommutative algebras over an infinite field of characteristic not 2 such that any element lies in some two-dimensional subalgebra.

Cohomology and deformations

1. Questions on Lie algebras of cohomological dimension 1 (last revised September 2009).
2. Leites' (super)questions (last revised September 2011).
3. Compute all deformations of W₁(1) ⊗ A (or even of W₁(n) ⊗ A), not only filtered ones.
4. Taking the "canonical" decompistion of loop algebras in Kac-Moody, one arrives at Serre-type relations (as explained at the end of [11], Invariants of Lie algebras extended over commutative algebras without unit). But there are other decompositions - see, for example, Andruskiewitch-Tiraboshi, p.394. What presentations we will get then?
5. Is it possible to define a (variant of) Poincaré duality for (co)homology Lie superalgebras? (For the ordinary case, see paper of Hazewinkel, Mat. Sbornik and the classical 1950 paper of Koszul. Look at other dualities - e.g., Alexander duality (for commutative-algebraic formulation of the latter, see a book by Miller and Sturmfels, Chapter 5)).
6. (M. Gerstenhaber) To construct a rigid finite-dimensional associative algebra with non-trivial second Hochschild cohomology.
7. Develop a framework in GAP for computing deformations (prolongations of given cocycles).
8. Toral rank conjecture. For a finite-dimensional Lie algebra L over a field of characteristic zero, dim H^*(L,K) is greater or equal than 2^{dim Z(L)} (known in many particular cases, see Cairns-Jessup 2008).

Varieties, identities

1. (S. Pikhtilkov, 2000) Does there exist a matrix Lie algebra which is not special?
2. (Yuri Bahturin, Dniester notebook, 2.9) Find a basis of identities of an (infinite-dimensional, characteristic 0) Lie algebra W_n (or even of W₁. (Some results for the W₁ case are due to Zaicev, 1994. See also reasonings at pp. 207-208 of [Yu.A. Bahturin and A.Yu. Olshanskii, Identities, Algebra II, Springer, 1991, 105-221]).
3. (Dniester notebook, 2.138 and Kourovka notebook, 4.72, also asked by Yuri Bahturin in his book about varieties of Lie algebras). Is it true that a variety of Lie algebras of characteristic zero that does not contain sl(2) is (locally) solvable? (Partial results belong to Vais and S.P. Mishchenko).
4. (I.V. Lvov, Dniester notebook, 2.74) Is an arbitrary associative algebra without zero divisors and maximality condition on subalgebras PI?
5. Describe identities of the 7-dimensional simple non-Lie Malcev algebra (see Bremner-Douglas, Conjecture 4.8).

Free objects, generators and relations

1. (Boris Shoikhet) Describe the structure of the "minus" Lie algebra of a free associative algebra.
2. (Oleg Belegradek) Can an infinite finitely presented associative ring be a skew field?
   (I.V. Lvov, Dniester notebook, 2.68) Is there a simple infinite-dimensional finitely-presented associative algebra over a field of positive characteristic? (Over a field of characteristic zero, Weyl algebras provide examples of such algebras).
   (Alexei Belov, D.Sci. dissertation) Is there an infinite-dimensional finitely-generated (associative) skew field? (There is no such among PI algebras).
3. (M. Slater, Dniester notebook, 1.125) Is it true that in a free alternative algebra A, [n,t](x,y,z) = 0 for any x,y,z,t in A, n in the associative center of A?
4. (Arturo Magidin) Whether there exist varieties of groups in which the relatively free group of rank 2 is finite, and the relatively free group of rank 3 is infinite?

Lie algebras

1. Do Lax operator algebras (Sheinman et al.) admit a nice realization in the spirit of affine Kac-Moody algebras?
2. Describe gradings on current and Kac-Moody Lie algebras. (For description of some of the gradings, see de Montigny J. Phys. A 1996).
3. (A.I. Sozutov, Dniester notebook, 3.74). Describe all finite-dimensional simple Lie algebras with a monomial basis.
4. Rajan proved that if two tensor products of irreducible representations of a simple classical Lie algebra are isomorphic, then the numbers of tensor factors on each side is equal, and the tensor factors themselves are isomorphic up to a permutation. The proof heavily involves structure theory. Give a proof using a linear algebra in tensor spaces, and general basic facts like Whitehead lemmas, etc.

Associative algebras

1. (Leonid Makar-Limanov, arXiv:0909.1462, Problem 13). The Makar-Limanov invariant, ML, of a ring is the intersection of kernels of all its locally nilpotent derivations. How ML(A ⊗ B) is expressed in terms of ML(A) and ML(B) for associative commutative rings A, B?
2. Noncommutative cancellation problem (Makar-Limanov, arXiv:0909.1462, Problem 14). Let the free product of two associative algebras is isomorphic to a free associative algebra. Does this imply that each of the tensor factors is isomorphic to a free associative algebra too? (For a very particular case, see Drensky-Yu). Does it related to associative algebras of cohomological dimension? (Bibliography: Cuntz-Quillen, Hochschild 1945).
3. (Jensen & Lenzig, Problem 13.8) Does elementary equivalence of polynomial rings over rings R[X], S[X] imply elementary equivalence of rings R, S?

Nonassociative algebras

1. (A.N. Grishkov, Dniester notebook, 2.33) Describe semisimple finite-dimensional binary-Lie algebras over an algebraically closed field of positive characteristic (in characteristic zero described by Grishkov, 1980).
2. (S.V. Pchelincev, Dniester notebook, 3.62) Do there exist simple non-alternative right-alternative Lie-admissible algebras?
3. (I.P. Shestakov, Dniester notebook, 2.134) Does there exist an infinite-dimensional simple noncommutative Jordan algebra satisfying an identity ([x,y],y,y) = 0 which is neither Jordan, nor alternative?
4. (A.A. Albert) Whether a finite-dimensional power-associative nil algebra is solvable? This is known to be so in a lot of particular cases (low dimension, index of nilpotency, etc.; see Arenas, Arenas-Correa et al.).

Varia

1. Consider a mechanical system in the usual 3-dimensional space, subject to the normal physical laws and described by some system of differential equations. Consider the same situation in two dimensions. Probably one may adequately formulate physical laws in a two-dimensional world which result in a similar description via differential equations. Now, project 3-dimensional system to a two-dimensional one. Does the operations of projection and "building the model" commute? If yes, this may probably give some means to verify "real" (3-dimensional) systems by their apriori much simpler 2-dimensional projections. Consider in this context a 3-body problem?
2. (P. Diaconis, Random walks on groups, characters and geometry, Groups St. Andrews 2001 in Oxford, 120-142, sect. 4, question 7). Study Markov Chains on finite rings. More specifically, study the following example: in GF(p), consider the walk which moves from x to x+1 or x² with probability 0.5.
3. • (Yu.I. Merzlyakov, Kourovka notebook, 15.83) Does there exist a rational number α such that |α| < 2 and the matrices

     ( 1 α ) and ( 1 0 )
     ( 0 1 )     ( α 1 ) 
     

     generate a free group?
   • For which algebraic integers α the matrices

     ( 1 2 ) and ( 1 0 )
     ( 0 1 )     ( α 1 ) 
     

     generate a free group ? (See, for example, here for references).
4. (Sottile, p.2 in the arXiv version) Prove (directly, via some linear algebra) that if the matrix

   (    α0     (b0-b1)-1 ... (b0-bn)-1 )
   ( (b1-b0)-1    α1     ... (b1-bn)-1 )
   (   ...       ...     ...   ...     )
   ( (bn-b0)-1 (bn-b1)-1 ...    αn     )
   

   has only real eigenvalues, then α₁, ..., α_n are real.
5. (The Casas-Alvero conjecture) Suppose that a polynomial f(x) over a field of charactersitic zero has common root with each of its derivative. Prove that f(x) = (x - a)ⁿ. (Bibliography: arXiv:math/0605090).
6. Provide an example of a binary quadratic operad which is not Koszul but whose Poincaré series is inverse to the Poincaré series of the dual operad, like in the Ginzburg-Kapranov criterion. (For associative algebras, such examples are due to J.-E. Roos and L. Positselski).
7. (Generalization of Craig-Sakamoto). Is it true that if A and B are real symmetric matrices such that det(E + tA + t²B) = det(E + tA) for any real t, then B = 0?